Abstract
We consider the initial-boundary value problem for the heat equation in the half space with an exponential nonlinear boundary condition. We prove the existence of global-in-time solutions under the smallness condition on the initial data in the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N_+)\). Furthermore, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We consider the initial-boundary value problem for the heat equation in the half space \({{\mathbb {R}}}^N_+=\{x=(x',x_N)\in {{\mathbb {R}}}^N : x_N>0\}\) with a nonlinear boundary condition
where \(N\ge 1\), \(\partial _t=\partial /\partial t\), \(\partial _\nu =-\partial /\partial x_N\), and \(\varphi \) is the given initial data. Here f(u) is the nonlinearity which has an exponential growth at infinity with \(f(0)=0\). More precisely, the condition for the nonlinearity (see (1.9)) covers certain limiting cases which are critical with respect to the growth of the nonlinearity and the regularity of the initial data. In this paper, under a smallness condition on the initial data, we prove the existence of global-in-time solutions to problem (1.1). Furthermore, we derive some decay estimates and the asymptotic behavior of small global-in-time solutions.
The nonlinear boundary value problem such as (1.1) can be physically interpreted as a nonlinear radiation law. The case of power nonlinearities \(f(u)=|u|^{p-1}u\) with \(p>1\), that is,
has been extensively studied in many papers (see e.g. [5, 6, 11, 13, 17,18,19,20,21,22, 25, 26] and the references therein). It is well-known that problem (1.2) satisfies a scale invariance property, namely, for \(\lambda \in {\mathbb {R}}_+\), if u is a solution to problem (1.2), then
is also a solution to problem (1.2) with initial data \(\varphi _\lambda (x):=\lambda ^{1/(p-1)}\varphi (\lambda x)\). In the study of the nonlinear boundary value problem (1.2), it seems that all function spaces invariant with respect to the scaling transformation (1.3) play an important role. In fact, for Lebesgue spaces, we can easily show that the norm of the space \(L^q({{\mathbb {R}}}^N_+)\) is invariant with respect to (1.3) if and only if \(q=q_c:=N(p-1)\), and, for the given nonlinearity \(|u|^{p-1}u\), the Lebesgue space \(L^{q_c}({{\mathbb {R}}}^N_+)\) plays the role of critical space for the local well-posedness and the existence of global-in-time solutions to problem (1.2) (see e.g. [13, 18, 20]).
On the other hand, the case of the Cauchy problem with the power nonlinearity, that is,
also satisfies a scale invariance property, namely, for \(\lambda \in {\mathbb {R}}_+\), if u is a solution to problem (1.4), then
is also a solution to problem (1.4) with the initial data \(\varphi _\lambda (x):=\lambda ^{2/(p-1)}\varphi (\lambda x)\). So we can easily show that the norm of the space \(L^q({\mathbb {R}}^N)\) is invariant with respect to (1.5) if and only if \(q={\tilde{q}}_c:=N(p-1)/2\), and it is well-known that the Lebesgue space \(L^{{\tilde{q}}_c}({\mathbb {R}}^N)\) plays the role of critical space for the well-posedness of problem (1.4) (see e.g. [3, 12, 27, 30, 31] and references therein). Furthermore, the scaling property (1.5) also holds for the nonlinear Schrödinger equation
and it is well known that the Sobolev space \(H^{s_c}({\mathbb {R}}^N)\) with \(s_c:=N/2-2/(p-1)\) plays the role of critical space for the well-posedness of problem (1.6) (see e.g. [4]). From these results, we have two critical growth rates of the nonlinearity, that is, \(p_h:=1+(2q)/N\) and \(p_s:=1+4/(N-2s)\), and these two critical exponents are connected by the Sobolev embedding, \(\dot{H}^s({\mathbb {R}}^N)\hookrightarrow L^q({\mathbb {R}}^N)\), where s and q satisfy \(0\le s<N/2\) and \(1/q=1/2-s/N\). The case \(s_c=N/2\) is a limiting case from the following points of view:
-
(i)
for \(s>N/2\), \(H^s({\mathbb {R}}^N)\) embeds into \(L^\infty ({\mathbb {R}}^N)\);
-
(ii)
any power nonlinearity is subcritical, since \(H^{N/2}({\mathbb {R}}^N)\) embeds into any \(L^q({\mathbb {R}}^N)\) space (for \(q\ge 2\));
-
(iii)
\(H^{N/2}({\mathbb {R}}^N)\) does not embed into \(L^\infty ({\mathbb {R}}^N)\), and thanks to Trudinger’s inequality [29] one knows that \(H^{N/2}({\mathbb {R}}^N)\) embeds into the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N)\).
For this limiting case, Nakamura and Ozawa [24] consider the nonlinear Schrödinger equation with an exponential nonlinearity of asymptotic growth \(f(u)\sim e^{u^2}\) and with a vanishing behavior at the origin, and they show the existence of global-in-time solutions under a smallness assumption of the initial data in \(H^{N/2}({\mathbb {R}}^N)\).
As a natural analogy to the results of [24], the third author of this paper and Ruf [28] and Ioku [14] consider the Cauchy problem of the semilinear heat equation with exponential nonlinearity of the form
and the initial data \(\varphi \) belonging to the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N)\) defined as
(see also Definition 2.1). They consider the corresponding integral equation
and prove the existence of local/global-in-time (mild) solutions to this equation (1.8) under the smallness assumption of initial data in \(\mathrm {exp}L^2({\mathbb {R}}^N)\). Furthermore, the authors of this paper and Ruf [10] show the equivalence between mild solutions (solution to the integral equation (1.8)) and weak solutions to the heat equation with the nonlinearity f(u) as in (1.7), and derive some decay estimates and the asymptotic behavior for small global-in-time solutions. The growth rate of (1.7) at infinity seems to be optimal in the framework of the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N)\). In fact, if \(f(u)\sim e^{|u|^r}\) with \(r>2\), there exist some positive initial data \(\varphi \in \mathrm {exp}L^2({\mathbb {R}}^N)\) such that problem (1.8) does not possess any classical local-in-time solutions (see [15]). For the fractional diffusion case and general power-exponential nonlinearities, see e.g. [8, 10, 23]. Furthermore, for \(\varphi \in \mathrm {exp}L^2({\mathbb {R}}^N)\), which implies \(\varphi \in L^p({\mathbb {R}}^N)\) for \(p\in [2,\infty )\), the decay rate of (1.7) near origin, that is, \(f(u)\sim |u|^{4/N}u\), is optimal in the framework of \(L^2({\mathbb {R}}^N)\). See e.g. [3, 30].
The above limiting case in \({\mathbb {R}}^N\) appears from the relationship between \(p_h\) and \(p_s\) by the Sobolev embedding. For problem (1.2), we can easily show that the norm of the space \(H^s({{\mathbb {R}}}^N_+)\) is invariant with respect to (1.3) if and only if \(s={\tilde{s}}_c:=N/2-1/(p-1)\), and we have two critical growth rate of the nonlinearity, that is, \({\tilde{p}}_h=1+q/N\) and \({\tilde{p}}_s=1+2/(N-2s)\). These two exponents are also connected by the Sobolev embedding, \(\dot{H}^s({{\mathbb {R}}}^N_+)\hookrightarrow L^q({{\mathbb {R}}}^N_+)\), where s and q satisfy the same conditions as in the case of \({\mathbb {R}}^N\). This means that the same limiting case appears for problem (1.2). On the other hand, as far as we know, there are no results which treat the exponential nonlinearity for the nonlinear boundary problem (1.1).
Based on the above, in this paper, we assume that the nonlinearity f satisfies the following: there exist \(C_f>0\) and \(\lambda >0\) such that
This assumption covers the case
which is one of the candidates for the optimal growth rate of the nonlinearity in the framework of the Orlicz space \(\mathrm {exp}L^2({{\mathbb {R}}}^N_+)\) and the optimal decay rate near origin in the framework of \(L^2({{\mathbb {R}}}^N_+)\) (see e.g. [18]). Following [10, 14, 28], for problem (1.1) with (1.9), we consider the corresponding integral equation, and prove the existence of global-in-time (mild) solutions under some smallness assumption of the initial data in \(\mathrm {exp}L^2({{\mathbb {R}}}^N_+)\). Furthermore, we obtain some decay estimates for the solutions in the following two cases
\(\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)\) only (slowly decaying case), and \(\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)\cap L^p({{\mathbb {R}}}^N_+)\) with \(p\in [1,2)\) (rapidly decaying case).
In particular, for the rapidly decaying case \(p=1\), we show that the global-in-time solutions with some suitable decay estimates behave asymptotically like suitable multiples of the Gauss kernel.
