Abstract
The aim of this paper is to study the following time-space fractional diffusion problem
where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary, \((-\Delta )^{\alpha }\) is the fractional Laplace operator with \(0<\alpha <1\), \(\partial _t^{\beta }\) is the Riemann-Liouville time fractional derivative with \(0<\beta <1\), \(\lambda \) is a positive parameter, \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function, and \(g\in L^2(0,\infty ;L^2(\Omega ))\). Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated.
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1 Introduction and the Main Results
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^N\) \((N\ge 1)\) with Lipschitz boundary. We consider the following time-space fractional problem
where \(0<\beta ,\alpha <1\), \(N>2\alpha \), \(\lambda \ne 0\) is a given parameter and \(g\in L^{2}(0,\infty ;L^2(\Omega ))\) is a given function. We assume that the reaction f is a continuous function on \(\Omega \times [0,\infty )\) and \(f(x,\xi )=0\) for all \(x\in \Omega \) and \(\xi \le 0\). Moreover, for all \(\xi >0\), f satisfies
- \((f_1)\):
-
\(f(x,\xi )=|\xi |^{p-2}\xi \) and \(2<p\le 2_\alpha ^*:=\frac{2N}{N-2\alpha },\)
or the following Lipschitz condition:
- \((f_2)\):
-
there exists \({\mathcal {L}}>0\) such that
$$\begin{aligned} |f(x,\xi _1)-f(x,\xi _2)|\le {\mathcal {L}} |\xi _1-\xi _2| \ \ \text {for all}\ \xi _1,\xi _2\in {\mathbb {R}}\ \ \mathrm{and}\ x\in \Omega . \end{aligned}$$
Here, the order \(\beta \) of Riemann-Liouville fractional operator \(\partial _t^{\beta }\) is defined by
where \(J^{1-\beta }\) denotes the \(1-\beta \) order Riemann-Liouville fractional integral operator and it is given by
Here \(\Gamma \) is the usual Gamma function. The fractional Laplace operator \((-\Delta )^\alpha \), up to a normalization constant, is defined by
for all \(\varphi \in C^\infty _0({\mathbb {R}}^{N})\). Here, \(B_\varepsilon (x)=\{y\in {\mathbb {R}}^N:|y-x|<\varepsilon \}.\) For more properties related to the fractional Laplacian and fractional Sobolev spaces as well as for applications of variational methods to fractional problems, we refer to [3].
The fractional operators and related differential equations have important applications in many areas such as physics [15], mechanics chemistry, population dynamic [4, 5], anomalous diffusion [29] and so on. Time fractional differential equations can be used to describe some problems with memory effects. Moreover, both time and space fractional differential equations have been exploited for anomalous diffusion or dispersion where particles spread at a rate inconsistent with Brown motion, see [9]. In the case of time fractional derivatives, particles with “memory effect” propagates slowly, which we call anomalous subdiffusion. Different from the former, spatial fractional diffusion equations are used to describe macroscopic transport and usually result in superdiffusion phenomenon. So far, the works on problems involving the fractional Laplacian and its variants are quite large, here we just list a few, see [8, 10, 20, 21, 31,32,33,34] and the references cited there.
To the best of our knowledge, until recently there has been still very little works on deal with the existence, decay estimates and blow-up of solutions for time-space fractional problems like (1.1). In [30], Vergara and Zacher considered the following time fractional diffusion problem
where \(k*(u-u_0)=\int _0^tk(t-\tau )(u(\tau )-u_0)d\tau \) and \(k\in L_{1,loc}({\mathbb {R}}_+)\). Some useful fundamental identities were obtained. Based on these identities, the existence and decay estimates of weak solutions were obtained. In particular, the decay estimates of weak solutions were given by using the sub-supersolution method. In [17], Li et al. studied the following time-space fractional Keller-Segel equation
where \(^C_0D_t^\beta \) denotes the Caputo derivative, \(B(\rho )=-s_{n,\gamma }\int _{{\mathbb {R}}^n} \frac{x-y}{|x-y|^{n-\gamma +2}}\rho (y)dy\) is the Riesz potential with a singular kernel. The authors obtained the existence and uniqueness of mild solutions. Moreover, the authors discussed the properties of the mild solutions, such as mass conservation and blow-up behaviors. In [1], Bekkai et al. studied the following Cauchy problem involving the Caputo derivative and the fractional Laplacian
\(N\ge 1,\) \(0<\alpha <1\), \(0<\beta \le 2\). First the existence of mild solutions of (1.2) was obatined by the Banach contraction mapping principle. Then the authors proved that the mild solution is also the weak solution. Furthermore, the authors showed the local weak solutions blow up in finite time by choosing suitable test function. See also [6, 23, 36, 39] for similar discussions of the blow-up properties of solutions. Very recently, Fu and Zhang [11] considered the following time-space fractional Kirchhoff problem
where \(M:[0,\infty )\rightarrow [0,\infty )\) is a continuous function. Under suitable assumptions, the authors obtained the global existence of solutions by using the Galerkin method. Furthermore, a decay estimate of solutions was established.