Before treating our main results, we introduce some notation and define a solution to problem (1.1). Throughout this paper we often identify \({{\mathbb {R}}}^{N-1}\) with \(\partial {{\mathbb {R}}}^N_+\). Let \(g_N=g_N(x,t)\) be the Gauss kernel on \({{\mathbb {R}}}^N\), that is,
Let \(G=G(x,y,t)\) be the Green function for the heat equation on \({{\mathbb {R}}}^N_+\) with the homogenous Neumann boundary condition, that is,
where \(y_*=(y',-y_N)\) for \(y=(y',y_N)\in {\overline{{{\mathbb {R}}}^N_+}}\). Then, we define a (mild) solution to problem (1.1).
Definition 1.1
Let \(\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)\), \(T\in (0,\infty ]\), and \(u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,T))\cap L^\infty (0,T;\mathrm {exp}L^2({{\mathbb {R}}}^N_+))\).
-
(i)
In the case when \(N\ge 2\), we call u a solution to problem (1.1) in \({{\mathbb {R}}}^N_+\times (0,T)\) if u satisfies
$$\begin{aligned} u(x,t)= & {} \int _{{{\mathbb {R}}}^N_+}G(x,y,t)\varphi (y)\,dy\nonumber \\&\quad + \int _0^t\int _{{\mathbb {R}}^{N-1}}G(x,y',0,t-s)f(u(y',0,s))\,dy'\,ds \end{aligned}$$(1.12)for \((x,t)\in {\overline{{{\mathbb {R}}}^N_+}}\times (0,T)\) and \(u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi \) in the weak\(^*\) topology.
-
(ii)
In the case when \(N=1\), we call u a solution to problem (1.1) in \((0,\infty )\times (0,T)\) if u satisfies
$$\begin{aligned} u(x,t)=\int _0^\infty G(x,y,t)\varphi (y)\,dy + \int _0^tG(x,0,t-s)f(u(0,s))\,ds \end{aligned}$$(1.13)for \((x,t)\in [0,\infty )\times (0,T)\) and \(u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi \) in the weak\(^*\) topology.
In the case when \(T=\infty \), we call u a global-in-time solution to problem (1.1).
We recall that \(u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi \) in weak\(^*\) topology if and only if
for any \(\psi \) belonging to the predual space of \(\mathrm {exp}L^2({{\mathbb {R}}}^N_+)\) (see Sect. 2).
In what follows, we denote by \(\Vert \cdot \Vert _{\mathrm {exp}L^2}\) the norm of \(\mathrm {exp}L^2:=\mathrm {exp}L^2({{\mathbb {R}}}^N_+)\) defined by (2.14), for \(r\in [1,\infty ]\), we write \(\Vert \cdot \Vert _{L^r}:=\Vert \cdot \Vert _{L^r({{\mathbb {R}}}^N_+)}\) and \(|\cdot |_{L^r}:=\Vert \cdot \Vert _{L^r({\mathbb {R}}^{N-1})}\) for simplicity. Furthermore, for a function \(\phi (x',x_N)\) with \(x' \in {{\mathbb {R}}}^{N-1}\) and \(x_N \in [0,\infty )\), we write \(| \phi |_{L^r}:=\Vert \phi (x',0)\Vert _{L^r({\mathbb {R}}^{N-1})}\).
Now we are ready to state the main results of this paper. First we show the existence of global-in-time solutions to problem (1.1) under the smallness assumption of the initial data in \(\mathrm {exp}L^2\).
Theorem 1.1
Let \(N\ge 1\) and \(\varphi \in \mathrm {exp}{L^2}\). Suppose that f satisfies (1.9). Then there exist positive constants \(\varepsilon =\varepsilon (N) >0\) and \(C=C(N)>0\) such that, if \(\Vert \varphi \Vert _{\mathrm {exp}{L^2}} <\varepsilon \), then there exists a unique global-in-time solution u to problem (1.1) satisfying
where \(h(t)=\min \{t^{N/4},1\}\), and for any \(q\in [2,\infty )\),
where \(\Gamma \) is the gamma function
Remark 1.1
-
(i)
By the definition of \({{\mathbb {R}}}^N_+\), if \(N\ge 2\), then the boundary of \({{\mathbb {R}}}^N_+\) is \({\mathbb {R}}^{N-1}\), namely, it is unbounded. On the other hand, if \(N=1\), then the boundary of \({\mathbb {R}}_+\) is \(x=0\), namely, it is only one point. From these differences, we need to divide the proof into two cases, \(N\ge 2\) and \(N=1\), and we have two estimates as in (1.15).
-
(ii)
Following [15], we denote by \(\mathrm {exp}L^2_0({{\mathbb {R}}}^N_+)\) the closure of \(C_0^\infty ({{\mathbb {R}}}^N_+)\) in \(\mathrm {exp}L^2({{\mathbb {R}}}^N_+)\). Then, by an argument similar to that in the proof of [15, Theorem 1.2], it seems likely to obtain the existence of local-in-time solutions to problem (1.1) for any \(\varphi \in \mathrm {exp}L^2_0({{\mathbb {R}}}^N_+)\) under the weaker condition
$$\begin{aligned} |f(u)-f(v)|\le C|u-v|(e^{\lambda u^2}+e^{\lambda v^2})\quad \text{ for } \text{ every }\quad u,v\in {\mathbb {R}}, \qquad f(0)=0, \end{aligned}$$where \(\lambda >0\) and \(C>0\). This has not been fully explored and it is left for further investigation.
From now, we focus on the unique solution u to problem (1.1) satisfying (1.14) and (1.15). The following result gives some decay estimates for the slowly decaying case, that is, \(\varphi \in \mathrm {exp}L^2\) only.
Theorem 1.2
Assume the same conditions as in Theorem 1.1. Furthermore, suppose that there exists a unique solution u to problem (1.1) satisfying (1.14) and (1.15). Then there exist some positive constants \(\varepsilon = \varepsilon (N) >0\) and \(C=C(N)>0\) such that, if \(\Vert \varphi \Vert _{\mathrm {exp}L^2} <\varepsilon \), then the solution u satisfies
for all \(q\in [2,\infty ]\).
Remark 1.2
-
(i)
By Theorem 1.1, if \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\) is small enough, then we can show that the assumption of Theorem 1.2 is not empty.
-
(ii)
We obtain the same decay estimate as the solution to the heat equation in \({{\mathbb {R}}}^N_+\) with the homogeneous Neumann boundary condition and initial data in \(L^2\). See \((G_1)\) in Sect. 2.
Next we consider the rapidly decaying case, that is, \(\varphi \in \mathrm {exp}L^2 \cap L^p\) with \(p\in [1,2)\). We can prove two kinds of results about decay estimates of solutions to problem (1.1). In Theorem 1.3, we only assume the smallness condition of the \(\mathrm {exp}L^2\) norm of the initial data. This means that we can allow the \(L^p\) norm of the same data to be large. On the other hand, under this mild assumption, we have an additional restriction about the range of \(L^q\) spaces for the case \(N\ge 3\). In Theorem 1.4, under a stronger assumption, that is, a smallness assumption not only for the \(\mathrm {exp}L^2\) but also for the \(L^p\) norm of the initial data, we obtain better decay estimates, with no additional restrictions about the range of \(L^q\) spaces even for the case \(N\ge 3\). In the following we denote for any \(r\ge 1\)
Theorem 1.3
Assume the same conditions as in Theorem 1.2. Furthermore, assume \(\varphi \in L^p({{\mathbb {R}}}^N_+)\) for some \(p\in [1,2)\). Put
Then there exist some positive constants \(\varepsilon =\varepsilon (N)>0 \), \(C=C(N)>0\) and a positive function \(F=F(N, p_1, \Vert \varphi \Vert _{L^{p_1}}, \lambda )\) such that, if
then the solution u satisfies
for all \(q\in [p_1,\infty ]\). In particular, if \(p_1\in (1,2)\), then
Theorem 1.4
Assume the same conditions as in Theorem 1.3. Then there exists a positive constant \(\varepsilon =\varepsilon (N)\) such that, if \(\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}<\varepsilon \), then (1.20) with \(p_1=p\) holds for all \(q\in [p,\infty ]\). In particular, for all \(q\in [p,\infty )\),
Furthermore, if \(p\in (1,2)\) or \(N\ge 3\), then (1.21) with \(p_1=p\) holds.
Remark 1.3
By (1.9) the nonlinearity f(u) behaves like \(|u|^{1+2/N}\) for \(u\rightarrow 0\). So, for the case \(N\ge 2\), since it follows from (1.15) that \(u\in L^\infty _{\mathrm{loc}}(0,\infty ; L^q(\partial {{\mathbb {R}}}^N_+))\) for \(q\ge 2\), the nonlinear term f(u) belongs to \(L^p(\partial {{\mathbb {R}}}^N_+)\) for \(p \ge (2N)/(N+2)\). For the case \(N=2\), this means that \(f(u)\in L^p(\partial {{\mathbb {R}}}^N_+)\) for all \(p\ge 1\), but this implies a true constraint for the case \(N\ge 3\). This is the reason why in Theorem 1.3 we have to introduce some parameters \(p_1\) (and \(p_2\), \(p_3\), and \(p_4\) in Lemmata 2.2, 5.1, and 5.6, respectively) which are meaningful only for the case \(N\ge 3\).
Finally we address the question of the asymptotic behavior of solutions to problem (1.1) when \(\varphi \in \mathrm {exp}L^2 \cap L^1\). We show that global-in-time solutions with suitable decay properties behave asymptotically like suitable multiples of the Gauss kernel.
Theorem 1.5
Let \(N\ge 1\) and \(\varphi \in \mathrm {exp}L^2 \cap L^1({{\mathbb {R}}}^N_+)\). Furthermore, let u be the global-in-time solution to problem (1.1) satisfying (1.22). Then there exists the limit
such that
Remark 1.4
For the case \(N\ge 2\), by (1.12) we see that
On the other hand, for the case \(N=1\), by (1.13) we have
The paper is organized as follows. In Sect. 2 we recall some properties of the kernel G and its associate semigroup. In Sect. 3, applying the Banach contraction mapping principle, we prove Theorem 1.1. In Sects. 4 and 5, modifying the arguments of [20], we derive decay estimates on the boundary, and prove Theorems 1.2, 1.3, and 1.4. In Sect. 6 we obtain the asymptotic behavior of solutions to problem (1.1).
2 Preliminaries
In this section we recall some properties of the kernel \(G=G(x,y,t)\) and its associate semigroup. Throughout this paper, by the letter C we denote generic positive constants that may have different values also within the same line.
We first recall the following properties of the kernel G (see e.g [13, 20, 22]):
-
(i)
\(\displaystyle {\int _{{{\mathbb {R}}}^N_+}G(x,y,t)dy=1}\) for any \(x\in {\overline{{{\mathbb {R}}}^N_+}}\) and \(t>0\);
-
(ii)
for any \((x,t), (z,s)\in {\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty )\), it holds that
$$\begin{aligned} \int _{{{\mathbb {R}}}^N_+}G(x,y,t)G(y,z,s)\,dy=G(x,z,t+s). \end{aligned}$$(2.1)
By (1.11) we have
Furthermore, it follows from (1.10) and (1.11) that
We denote by \(S_1(t)\varphi \) the unique bounded solution to the heat equation in \({{\mathbb {R}}}^N_+\) with the homogeneous Neumann boundary condition and the initial datum \(\varphi \), that is,
and denote by \(e^{t\Delta '}\psi \) the unique bounded solution to the heat equation in \({{\mathbb {R}}}^{N-1}\) with the initial datum \(\psi \), that is,
In the case where \(N\ge 2\), we put
for \(\psi \in L^r({\mathbb {R}}^{N-1})\) with some \(r\in [1,\infty ]\). Since it holds that, for any \(r\in [1,\infty ]\),
by (2.2), (2.3), and applying Young’s inequality to (2.4) and (2.5) we have the following.
- \((G_1)\):
-
There exists a constant \(c_1\), which depends only on N, such that
$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{L^r} \le {c_1}t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})}\Vert \varphi \Vert _{L^q}, \qquad t>0, \end{aligned}$$(2.8)for \(\varphi \in L^q({{\mathbb {R}}}^N_+)\) and \(1\le q\le r\le \infty \). Furthermore, there exists a constant \(c_2\), which depends only on N, such that, for the case \(N\ge 2\),
$$\begin{aligned} |S_1(t)\varphi |_{L^r} \le c_2 t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2r}}\Vert \varphi \Vert _{L^q}, \qquad t>0, \end{aligned}$$(2.9)and, for the case \(N=1\),
$$\begin{aligned} |[S_1(t)\varphi ](0)| \le c_2t^{-\frac{1}{2q}}\Vert \varphi \Vert _{L^q}, \qquad t>0{.} \end{aligned}$$(2.10) - \((G_2)\):
-
For any \(\psi \in L^q({\mathbb {R}}^{N-1})\) and \(1\le q\le r\le \infty \), it holds that
$$\begin{aligned}&\Vert S_2(t)\psi \Vert _{L^r} \le Ct^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}(1-\frac{1}{q})}|\psi |_{L^q}, \qquad t>0, \end{aligned}$$(2.11)$$\begin{aligned}&|S_2(t)\psi |_{L^r} \le Ct^{-\frac{N-1}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}|\psi |_{L^q}, \qquad t>0{.} \end{aligned}$$(2.12) - \((G_3)\):
-
Let \(\varphi \in L^q({{\mathbb {R}}}^N_+)\) with \(1\le q\le \infty \). Then, for any \(T>0\), \(S_1(t)\varphi \) is bounded and smooth in \(\overline{{{\mathbb {R}}}^N_+}\times (T,\infty )\).
We recall now the definition and the main properties of the Orlicz space \(\mathrm {exp}L^2\).
Definition 2.1
We define the Orlicz space \(\mathrm {exp}L^2\) as
where the norm is given by the Luxemburg type
The space \(\mathrm {exp}L^2\) endowed with the norm \(\Vert u\Vert _{\mathrm {exp}L^2}\) is a Banach space, and admits as predual the Orlicz space defined by the complementary function of \(A(t)=e^{t^2}-1\), denoted by \({\tilde{A}}(t)\). This complementary function is a convex function such that \({\tilde{A}}(t)\sim t^2\) as \(t\rightarrow 0\) and \({\tilde{A}}(t)\sim t\log ^{1/2}t\) as \(t\rightarrow \infty \). (see e.g. [2, Section 8].) Furthermore, it follows from (2.13) that
and we have
(In the case where \(\Omega ={\mathbb {R}}^N\), see e.g. [15, 23].) On the other hand, it is well known that, for any \(2\le p<\infty \),
(See e.g. [15, Proposition 2.1].) Then, applying the same argument as in the proof of [14, Lemma 2.2] with (2.17), we have
Next we recall the following property of the Gamma function.
Lemma 2.1
[10, Lemma 3.3] For any \(q\ge 1\) and \(r\ge 1\), there exists a positive constant \(C>0\), which is independent of q and r, such that
Applying this lemma, we prepare the following estimate for the nonlinear term f for the case \(N\ge 2\).
Lemma 2.2
Let \(N\ge 2\) and \(m>0\). Suppose that, for any \(q\in [2,\infty )\), the function \(u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\) satisfies the condition
Let f be the function satisfying the condition (1.9), and put
Then, for all \(r\in [p_2,\infty )\), there exists a positive constant \(\varepsilon =\varepsilon (r,\lambda ) >0\) such that, if \(m<\varepsilon \), then
where C is independent of r, N, and m.
Proof
For any \(k\in {{\mathbb {N}}}\cup \{0\}\), we put
Then, since it holds from \(N\ge 2\) and \(r\ge p_2\) with (2.20) that
Applying Lemma 2.1 with the monotonicity property of the Gamma function \(\Gamma (q)\) for \(q\ge 3/2\) (see, e.g. [1]), we see that
This together with (2.23) implies that
Therefore, taking a sufficiently small \(m <\varepsilon (r,\lambda )\) if necessary (e.g. \(m^2\le 1/(4\lambda r)\)), we get
This implies (2.21), and the proof of Lemma 2.2 is complete. \(\square \)
Similarly to the case \(N\ge 2\), we prepare the following lemma, which is the one dimensional counterpart of Lemma 2.2.
Lemma 2.3
Let \(m>0\). Suppose that, for any \(q\in [2,\infty )\), the function \(u\in C(0,\infty )\) satisfies the condition
Let f be the function satisfying the condition (1.9). Then there exists a positive constant \(\varepsilon =\varepsilon (\lambda ) >0\) such that, if \(m<\varepsilon \), then
and
where C is independent of m and r.
Proof
We first prove (2.25). For any \(k\in {{\mathbb {N}}}\cup \{0\}\), let \(\ell _k\) be the constant defined by (2.22) with \(N=1\), namely, \(\ell _k=2k+3\). Then, by (1.9) and (2.24) with \(q=\ell _k\) we have
Since it holds from the monotonicity property of the Gamma function that
by (2.27) we have
Therefore, taking a sufficiently small \(m <\varepsilon (\lambda )\) if necessary, we get
This implies (2.25).
Next we prove (2.26). For any \(k\in {\mathbb {N}}\cup \{0\}\), put \({\tilde{\ell }}_k=2k+2\). Then, similarly to the proof of (2.25), we have
and then, by (2.24) with \(q=r\) and also \(q={\tilde{\ell }}_k\) and taking a sufficiently small \(m <\varepsilon (\lambda )\) if necessary, we have
This implies (2.26), and the proof of Lemma 2.3 is complete. \(\square \)
3 Existence
In this section we prove Theorem 1.1. We first consider the case \(N\ge 2\). We introduce some notation. Let \(M>0\). Set
equipped with the metric
Then \((X_M,d_X)\) is a complete metric space. For the proof of Theorem 1.1 we apply the Banach contraction mapping principle in \(X_M\) to find a fixed point of
where \(S_1(t)\) is as in (2.4) and
Here \(S_2(t)\) is as in (2.6) and f satisfies (1.9). We remark that, for \(u\in X_M\), the function f(u) belongs to \(C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\). Therefore, by Lemma 2.2 we have that \(f(u(\cdot ,0,s))\in L^r({\mathbb {R}}^{N-1})\) with \(r\in [p_2,\infty )\), and we can define \(S_2(t-s)f(u(s))\) for \(t>s>0\). More precisely, with an abuse of notation we denote by \(S_2(t-s)f(u(s))\) the operator \(S_2(t-s)\) applied to the function \(f(u(x',0,s))\). In particular, we have
Hence any fixed point of the integral operator \( \Phi \) satisfies the equation (1.12).