On the other hand, when \(\alpha \), \(\beta \) and s limit to 1, the Eq. (1.1) reduces to the following equation
which is called pseudo-parabolic equation. Equations like (1.3) can be used to describe many important physical processes, such as unidirectional propagation of nonlinear, long waves [2, 27], the aggregation of population [25] and semiconductors [14]. The study of Eq. (1.3) received much more attention in the past years, see [12, 18, 35].
Recently, Tuan et al. [28] studied the initial boundary value problem and Cauchy problem of Caputo time-fractional pseudo-parabolic equations
where \({\mathbb {D}}_t^\alpha \) denotes the Caputo time fractional derivative. The local well-posedness of Eq. (1.4) was established. Further, the finite time blow-up of solutions was also obtained. In [24], Nguyen et al. considered a class of pseudoparabolic equations with the nonlocal condition and the Caputo derivative and obtained the existence and uniqueness of the mild solution. In [7], Chaoui and Rezgui dealt with a time fractional pseudoparabolic equation with fractional integral condition. By the Rothe time discretization scheme, the existence of weak solution was obtained. Moreover, the uniqueness of weak solution as well as some regularity results were obtained.
Inspired by the above papers, we discuss in this work the existence, uniqueness, decay estimates of weak solutions and solutions that blow up in finite time for problem (1.1) involving the time-space fractional operators. Since our problem is nonlocal, our discussion is more elaborate than the papers in the literature. Comparing with the papers in the literature, the main feature of this paper is that the problem (1.1) contains the Riemann-Liouville time fractional derivative and the fractional Laplacian. Definitely, this paper is the first time to deal with the local existence and global nonexistence of solutions for problems involving the fractional Laplacian and the Riemann-Liouville time fractional derivative.
Definition 1.1
We say that \(u\in L^{\infty }(0,T;H_0^\alpha (\Omega ))\) with \(\partial _t^{\beta }u \in L^2(0,T;L^2(\Omega ))\) is a weak subsolution (supsolution) of problem (1.1) if \(u(x,0)\le (\ge )u_0(x)\) and
for any \(0\le \varphi \in H_0^\alpha (\Omega )\) and a.e. \(t\in (0,T)\). u is a weak solution if and only u is both a subsolution and a supsolution. Here, we call u is a global weak solution of problem (1.1), if the equality in above holds for any \(0<T<\infty \); u is a local weak solution, if there exists \(T_0>0\) such that the equality in Definition 1.1 holds for \(0<T\le T_0\).
The proof of the following existence results relies on the contract mapping theorem and the Galerkin method.
Theorem 1.2
Assume that \(0\le u_{0}\in H_{0}^{\alpha }(\Omega )\), \(g\in L^2(0,\infty ;L^2(\Omega ))\) and \(f(x,\xi )=0\) for all \(x\in \Omega \) and \(\xi \le 0\). If f satisfies \((f_1)\), then problem (1.1) admits a local nonnegative weak solution. If f satisfies \((f_2)\), then problem (1.1) has a unique global weak solution.
The following theorem shows the asymptotic behavior of global solutions to problem (1.1).
Theorem 1.3
Assume that \(g\equiv 0\) and \(f(x,\xi )=0\) for all \(x\in \Omega \) and \(\xi \le 0\). If \(0\le u_0\in H_0^{\alpha }(\Omega )\) and \(u_0(x)\le \eta _0\varphi _1(x)\) with \(\eta _0>0\) for all \(x\in \Omega \), and f satisfies \((f_2)\), then the unique solution of problem (1.1) satisfies the following decay estimates
where \(c_2>0\) and \( \varphi _1>0\) is the eigenfunction corresponding to the first eigenvalue of the fractional Laplacian.
We also discuss the global nonexistence of local solutions for problem (1.1).