Furthermore, we have the following estimates for the function D[u].
Lemma 3.1
Let \(N\ge 2\) and \(u\in X_M\). Then there exists a positive constant \(\varepsilon _*=\varepsilon _*(N, \lambda )>0\) such that, if \(M<\varepsilon _*\), then, for any \(q\in [2,\infty )\),
where C is independent of q and M. Furthermore, D[u] is continuous in \(\overline{{{\mathbb {R}}}^N_+}\times (0,\infty )\).
Proof
We first prove (3.4). Let \(p_2\) be the constant given in (2.20). Then, it holds that
By (2.11) with \((q,r)=(p_2,2)\) and (3.3) we have
Since \(u\in X_M\), taking a sufficiently small \(\varepsilon _1=\varepsilon _1(p_2, \lambda )>0\) such that, for \(M<\varepsilon _1\), we can apply Lemma 2.2, and it holds that
Substituting (3.6) to (3.5), we see that
where B is the beta function, namely
Furthermore, similarly to (3.5), by (2.11) with \((q,r)=(N,\infty )\) and (3.3) we have
Since \(N\ge 2\ge p_2\), similarly to (3.6), taking a sufficiently small \(\varepsilon _2=\varepsilon _2(N, \lambda )>0\) such that, for \(M<\varepsilon _2\), we get
Substituting (3.9) to (3.8), we see that
On the other hand, for fixed \(q\in [2,\infty )\), we put
Then, it holds that \(p_2\le q_*< q\) and
By (2.12) with \((q,r)=(q_*,q)\) and (3.3) we have
Since \(p_2\le q_*\le N\), similarly to (3.6) again, taking a sufficiently small \(\varepsilon _3=\varepsilon _3(N, \lambda )>0\) such that, for \(M<\varepsilon _3\), we have
This together with (3.11) and (3.12) yields that
where the constant C depends only on N since \(p_2\le q_*\le N\). Thus, taking \(\varepsilon _*=\min \{\varepsilon _1, \varepsilon _2, \varepsilon _3\}\) with (3.7), (3.10), and (3.13), we obtain (3.4).
Next we prove the continuity of D[u](x, t). Let T be an arbitrary positive constant. Then, it follows from (2.1) that
for \(x\in {\overline{{{\mathbb {R}}}^N_+}}\) and \(0<T<t<\infty \). Then, by (3.4) and \((G_3)\) we see that
is continuous in \({\overline{{{\mathbb {R}}}^N_+}}\times (T,\infty )\). Furthermore, since it follows from \(u(t)\in L^\infty ({\overline{{{\mathbb {R}}}^N_+}})\) for \(t\ge T/2\) that \(f(u(t))\in L^\infty (\partial {{\mathbb {R}}}^N_+)\) for \(t\ge T/2\), we apply the same argument as in [9, Section 3, Chapter 1] to see that
is also continuous in \({\overline{{{\mathbb {R}}}^N_+}}\times (T,\infty )\). (See also [7, Proposition 5.2] and [16, Lemma 2.1].) Therefore we deduce that D[u] is continuous in \(\overline{{{\mathbb {R}}}^N_+}\times (T,\infty )\). Thus Lemma 3.1 follows from arbitrariness of T. \(\square \)
Lemma 3.2
Let \(N\ge 2\) and \(u,v\in X_M\). Then there exist some positive constants \(C=C(N)\) and \(\varepsilon ^*=\varepsilon ^*(N,\lambda )>0\) such that, if \(M<\varepsilon ^*\), then
Proof
For any \(k\in {\mathbb {N}}\cup \{0\}\), we put
Then, by (1.9) we recall that
Since \(h(t)\le 1\), by (2.11) with \((q,r)=(N,\infty )\), (3.3), and (3.16), for any \(t>0\), we have
Since it follows from Hölder’s inequality that
by (3.1), (3.15), and (3.17) we see that, for \(u,v\in X_M\),
For \(k=0\), by (3.15) we have \(\Gamma ({{\tilde{\ell }}_0 N}+1)=\Gamma (3)\). Furthermore, applying Lemma 2.1 with (3.15) and by the monotonicity property of the Gamma function, for \(k\ge 1\), we see that
These together with (3.18) implies that
Then, taking a sufficiently small \(\varepsilon ^*=\varepsilon ^*(N,\lambda )>0\) such that, for \(M<\varepsilon ^*\), in a similar way as in Lemma 2.2, it holds that
On the other hand, similarly to (3.12), by (2.12) with \((q,r)=((2N)/3,2N)\), (3.3), and (3.16) we have
Therefore, applying the same argument as in the proof of (3.19), for \(M<\varepsilon ^*\), it holds that
This implies that
Combining (3.19) and (3.20), we have (3.14), thus Lemma 3.2 follows. \(\square \)
Remark 3.1
In the proof of Lemma 3.2, the estimate for \(\sup _{t>0}t^{1/(4N)}|\cdot |_{L^{2N}}\) is closed by itself. We need the term \(\sup _{t>0} h(t)\Vert \cdot \Vert _{L^\infty }\) in the definition of the metric \(d_X\) in order to ensure the uniform convergence of the Cauchy sequence so that the solution is continuous.
Now we are ready to complete the proof of Theorem 1.1 for the case \(N\ge 2\).
Proof of Theorem 1.1
(\(N\ge 2\)). Let
where \(c_1\) and \(c_2\) are constant given in \((G_1)\). Then, by (2.8), (2.9), (2.17), and (2.18) we see that
Let \(u\in X_M\). Then, by Lemma 3.1 with (2.16) we can take a sufficiently small \(\varepsilon _4=\varepsilon _4(N, \lambda )>0\) such that, for \(M<\varepsilon _4\), it holds \(CM^{2/N}<1/2\) and so
This together with property (\(G_3\)), Lemma 3.1, (3.2), and (3.21) yields that \(\Phi \) is a map on \(X_M\) to itself. Furthermore, since it follows from (3.1) and (3.2) that
for \(u,v\in X_M\), taking a sufficiently small \(\varepsilon _5=\varepsilon _5(N)>0\) if necessary, for \(M<\varepsilon _5\), we can apply Lemma 3.2, and it holds that
Then, applying the contraction mapping theorem ensures that there exists a unique \(u\in X_M\) with
Thus we see that u is the unique global-in-time solution of problem (1.12) satisfying (1.14) and (1.15). Furthermore, by the same argument as in the proof of [14, (1.7)] with Lemma 3.1, we can prove that \(u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi \) in the weak\(^*\) topology, and the proof of Theorem 1.1 for the case \(N\ge 2\) is complete. \(\square \)
Next we consider the case \(N=1\). Similarly to the case \(N\ge 2\), let \(M>0\), and we set
equipped with the metric
Then \((Y_M,d_Y)\) is a complete metric space. Similarly to the proof of Theorem 1.1 for the case \(N\ge 2\), we apply the Banach contraction mapping principle in \(Y_M\) to find a fixed point of
where
Here \(g_1\) is as in (1.10) and f satisfies (1.9).
Applying Lemma 2.3, we have the following.
Lemma 3.3
Let \(u\in Y_M\). Then there exists a positive constant \(\varepsilon _*=\varepsilon _*(\lambda )>0\) such that, if \(M<\varepsilon _*\), then, for any \(q\in [2,\infty )\),
where C is independent of q and M. Furthermore, \({\tilde{D}}[u]\) is continuous in \([0,\infty )\times (0,\infty )\).
Proof
By (2.7) with \((N,r)=(1,2)\) and (3.23) we have
Since \(u\in Y_M\), taking a sufficiently small \(\varepsilon _*=\varepsilon _*(\lambda )>0\) such that, for \(M<\varepsilon _*\), we can apply Lemma 2.3, and it holds from (2.26) with \(r=2\) and (3.26) that
Similarly, by (2.7) with \((N,r)=(1,\infty )\), (2.26), and (3.23), for any \(q\in [2,\infty )\), it holds that
where C is independent of q and M. This implies that
Thus, by (3.27) and (3.28) we obtain (3.24) and (3.25). Furthermore, applying the same argument as in the proof of Lemma 3.1, we see that \({\tilde{D}}[u]\) is continuous in \([0,\infty )\times (0,\infty )\), and the proof of Lemma 3.3 is complete. \(\square \)
Lemma 3.4
Let \(u,v\in Y_M\). Then there exists a positive constant \(\varepsilon ^*=\varepsilon ^*(\lambda )>0\) such that, if \(M<\varepsilon ^*\), then
where C is independent of M.