Theorem 1.4
Assume that \(0\le u_{0}\in H_{0}^{\alpha }(\Omega )\), \(g=0\) and \(f(x,\xi )=0\) for all \(x\in \Omega \) and \(\xi \le 0\). Suppose that f satisfies \((f_1)\), and \(\int _\Omega u_0(x)\varphi _1(x)dx>\left( \frac{\lambda _1}{\lambda }\right) ^{1/(p-2)}\), where \(c_2>0\) and \(\varphi _1>0\) is the eigenfunction corresponding to the first eigenvalue \(\lambda _1\). Then the nonnegative weak solutions of problem (1.1) blow up in finite time.
In what follows, the letters c, \(c_{i}\), C, \(C_{i}\), \(i=1, 2, \ldots , \) denote positive constants which vary from line to line, but are independent of terms that take part in any limit process. Furthermore, for any \(p\ge 1\) we denote \(\Vert u\Vert _{p}=\Vert u\Vert _{L^{p}(\Omega )}.\)
2 Preliminaries
In this section, we provide some basic results which will be used in the next sections. The fractional Sobolev space \(H^\alpha ({\mathbb {R}}^N)\) is defined as
endowed with the norm
\(H_0^\alpha (\Omega )\) is defined as
in the sequel, we take
\(H_0^\alpha (\Omega )\) is a Hilbert space in which a scalar product is given by
for any \(u,v\in H_0^\alpha (\Omega )\).
Denote by
the distinct eigenvalues of the fractional Laplace operator and let \(\omega _k\) be the eigenfunction corresponding to \(\lambda _k\) of the following eigenvalue problem
We obtain for \(k\in {\mathbb N}\),
where \(P_1=H_0^\alpha (\Omega )\) and
Lemma 2.1
([3]). Let \(2_\alpha ^*=\frac{2N}{N-2\alpha }\). For any \(q\in [1,2_\alpha ^*]\), the embedding \(H_0^\alpha (\Omega )\hookrightarrow L^q(\Omega )\) is continuous. Furthermore, the embedding is compact if \(q\in [1,2_\alpha ^*)\).
The Yosida approximation of the time-fractional derivative operator is an useful tool to deal with problems with Caputo fractional derivative operators. For more details, we refer to [30, 37, 38]. Let \(1\le p< \infty , 0<\beta <1\) and X be a real Banach space. Define fractional derivative operator
where \(g_{1-\beta }\) is given by
Its Yosida approximation \(B_n\) defined by \(B_n=nB(n+B)^{-1}\) (\(n\in {\mathbb {N}}\)) possesses the property that for any \(u\in D(B)\), \(B_nu\rightarrow Bu\) strongly in \(L^p([0,T];X)\) as \(n\rightarrow \infty \). Here, we collect some important properties of \(g_{1-\beta }\) and \(B_n\) which are listed in the following:
-
The kernel \(g_{1-\beta ,n}\) is nonnegative and nonincreasing for all \(n\in N\), and \(g_{1-\beta ,n}\in W^{1,1}([0,T])\).
-
\(g_{1-\beta ,n}\rightarrow g_{1-\beta }\) in \(L^1([0,T])\) and \(B_nu \rightarrow Bu\) in \(L^p([0,T];X)\) as \(n\rightarrow \infty \).
Lemma 2.2
([30]). Assume that H is a real Hilbert space and \(T>0\) is a real number. Then for any \( k\in W^{1,1}([0,T])\) and \(u\in L^2(0,T;H)\), the following identity holds
Remark 1
Obviously, if k is a nonincreasing and nonnegative function in \(W^{1,1}([0,T])\), then we obtain for any \(u\in L^2(0,T;H)\) that
Lemma 2.3
([30]). Let \(H\in C^1({\mathbb {R}})\) and \(k\in W^{1,1}([0,T])\), for a sufficiently smooth function u, then there holds for a.e. \(t\in (0,T)\)
Definition 2.4
(see [19]). Let \(q>0\) be a real number and \(0<T\le \infty \). We say that a function \(\omega :{\mathbb {R}}^+\rightarrow {\mathbb {R}}\) \(({\mathbb {R}}^+=[0,\infty ))\) satisfies a condition (q), if
where R(t) is a continuous and nonnegative function.
Clearly, if \(\omega (u)=u^r,r>0\), then \(\omega \) satisfies the condition (q) with any \(q>1\), and \(R(t)=e^{(r-1)qt}\).
Lemma 2.5
(see [19, Theorem 1]). Let a be a nondecreasing, nonnegative \(C^1\)-function on [0, T), F be a continuous, nonnegative function on [0, T), \(\omega : {\mathbb {R}}^+\rightarrow {\mathbb {R}}\) be a continuous, nondecreasing function, \(\omega (0)=0, \omega (u)>0\) on [0, T), and u be a continuous, nonnegative function on [0, T) with
where \(\beta >0\). Then the following assertions hold:
(i) Suppose that \(\beta >1/2\) and \(\omega \) satisfies the condition (q) with \(q=2\). Then
where
\(\Gamma \) is the gamma function, \(\Omega (v)=\int _{v_0}^v\frac{dy}{\omega (y)}\), \(v_0>0\), \(\Omega ^{-1}\) is the inverse of \(\Omega \), and \(T_1\in {\mathbb {R}}^+\) is such that \(\Omega (2a(t)^2)+g_1(t)\in Dom(\Omega ^{-1})\) for all \(t\in [0,T_1]\).
(ii) Let \(\beta \in (0,1/2]\) and \(\omega \) satisfies the condition (q) with \(q=z+2\), where \(z=\frac{1-\beta }{\beta }\). Then
where
\(T_1\in {\mathbb {R}}^+\) is such that \(\Omega (2^{q-1}a(t)^{q})+g_2(t)\in Dom (\Omega ^{-1})\) for all \(t\in [0,T_1]\).
In particular, if \(w(u)=u\), then there holds
Lemma 2.6
(see [19, Theorem 2]). Let \(0<T\le \infty \), a(t), F(t) be as in Lemma 2.5, and let u(t) be a continuous, nonnegative function on [0, T) with
where \(\beta >0\). Then the following assertions hold
(i) If \(\beta >1/2\), then
(ii) If \(\beta =\frac{1}{z+1}\) for some \(z\ge 1\), then
where \(K_z\) is defined by \(K_z=[\frac{\Gamma (1-\gamma p)}{p^{1-\gamma p}}]^{1/p}\), \( p=\frac{z+2}{z+1},\) \(q=z+2.\)
Lemma 2.7
Define an operator
for any \(u\in L^2(0,T;L^2(\Omega ))\). If \(\beta \ge 1/2\), then \(I_{1-\beta }:L^2(0,T;L^2(\Omega ))\rightarrow L^2(0,T;L^2(\Omega ))\) is a bounded linear operator; If \(0<\beta <1/2\), then \(I_{1-\beta }:L^r(0,T;L^r(\Omega ))\rightarrow L^r(0,T;L^r(\Omega ))\) is a bounded linear operator, where \(r>\frac{1}{\beta }\).
Proof
If \(\beta \ge \frac{1}{2}\), then the Hölder inequality implies that
Thus,
which yields the desired result.
Now we consider the case \(0<\beta <1/2\). By the Hölder inequality, we have
where \(q>1\) satisfying \(1-(1-\beta )q>0\), and \(\frac{1}{q}+\frac{1}{q^\prime }=1\). Observe that \(1-(1-\beta )q>0\). Thus, we get
which ends the proof. \(\square \)
Lemma 2.8
(see [26] Fractional integration by parts). Let \(\alpha >0\), \(p\ge 1\), \(q\ge 1\) and \(\frac{1}{p}+\frac{1}{q}\le 1+\alpha \) (\(p\ne 1,\ q\ne 1\) in the case when \(\frac{1}{p}+\frac{1}{q}=1+\alpha \)). If \(\varphi \in L^p(a,b)\) and \(\psi \in L^q(a,b)\), then
where
and
Using Lemma 2.8, we can obtain the following result.
Lemma 2.9
Let \(\alpha >0\) and \(1\le p \). Assume that \(\varphi \in C_0^1(0,T)\) and \(\psi \in L^p(0,T)\). Then
In order to show the existence of solutions to problem (1.1), we give some properties of the operator \(L: H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'\) defined by
for all \(u,v\in H_0^\alpha (\Omega )\).
Lemma 2.10
The operator \(L: H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'\) is a monotone and linear bounded functional. Moreover, \(\Vert L(u)\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }\) for all \(u\in H_{0}^{\alpha }(\Omega ).\)
Proof
Let \(u,v\in H_{0}^{\alpha }(\Omega )\), we have
Thus, L is monotone. Clearly, L is a linear functional. It remains to show that \(\Vert Lu\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }.\) It follows from the Hölder inequality, we have
which means that \(\Vert L(u)\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }\). The proof is now complete. \(\square \)
Lemma 2.11
The operator \(L:H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'\) is hemicontinuous.