Proof
For any \(k\in {\mathbb {N}}\cup \{0\}\), let \({\tilde{\ell }}_k\) be the constant defined by (3.15) with \(N=1\). Then, similarly to (3.18), by (2.7) with \((N,r)=(1,\infty )\), (3.16), (3.22), and (3.23), for \(u,v\in Y_M\), we have
Then, we can take a sufficiently small \(\varepsilon ^*=\varepsilon ^*(\lambda )>0\) such that, for \(M<\varepsilon ^*\), it holds that
This implies (3.29), thus Lemma 3.4 follows. \(\square \)
Proof of Theorem 1.1
(\(N=1\)). By Lemmata 3.3, 3.4, and applying the same arguments as in the proof of Theorem 1.1 for the case \(N\ge 2\), we can prove Theorem 1.1 for the case \(N=1\). \(\square \)
4 Slowly decaying initial data
In this section we prove Theorem 1.2. Similarly to Sect. 3, we first consider the case \(N\ge 2\). Let u be the unique solution to problem (1.1) satisfying (1.14) and (1.15). Put
Then, it follows from (1.12) and (2.1) that the function v satisfies
where D[v] is the function defined by (3.3). Since it follows from (1.14) and (2.17) that
by (2.9) with \(q=r\), for any \(q\in [2,\infty ]\), we have
Here \(c_*\) is a constant independent of q and \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\). Furthermore, since it follows from the continuity of the function D[u](x, t) that \(|D[u](t)|_{L^\infty }\le \Vert D[u](t)\Vert _{L^\infty }\), applying the same argument as in the proof of Lemma 3.1 with (1.15) and (4.1), we see that, for any \(q\in [2,\infty ]\),
Then, we can take a sufficiently small \(\varepsilon >0\) such that, for \(\Vert \varphi \Vert _{\mathrm {exp}L^2}<\varepsilon \), it follows from (4.2), (4.4), and (4.5) that
On the other hand, we have the following.
Lemma 4.1
Let \(N\ge 2\), \(T>0\), and \(A>0\). Suppose that, for any \(q\in [2,\infty ]\), the function \(v\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\) satisfies
Let f be a function satisfying (1.9). Then, there exists \(\varepsilon _*>0\), independent of T, such that, if \(A<\varepsilon _*\), then, for any \(r\in [p_2,\infty ]\),
where \(C_f\) and \(p_2\) are given in (1.9) and (2.20), respectively.
Proof
Let \(k\in {{\mathbb {N}}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22). Then, for any \(r\in [p_2,\infty ]\), by (1.9) and (4.7) we have
We can take a sufficiently small \(\varepsilon _*=\varepsilon _*(\lambda )>0\) so that, for \(A< \varepsilon _* \), it holds that
This together with (4.9) implies (4.8). Thus Lemma 4.1 follows. \(\square \)
Now we are in position to prove Theorem 1.2 for the case \(N\ge 2\).
Proof of Theorem 1.2
(\(N\ge 2\)). Following the idea of the proof of [20, Lemma 2.4], we prove this theorem.
Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15), and let v be the function defined by (4.2). Then, applying arguments similar to that in the proof of [20, Lemma 2.1] with (4.6), we see that
Let \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\) be a sufficiently small to be chosen later. Put
and
where \(c_*\) and \(c_2\) are given in (4.3) and (2.9), respectively. Then, by (4.6) and (4.12) we have \(T\ge 1\).
We prove \(T=\infty \). The proof is by contradiction. We assume that \(T<\infty \). Then, by (4.11) we see that
On the other hand, by (2.9) with \((q,r)=(2,q)\), (4.3) and (4.12) we have
Furthermore, by the definition of T, taking a sufficiently small \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\) if necessary, we can apply Lemma 4.1, and it holds that, for any \(r\in [p_2,\infty ]\),
where \(C_f\) and \(p_2\) are given in (1.9) and (2.20), respectively. Let D[v] be the function defined by (3.3). Then, we put
For the term \(I_1\), since \(T\ge 1\) and \(N(1/2-1/p_2)=-1\), by (2.12) with \((q,r)=(p_2,q)\) and (4.15) we obtain
where \(D_1\) is a positive constant independent of q and \(\delta \). Furthermore, for the term \(I_2\), since \(T\ge 1\), by (2.12) with \(q=r\) and (4.15) we have
where \(D_2\) is a positive constant independent of q and \(\delta \). Then, combining (4.17) and (4.18), we see that
Taking a sufficiently small \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\) if necessary, we have
This together with (4.2), (4.14), and (4.19) implies that
This contradicts (4.13), and we see \(T=\infty \). Therefore, for any \(q\in [2,\infty ]\), it holds that
It remains to show that, for any \(q\in [2,\infty ]\),
By (2.8), (4.3), and (4.12) we see that
On the other hand, by (4.20), similarly to (4.15), it holds that, for any \(r\in [p_2,\infty ]\),
Similarly to (4.16), by (3.3) we put
Then, for the term \(J_1\), it holds from (2.11) with \((q,r)=(p_2,q)\) and (4.23) that
Furthermore, for the term \(J_2\), by (2.11) with \(q=r\) and (4.23) we have
Then, combining (4.24), (4.25), and (4.26), we see that
This together with (4.2) and (4.22) yields (4.21). Therefore, by (4.1), (4.20), and (4.21) we have (1.16), and the proof of Theorem 1.2 for the case \(N\ge 2\) is complete. \(\square \)
We next consider the case \(N=1\). Let v be the function defined by (4.1). Then, it follows from (1.13) and (2.1) that the function v satisfies
where \({\tilde{D}}[v]\) is the function defined by (3.23). Then, by (2.10) and (4.3) we have
Here \(d_*\) is a constant independent of \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\). Furthermore, similarly to (4.5), applying the same argument as in the proof of Lemma 3.3 with (1.15) and (4.1), we see that
Then, we can take a sufficiently small \(\varepsilon >0\) such that, for \(\Vert \varphi \Vert _{\mathrm {exp}L^2}<\varepsilon \), it follows from (4.27), (4.28), and (4.29) that
On the other hand, we have the following, which is the one dimensional counterpart of Lemma 4.1.
Lemma 4.2
Let \(N=1\), \(T>0\), and \(A>0\). Suppose that the function \(v\in C(0,\infty )\) satisfying
Let f be a function satisfying (1.9). Then, there exists \(\varepsilon _*>0\), independent of T, such that, if \(A<\varepsilon _*\), then
where \(C_f\) is constant given in (1.9).
Proof
Let \(k\in {{\mathbb {N}}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22) with \(N=1\), namely, \(\ell _k=2k+3\). Then, by (1.9) and (4.31) we have
This together with (4.10) implies (4.32). Thus Lemma 4.2 follows. \(\square \)
Proof of Theorem 1.2
(\(N=1\)). Let v be the function defined by (4.27). Then, since \(\Vert \varphi \Vert _{\mathrm {exp}L^2}\) is sufficiently small, by (4.30), Lemma 4.2, and applying the same argument as in the proof of Theorem 1.2 for the case \(N\ge 2\), we can prove that
and
Let \(q\in [2,\infty ]\). Then, by (2.7) with \((N,r)=(1,q)\), (3.23), and (4.35) we have
This together with (4.22) and (4.27) implies
Therefore, by (4.1), (4.34), and (4.36) we have (1.16), and the proof of Theorem 1.2 for the case \(N=1\) is complete. \(\square \)
5 Rapidly decaying initial data
In this section we prove Theorems 1.3 and 1.4. Let
We can assume, without loss of generality, that \(L<1\). Let \(p_1\) be the constant given in (1.18). For \(\Vert \varphi \Vert _{L^{p_1}}>0\), we denote
and
where \(c_1\) and \(c_2\) are given in \((G_1)\). Since we assume \(L<1\) and thanks to (1.17) we have
Then we first show the following lemma, which is analogous to Lemma 4.1.
Lemma 5.1
Let \(N\ge 2\), \(T>0\), and \(p\in [1,2)\). Furthermore let \(p_1\) be the constant given in (1.18). Suppose that, for any \(q\in [p_1,\infty ]\), the function \(u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\) satisfies
where D is independent of q and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, for \({\tilde{K}}\) as in (5.3), there exists a sufficiently large constant \(T_1=T_1({\tilde{K}},p_1,\lambda ,D)\ge 1\) such that if \(T\ge T_1\) it follows that, for any \(r\in [p_3,\infty ]\),
where \(C_f\) is given in (1.9) and
Proof
Let \(k\in {{\mathbb {N}}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22). Since
for any \(r\in [p_3,\infty ]\), by (1.9) and (5.5) we have
for all \(t>0\). We can take a sufficiently large constant \(T_1\ge 1\) such that, for all \(t>T_1\), it holds that
It is enough to choose
This together with (5.8) implies (5.6). Thus Lemma 5.1 follows. \(\square \)
Similarly, for the case \(N=1\), we have the following.