Proof
We are going to prove that the map \(t\mapsto \langle L(u+tv), w\rangle \) is continuous on [0, 1] for all \(u,v,w\in H_{0}^{\alpha }(\Omega )\), i.e,
for all \(w\in H_{0}^{\alpha }(\Omega ).\) We have,
We define \(G_t:[0,1]\rightarrow {\mathbb {R}}\) by
and set
Obviously, \(\lim _{t\rightarrow 0}G_{t} (x,y)=G(x,y)\) and there exists \(h\in L^{1}({\mathbb {R}}^{2N})\) such that \(|G_{t}(x,y)|\le h(x,y)\). Thus, by the Lebesgue dominated convergence theorem, we obtain the desired result. \(\square \)
To discuss the compactness of approximate solutions, we need the following Lions-Aubin lemma.
Proposition 2.12
([16, Theorem 4.1]). Let \(T>0, \beta \in (0,1)\) and \(p\in [1,\infty ).\) Let \(B_{0}, B, B_{1}\) be Banach spaces. Assume that \(B_{0}\hookrightarrow B\) is compact and \(B\hookrightarrow B_{1}\) is continuous. Suppose that \(W\subset L_{loc}^1(0,T;B_0)\) satisfies;
-
(i)
There exists \(C_1>0\) such that \(\forall u\in W\),
$$\begin{aligned} \sup _{t\in (0,T)}\frac{1}{\Gamma (\beta )}\int _0^t(t-s)^{\beta -1} \Vert u\Vert _{B_0}^p(s)ds\le C_1. \end{aligned}$$ -
(ii)
There exist \(r\in (\frac{p}{1+p\gamma },\infty )\bigcap [1,\infty )\) and \(C_3>0\) such that \(\forall u\in W\), there is an assignment of initial value \(u_0\) for u such that the weak Caputo derivative satisfies
$$\begin{aligned} \Vert D_c^\beta u\Vert _{L^r(0,T; B_1)}\le C_3. \end{aligned}$$Then W is relatively compact in \(L^{p}(0,T,B)\).
Remark 2
The Caputo derivative of an absolutely continuous function u is defined as follows
In view of the definition of the Riemann-Liouville derivative, we know that the Riemann-Liouville derivative and the Caputo derivative have the relationship
3 Existence and Uniqueness of Weak Solutions
In this section by means of the Galerkin method, we establish the existence of local solutions to the problem (1.1). Assume that \(\{\omega _k\}\) is an orthonormal basis in \(L^2(\Omega )\) and
then we shall find Galerkin approximation solutions \(u_m=u_m(t)\) of the following form
where \(a_{mj}\) satisfies that
for \(j=1,2,\ldots ,m.\) Here, \((\cdot ,\cdot )\) denotes the inner product of \(L^2(\Omega )\). Problem (3.1) is a nonlinear fractional ordinary differential system. Next, we show that problem (3.1) has a unique local solution for every \(m\in N\).
First, by Lemmas 2.2 and 2.3, we give a prior estimate for problem (3.1).