Lemma 5.2
Let \(N=1\), \(T>0\), and \(p\in [1,2)\). Suppose that the function \(u\in C(0,\infty )\) satisfies
where \(D>0\) and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, for \({\tilde{K}}\) as in (5.3), there exists a sufficiently large constant \({\tilde{T}}_1={\tilde{T}}_1({\tilde{K}},p,\lambda ,D)\) such that, if \(T\ge {\tilde{T}}_1\), then it follows that
where \(C_f\) is given in (1.9).
Proof
Let \(k\in {{\mathbb {N}}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22) with \(N=1\), namely, \(\ell _k=2k+3\). Furthermore, let \({\tilde{T}}_1\) be a sufficiently large constant satisfying (5.9) with \((N,p_1)=(1,p)\). Then, by (1.9) and (5.10) we have
This implies (5.11). Thus Lemma 5.2 follows. \(\square \)
Next we prove (1.20) for small times.
Lemma 5.3
Let \(N\ge 1\) and u be the unique solution to problem (1.1) satisfying (1.14) and (1.15). Suppose \(\varphi \in L^p\) for \(p\in [1,2)\). Let \(p_1\) and K be the constants given in (1.18) and (5.2), respectively. Then, for any fixed \(T_*\ge 1\), there exists a constant \(\varepsilon = \varepsilon (p_1, T_*) >0\) such that, if \(L <\varepsilon \) (where L is the constant given in (5.1)) then, for any \(q\in [p_1,\infty ]\),
where \(C_*\) is independent of q, K, and \(T_*\).
Proof
We first prove (5.12). Let \(N\ge 2\). By (1.12) we consider
where D[u] is the function defined by (3.3). For the linear part, by (2.8), (2.9), and (5.2), for any \(q\in [p_1,\infty ]\), we have
For the nonlinear part D[u], let \(p_2\) be the constant given in (2.20). Then, by (2.11) with \((q,r)=(2N,\infty )\) and (3.3) we see that
On the other hand, for \(r\in [p_2,\infty )\), by (1.15) and taking a sufficiently small \(\varepsilon =\varepsilon (r)>0\), for \(L<\varepsilon \), we can apply Lemma 2.2, and it holds that
where \(C>0\) is independent of r, N, and L. Since \(2N\ge p_2\), by (5.16) and (5.17) we obtain
Furthermore, since it follows from \(p\in [1,2)\) with (1.18) and (2.20) that \(N(1/p_2-1/p_1)<1\), by (2.11) with \((q,r)=(p_2,p_1)\), (3.3), and (5.17) we have
Similarly, by (2.12) with \((q,r)=(p_2,p_1)\) we obtain
If we choose L small enough such that
then, by (5.4), (5.18), (5.19), and (5.20), for any \(q\in [p_1,\infty ]\), we get
Since \(T_*\ge 1\), by (5.15) and (5.21), for any \(q\in [p_1,\infty ]\), we obtain
where \( C_*\) is independent of q, K, and \(T_*\). This implies (5.12).
Next we prove (5.13). Let \(N=1\). Then, we recall that \(p_1=p\). By (1.13) we consider
where \({\tilde{D}}[u]\) is the function defined by (3.23). For the linear part, by (2.8), (2.10), and (5.2), for any \(q\in [p,\infty ]\), we have
and
On the other hand, by (1.15) and taking a sufficiently small \(\varepsilon >0\), for \(L<\varepsilon \), we can apply Lemma 2.3, and we have
Then, for the nonlinear part \({\tilde{D}}[u]\), it holds from (2.7), (3.23), and (5.25) that, for any \(q\in [p,\infty ]\),
Similarly, we have
If we choose \(L<T_*^{-1/(4p)}\), then, by (5.4) and (5.26), for any \(q\in [p,\infty ]\), we get
Furthermore, by (5.4) and (5.27), it holds that
Combining (5.23) and (5.28), we have
Similarly, by (5.24) and (5.29), we obtain
These imply (5.13), thus Lemma 5.3 follows. \(\square \)
For the case \(N\ge 2\), applying Lemmata 5.1 and 5.3, we show the decay estimate of \(|u(t)|_{L^q}\).
Lemma 5.4
Assume the same conditions as in Lemma 5.3 for the case \(N\ge 2\). Then, for \({\tilde{K}}\) as in (5.3), there exists a positive function \(F=F(N,p_1,{\tilde{K}}, \lambda )\) such that, if \(L<F\) and L is small enough, then, for any \(q\in [p_1,\infty ]\),
where C depends only on N.
Proof
Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Then, similarly to (4.11), applying arguments similar to that in the proof of [20, Lemma 2.1] with (1.15), we see that
By Lemma 5.3, for any \(T_*\ge 1\), there exists \(\varepsilon =\varepsilon (p_1,T_*)\) such that, if \(L<\varepsilon \), then
where \(C_*\ge 1\) is independent of q, K and \(T_*\). Let us fix \(T_*\) large enough to be chosen later, put
Then, since \(T_*\ge 1\), by (5.32) we have \(T\ge 2T_*\ge 2\).
We prove \(T=\infty \). The proof is by contradiction. We assume that \(T<\infty \). Then, by (5.31) we see that
On the other hand, by (2.9) with \((q,r)=(p_1,q)\) and (5.2) we have
Let \(T_1\) be the constant given in Lemma 5.1 with \(D=2C_*\), and let us assume that
Furthermore, let \(I_1\) and \(I_2\) be functions given in (4.16), and let \(p_2\) be the constant given in (2.20). Then, for the term \(I_1\), since \(T\ge 2T_*\), by (2.12) with \((q,r)=(p_2,q)\) we get
Since \(p_1\ge p_2\ge 1\) and \(T\ge 1\), due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2 to the term A(T), and we obtain
Furthermore, let \(p_3\) be the constant given in (5.7). Then, since \(T_*\ge T_1\) and it follows from \(p_1<2\) that \(p_2\ge p_3\) we can apply Lemma 5.1 to the term B(T), and we have
where C is independent of q, L, K, and \(T_*\). Moreover, for the term \(I_2\), since \(q\ge p_1\ge p_3\) and \(p_1<2\), by (2.12) with \(q=r\) and (5.6) we see that
where C is independent of q, L, K, and \(T_*\). This together with (4.16), (5.36), (5.37), and (5.38) implies that
where \(D_*\) is a constant independent of L, K, and \(T_*\). Since \(p_1<2\), we can take a sufficiently large constant \(T_*\ge 1\) so that
which means
This together with (5.35) implies that \(T_*\) depends on \(\lambda \), \({\tilde{K}}\), and \(p_1\) but not on L. Then we can also take a sufficiently small constant L so that
and this means
By (5.4), (5.39), (5.40), and (5.42) we have
Combining (5.14), (5.34), and (5.44), we see that
This contradicts (5.33), and we see \(T=\infty \). In order to make clear the dependence of the choice we made on \(T_*\) and L, we collect below all the conditions (5.35), (5.41), and (5.43)
where \(T_1\) satisfies (5.9) with \(D=2C_*\), namely
Here \(C_*\) and \(D_*\) are constants depending at most on N and \(p_1\). Then we can find a function F depending on N, \(p_1\), \({\tilde{K}}\), and \(\lambda \) such that the conditions on L can be written as \(L< F(N, p_1, { {{\tilde{K}}}}, \lambda )\) and L small enough. Thus Lemma 5.4 follows. \(\square \)
Similarly, for the case \(N=1\), applying Lemmata 5.2 and 5.3, we have the following.
Lemma 5.5
Assume the same conditions as in Lemma 5.3 for the case \(N=1\). Then, for \({\tilde{K}}\) as in (5.3), there exists a positive function \(F=F(p,{\tilde{K}}, \lambda )\) such that, if \(L<F\) and L is small enough, then,
where C is independent of p and K.
Proof
Applying the same argument as in the proof of Lemma 5.4, we can prove this lemma. For reader’s convenience, we give it here.
Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Then, similarly to (5.31), we can easily show that
By Lemma 5.3, for any \(T_*\ge 1\), there exists \(\varepsilon =\varepsilon (p,T_*)\) such that, if \(L<\varepsilon \), then
where \(C_*\ge 1\) is independent of K and \(T_*\). Let us fix \(T_*\) large enough to be chosen later, put
Then, since \(T_*\ge 1\), by (5.46) we have \(T\ge 2T_*\ge 2\).
We prove \(T=\infty \). The proof is by contradiction. We assume that \(T<\infty \). Then, by (5.45) we see that
On the other hand, by (2.10) with \(q=p\) and (5.2) we have
Let \({\tilde{T}}_1\) be the constant given in Lemma 5.2 with \(D=2C_*\), and let us assume
Furthermore, since \(T\ge 2T_*\), by (3.23) we put
Since \(p\ge 1\) and \(T\ge 2T_*\), due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.3 to the term \({\tilde{I}}_1(T)\), and we obtain
Furthermore, since \(T_*\ge {\tilde{T}}_1\) and \(p<2\), for the terms \({\tilde{I}}_2(T)\) and \({\tilde{I}}_3(T)\), we can apply Lemma 5.2, and we have
and
where C is a constant independent of p, L, K, and \(T_*\). These together with (5.50) and (5.51) imply that
where \(D_*\) is a constant independent of L, K, and \(T_*\). Since \(p<2\), we can take a sufficiently large constant \(T_*\ge 1\) so that
which means
This together with (5.49) implies that \(T_*\) depends on \(\lambda \), \({\tilde{K}}\), and p but not on L. Then we can also take a sufficiently small constant L so that
and this means
By (5.4), (5.52), (5.53), and (5.55) we have
This together with (5.22) and (5.48) implies
This contradicts (5.47), and we see \(T=\infty \). In order to make clear the dependence of the choice we made on \(T_*\) and L, we collect below all the conditions (5.49), (5.54), and (5.56)
where \({\tilde{T}}_1\) satisfies (5.9) with \((N,p_1)=(1,p)\) and \(D=2C_*\), namely
Here \(C_*\) and \(D_*\) are constants depending at most on p. Then we can find a function F depending on p, \({\tilde{K}}\), and \(\lambda \) such that the condition L can be written as \(L< F(p, {\tilde{K}}, \lambda )\) and L small enough. Thus Lemma 5.5 follows. \(\square \)
Now we ready to prove Theorem 1.3. We first prove it for the case \(N\ge 2\).
Proof of Theorem 1.3
(\(N\ge 2\)). Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Let T be a sufficiently large constant to be chosen later, which satisfies \(T\ge T_1\), where \( T_1\) is the constant given in Lemma 5.1 with \(D=C_*\). Suppose that L is small enough such that Lemmata 5.3 and 5.4 hold. Then, by (5.12) and (5.30), in order to prove (1.20), it suffices to prove the decay estimate of \(\Vert u(t)\Vert _{L^q}\) for \(t\ge 2T\).
Let \(p_1\) be the constant given in (1.18) and \(q\in [p_1,\infty ]\). For the linear part, by (2.8) with \((q,r)=(p_1,q)\) and (5.2) we have
For the nonlinear part, let \(J_1\) and \(J_2\) be functions given in (4.24), and let \(p_2\) be the constant given in (2.20). Then, for the term \(J_1\), similarly to (5.36), by (2.11) with \((q,r)=(p_2,q)\) we put
For the term \({\tilde{A}}(t)\), since \(p_1\ge p_2\ge 1\), by (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2, and we have
Furthermore, let \(p_3\) be the constant given in (5.7). Then, for the term \({\tilde{B}}(t)\), since \(T\ge T_1\) , and \(p_2\ge p_3\), we can apply Lemma 5.1, and it follows from \(p_2\ge 1\) that
For \(p\in (p_2,2)\) (which implies \(p_1=p\)), since \(p_1<2\), we can choose \(\sigma _1\in (0,1)\) satisfying
Then, by (5.59) and (5.60) we have
and
This together with (5.4) and (5.58) implies that
Choosing T large enough such that
namely
and L small enough such that
thanks to (5.4) we get
On the other hand, for \(p\le p_2\), namely \(p_1=p_2\), we consider two cases, \(N=2\) and \(N\ge 3\). For the case \(N\ge 3\), since \(p_1\in (1,2)\), we can choose \(\sigma _2\in (0,1)\) satisfying
Then, by (5.59) and (5.60) we see that
and
This together with (5.4) and (5.58) implies that
Choosing T large enough such that
and L small enough such that
thanks to (5.4) we get
For the case \(N=2\), since \(p_1=p_2=1\) (which implies \(p=1\)), by (5.59) and (5.60) again we see that
and
This together with (5.4) and (5.58) implies that
Choosing T large enough such that
and L small enough such that
thanks to (5.4) we get
Therefore, by (5.61), (5.62), and (5.63), for \(N\ge 2\), we have
Let us come back to the \(J_2(t)\) term. Since \(T\ge T_1\) and \(q\ge p_1\ge p_3\), we can apply Lemma 5.1, and by (2.11) with \(q=r\) and (5.6) we have
Since \(p_1<2\), we can choose \(\sigma _3 >0\) satisfying \(0<\sigma _3 <1/p_1-1/2\), and we get
Choosing T large enough such that
we have
Combining (5.57), (5.64), and (5.65), we obtain
thus (1.20) follows.
Finally we prove (1.21) by the same arguments as in the proof of [10, Theorem 2.2]. Indeed, let \(p_1\in (1,2)\). By (5.61), (5.62), and (5.65) we have
Now, by density, let \(\{\varphi _n\}\subset C_0^\infty \) such that \(\varphi _n\rightarrow \varphi \) in \(L^{p_1}\). Then, by (2.8), it holds that
Since \(p_1>1\), this proves that
and so
Thus the proof of Theorem 1.3 for the case \(N\ge 2\) is complete. \(\square \)
Next, applying the same argument as in the prof of Theorem 1.3 for the case \(N\ge 2\), we prove Theorem 1.3 for the case \(N=1\).
Proof of Theorem 1.3
(\(N=1\)). Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Let T be a sufficiently large constant to be chosen later, which satisfies \(T\ge {\tilde{T}}_1\), where \({\tilde{T}}_1\) is the constant given in Lemma 5.2 with \(D=C_*\). Suppose that L is sufficiently small so that Lemmata 5.3 and 5.5 hold. Then, it is enough to prove the decay estimate of \(\Vert {\tilde{D}}(t)\Vert _{L^q}\) for \(t\ge 2T\) in order to obtain (1.20).
Let \(q\in [p,\infty ]\). Then, similarly to (5.50), by (2.7) and (3.23) we put
For the term \({\tilde{J}}_1\), by (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.3, and we have
Furthermore, for the term \({\tilde{J}}_2(t)\), since \(T\ge {\tilde{T}}_1\), we can apply Lemma 5.2, and it holds that
For \(p\in (1,2)\), we can choose \({\tilde{\sigma }}_1\in (0,1)\) satisfying
Then, for \(t\ge 2T\) we have
Then, by (5.67) and (5.68) we have
Now, choosing T large enough such that
and then L small enough so that
thanks to (5.4) we get
On the other hand, for \(p=1\), by (5.67) and (5.68) again we see that
This together with (5.69) implies for all \(p\in [1,2)\)
For the \({\tilde{J}}_3(t)\) term, since \(T\ge {\tilde{T}}_1\), we can apply Lemma 5.2, and it holds that
Since \(p<2\), we can choose \({\tilde{\sigma }}_2 >0\) satisfying \(0<{\tilde{\sigma }}_2 <1/p-1/2\), and we get
Now, choosing T large enough such that
we get
Combining (5.66), (5.70), and (5.71), we see that
thus (1.20) follows. Furthermore, applying the same arguments as in the proof of Theorem 1.3 for the case \(N\ge 2\) with (5.69) and (5.71), we obtain (1.21). Thus the proof of Theorem 1.3 for the case \(N=1\) is complete. \(\square \)
Remark 5.1
Similarly to the case of the Cauchy problem for the semilinear heat equation with (1.7), the nonlinear boundary problem (1.1) with (1.9) has no scaling invariance and the \(L^p\) and \(\mathrm {exp}L^2\) norms have no relationship between each other. In order to have initial data which fulfill condition (1.19), let us choose a function \(\varphi \in L^p({{\mathbb {R}}}^N_+) \cap L^\infty ({{\mathbb {R}}}^N_+)\) with \(p\in [1,2)\). Then, by (2.15) we see that \(\varphi \in \mathrm {exp}L^2\). Then, let us consider a dilation \(\varphi _\lambda (x)= \lambda ^{N/p} \varphi (\lambda x)\) so that \(\Vert \varphi _\lambda \Vert _{L^p}=\Vert \varphi \Vert _{L^p}\). Since \(\Vert \varphi _\lambda \Vert _{L^2}= \lambda ^{N(1/p -1/2)}\Vert \varphi \Vert _{L^2}\) and \(\Vert \varphi _\lambda \Vert _{L^\infty }= \lambda ^{N/p}\Vert \varphi \Vert _{L^\infty }\), it follows
This implies that there is \(\lambda >0\) so that \(\varphi _\lambda \) fulfills condition (2.4), even though its \(L^p\) norm might be large.
In the end of this section we prove Theorem 1.4. In the following Lemmata, we assume \(\Vert u(t)\Vert _{L^q}\) bounded at the origin and decaying at infinity, and we can deduce that also \(\Vert f(u(t))\Vert _{L^r}\) is bounded and decays at infinity for \(r\ge p_3\), where \(p_3\) is given in (5.7).
Lemma 5.6
Let \(N\ge 2\), \(p\in [1,2)\), and \( K>0\). Suppose that \(u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\) and for any \(q\in [p,\infty ]\),
where C is independent of q and K. Let f be a function satisfying (1.9). Then, there is \(\varepsilon >0\) depending only on \(\lambda \) such that, if \( K<\varepsilon \), then, for any \(r\in [p_4,\infty ]\),
where \(C_f\) is given in (1.9) and
Proof
Let \(k\in {\mathbb {N}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22). Then, since it follows from (5.74) that
similarly to (4.9), for any \(r\in [p_4,\infty ]\), it follows from (1.9) and (5.72) that
We can take a sufficiently small \(\varepsilon =\varepsilon (\lambda )>0\) so that, for \( K\le \varepsilon \), it holds that
This together with (5.75) implies (5.73). Thus Lemma 5.6 follows. \(\square \)
Lemma 5.7
Let \(N=1\), \(p\in [1,2)\), and \(K>0\). Suppose \(u\in C((0,\infty ))\) and
where C is independent of K. Let f be a function satisfying (1.9). Then, there is \(\varepsilon >0\) such that, if \(K<\varepsilon \), then,
where \(C_f\) is given in (1.9).
Proof
Let \(k\in {\mathbb {N}}\cup \{0\}\) and \(\ell _k\) be the constant given in (2.22) with \(N=1\), namely, \(\ell _k=2k+3\). Furthermore, let \(\varepsilon \) be a sufficiently small constant given in Lemma 5.6. Then, similarly to (4.33), it follows from (1.9), (5.76), and (5.77) that
This implies (5.78), thus Lemma 5.7 follows. \(\square \)
Proof of Theorem 1.4
Put \( K=\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}\). Applying the same arguments as in the proofs of Theorems 1.2 and 1.3 with Lemmata 5.6 and 5.7, we can prove Theorem 1.4. \(\square \)
6 Asymptotic behavior
Let us come to the asymptotic behavior of the solution u as stated in Theorem 1.5.
Proof of Theorem 1.5
Let u be the global-in-time solution to problem (1.1) satisfying (1.22). Furthermore, let \(\varepsilon >0\) be a sufficiently small constant chosen later. Then, by (1.22) and (2.16) we can take a sufficiently large \(T=T(\varepsilon ,N)>0\) so that
Therefore, applying the semigroup property of the kernel G, namely (2.1), we can assume, without loss of generality, that \(\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^1}<\varepsilon \).
We first consider the case \(N\ge 2\). By (1.22), taking a sufficiently small \(\varepsilon >0\) if necessary, and applying the same argument as in the proof of Lemmata 2.2 and 5.1 with \(p_1=p_2=p_3=1\), we have
Therefore we can define a mass of u(t) denote by m(t), that is,
Furthermore, it holds that
This implies that there exists the limit of m(t), which we denote by \(m_*\), such that
Furthermore, similarly to (6.1), we obtain
Therefore, applying an argument similar to the proof of [20, Theorem 1.1] (see also [22]) with (1.22), we have (1.23) for the case \(N\ge 2\).
Next we consider the case \(N=1\). By (1.22) and taking a sufficiently small \(\varepsilon >0\) if necessary, we can apply Lemmata 2.3 and 5.2, and we have
Therefore we can define a mass of u(t) denote by m(t), that is,
Furthermore, it holds that
This implies that there exists the limit of m(t), which we denote by \(m_*\), such that
and it holds that
Therefore, applying the same argument as in the proof of (1.23) for the case \(N\ge 2\), we have (1.23) for the case \(N=1\). Thus the proof of Theorem 1.5 is complete. \(\square \)
Change history
08 September 2022
Missing Open Access funding information has been added in the Funding Note.
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C. For sale by the Superintendent of Documents (1964)
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam (2003)
Brezis, H., Cazenave, T.: A nonlinear heat equation with singular initial data. J. Anal. Math. 68, 277–304 (1996)
Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in Hs. Nonlinear Anal. 14(10), 807–836 (1990)
Chlebík, M., Fila, M.: From critical exponents to blow-up rates for parabolic problems. Rend. Mat. Appl. (7) 19 4, 449–470 (1999)
Deng, K., Fila, M., Levine, H.A.: On critical exponents for a system of heat equations coupled in the boundary conditions. Acta Math. Univ. Comenian. (N.S.) 63(2), 169–192 (1994)
Fila, M., Ishige, K., Kawakami, T.: Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Commun. Pure Appl. Anal. 11(3), 1285–1301 (2012)
Fino, A.Z., Kirane, M.: The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Commun. Pure Appl. Anal. 19(7), 3625–3650 (2020)
Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall Inc, Englewood Cliffs (1964)
Furioli, G., Kawakami, T., Ruf, B., Terraneo, E.: Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity. J. Differ. Equ. 262(1), 145–180 (2017)
Galaktionov, V.A., Levine, H.A.: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J. Math. 94, 125–146 (1996)
Haraux, A., Weissler, F.B.: Nonuniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31(2), 167–189 (1982)
Hisa, K., Ishige, K.: Solvability of the heat equation with a nonlinear boundary condition. SIAM J. Math. Anal. 51(1), 565–594 (2019)
Ioku, N.: The Cauchy problem for heat equations with exponential nonlinearity. J. Differ. Equ. 251(4–5), 1172–1194 (2011)
Ioku, N., Ruf, B., Terraneo, E.: Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in R2. Math . Phys. Anal. Geom. 18(1), 29, 19 (2015)
Ishige, K., Kawakami, T., Kobayashi, K.: Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete Contin. Dyn. Syst. Ser. S 7(4), 767–783 (2014)
Ishige, K., Kawakami, T.: Global solutions of the heat equation with a nonlinear boundary condition. Calc. Var. Partial Differ. Equ. 39(3–4), 429–457 (2010)
Ishige, K., Sato, R.: Heat equation with a nonlinear boundary condition and uniformly local Lr spaces. Discrete Contin. Dyn. Syst. 36(5), 2627–2652 (2016)
Ishige, K., Sato, R.: Heat equation with a nonlinear boundary condition and growing initial data. Differ. Integral Equ. 30(7–8), 481–504 (2017)
Kawakami, T.: Global existence of solutions for the heat equation with a nonlinear boundary condition. J. Math. Anal. Appl. 368(1), 320–329 (2010)
Kawakami, T.: Entropy dissipation method for the solutions of the heat equation with a nonlinear boundary condition. Adv. Math. Sci. Appl. 20(1), 169–192 (2010)
Kawakami, T.: Higher order asymptotic expansion for the heat equation with a nonlinear boundary condition. Funkcial. Ekvac. 57(1), 57–89 (2014)
Majdoub, M., Tayachi, S.: Well-posedness, global existence and decay estimates for the heat equation with general power-exponential nonlinearities. In: Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, pp. 2413–2438 (2018)
Nakamura, M., Ozawa, T.: Nonlinear Schrödinger equations in the Sobolev space of critical order. J. Funct. Anal. 155(2), 364–380 (1998)
Quittner, P., Rodríguez-Bernal, A.: Complete and energy blow-up in parabolic problems with nonlinear boundary conditions. Nonlinear Anal. 62(5), 863–875 (2005)
Quittner, P., Souplet, P.: Bounds of global solutions of parabolic problems with nonlinear boundary conditions. Indiana Univ. Math. J. 52(4), 875–900 (2003)
Quittner, P., Souplet, P.: Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel. Blow-up, global existence and steady states (2007)
Ruf, B., Terraneo, E.: The Cauchy problem for a semilinear heat equation with singular initial data. Evol. Equ. Semigroups Funct. Anal. 2002, 295–309 (2000). ((Milano: Progr. Nonlinear Differential Equations Appl., vol. 50. Birkhäuser, Basel)
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in Lp. Indiana Univ. Math. J. 29(1), 79–102 (1980)
Weissler, F.B.: Existence and nonexistence of global solutions for a semilinear heat equation. Israel J. Math. 38(1–2), 29–40 (1981)
Acknowledgements
Open access funding provided by Universitá degli studi di Bergamo within the CRUI-CARE Agreement. The authors would like to express their sincere gratitude to the anonymous referees for their careful reading and useful comments. The work of the second author (T.K.) was supported by JSPS KAKENHI Grant Numbers JP 19H05599 and 20K03689.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This article is part of the topical collection “Qualitative properties of solutions to nonlinear parabolic problems: In honor of Professor Eiji Yanagida on the occasion of his 65th birthday” edited by Senjo Shimizu, Tohru Ozawa, Kazuhiro Ishige, Marek Fila (Guest Editor).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Furioli, G., Kawakami, T. & Terraneo, E. Heat equation with an exponential nonlinear boundary condition in the half space. Partial Differ. Equ. Appl. 3, 36 (2022). https://doi.org/10.1007/s42985-022-00170-7
Published:
DOI: https://doi.org/10.1007/s42985-022-00170-7
Keywords
- Global existence
- Asymptotic behavior
- Initial-boundary value problem
- Nonlinear boundary condition
- Exponential nonlinearity
- Orlicz space