Lemma 3.1
Suppose that \(u_m= \sum _{j=1}^ma_{mj}w_j\) solves problem (3.1). If \(f(x,\xi )=|\xi |^{p-2}\xi \) and \(2<p\le 2_\alpha ^*\), then there exist \(T^*>0\) and \({\mathcal {C}}_1>0\) such that
Proof
Multiplying (3.1) by \(\partial _t^{\beta }a_{mj}\) and summing j from 1 to m, we have
By the Yosida approximation of time Riemamm-Liouville fractional derivative, Lemma 2.2 and Hölder’s inequality, one can deduce that
where
By using the Hölder inequality and Young inequality, we deduce
where \(q=2_\alpha ^*,\quad q^\prime =\frac{q}{q-1}=\frac{2N}{N+2\alpha }\) and \({\mathcal {S}}_*\) is the embedding constant from \(H_0^\alpha (\Omega )\) to \(L^{2_\alpha ^*}(\Omega )\). Here, we have used the fact that \((p-1)q^\prime \le 2_\alpha ^*\), thanks to \(p\le 2_\alpha ^*.\) We also get
Therefore, we obtain
Choose \(\varepsilon \) small enough such that \(S_* \lambda \varepsilon <1\). Then it follows from (3.4) that
Convolving (3.5) with \(g_{\beta }\) and letting n go to \(\infty \) and selecting an appropriate subsequence(if necessary), it leads to
where \(C_1=2\lambda {\mathcal {S}}_*C_{\varepsilon }\),
In Lemma 2.5, let \(u(t)=\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2\), \(w(u)=u^{p-1}\), \(F(s)=\frac{C_1}{\Gamma (\beta )}\), \(a(t)=\sup _{m\ge 1}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2.\) When \(\beta >\frac{1}{2}\), \(R(t)=e^{2(p-2)t}\),
where \(C_2=\frac{4^{\beta -1}C_1^2}{2(p-2)\Gamma (2\beta -1)\Gamma ^2(\beta )}>0\), \(C_3=\frac{1}{(p-2) v_0^{p-2}}>0\). Then we deduce from (3.6) that
Considering the definition of \(\Omega ^{-1}\) and using \(\Omega (2a^2(t))+g_1(t)\in Dom \Omega ^{-1}\), we get
which implies that
where \(C_0=\sup _{m\ge 1}\Vert u_{0m}\Vert _{H_0^\alpha (\Omega )}^2.\) Then we have \(t<\frac{\ln C_4}{2(p-2)}:=T_1\), where
Therefore, for the case \(\beta >\frac{1}{2}\), there exists \(T_1>0\) such that
for all \(0<t<T_1\).
When \(\beta \in (0,\frac{1}{2})\), \(R(t)=e^{(p-2)(z+2)t}\),
then we have
where \(z=\frac{1-\beta }{\beta }\), \(q=z+2\), \(K_z=(\frac{\Gamma (1-\gamma r)}{r^{1-\gamma r}})^{\frac{1}{r}}\), \(\gamma =\frac{z}{z+1}\), \(r=\frac{z+2}{z+1}\), \(C_6=\frac{C_1^qK_z^q2^{q-1}}{\Gamma ^q(\beta )(p-2)(z+2)}>0\).
Consider the definition of \(\Omega ^{-1}\), we obtain
which implies that \(t<\frac{\ln C_7}{(p-2)(z+2)}:=T_2\). Here, \(C_7=\frac{1}{C_6(p-2)(2^{q-1}C_0^q)^{p-2}}+1>0\).
Therefore, for \(\beta \in (0,\frac{1}{2})\), there exists \(T_2>0\) such that \(\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2 \le C_8\) for all \(0<t<T_2\).
In conclusion, there exist \(T^*=\min \{T_1,T_2\}\) and \({\mathcal {C}}>0\) such that \(\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}\) for all \(0<t<T^*\). \(\square \)
Lemma 3.2
Suppose that \(u_m= \sum _{j=1}^ma_{mj}w_j\) solves problem (3.1). If f satisfies Lipschitz condition \((f_1)\), then for any \(T>0\) there exists \({\mathcal {C}}_{2}>0\) such that
Proof
Since f satisfies the Lipschitz condition, there exists a positive constant C such that
Then a similar discussion as in Lemma 3.1 gives that
By Lemma 2.6, we get (i) If \(\beta >1/2\), then
where \(a(t)=\sup _{m\ge 1}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +\frac{C_1 t^{1-\beta }}{\Gamma (1-\beta )} +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2.\)
(ii) If \(\beta =\frac{1}{z+1}\) for some \(z\ge 1\), then
where \(K_z=[\frac{\Gamma (1-\gamma p)}{p^{1-\gamma p}}]^{1/p}\), \(\gamma =\frac{z}{z+1}\), \( p=\frac{z+2}{z+1} \) and \(q=z+2.\)
In conclusion, for any \(T>0\) there exists \({\mathcal {C}}_2>0\) such that \(\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}_2\) for all \(t\in [0, T]\) and \(m\ge 1\). \(\square \)
Based on Lemmas 3.1 and 3.2, we obtain the following estimate.
Lemma 3.3
Suppose that \(u_m= \sum _{j=1}^ma_{mj}w_j\) solves (3.1), then there exists \({\mathcal {C}}_3\) such that
Proof
Choosing \(\varepsilon =\frac{1}{2{\mathcal {S}}_* \lambda }\) in (3.4), we get
Integrating above inequality from 0 to T and letting \( n\rightarrow \infty \), we obtain
By Lemma 3.1, it yields
For the case f satisfies the Lipschitz condition, by Lemma 3.2 and a similar discussion as above, one can obtain the desired result. \(\square \)
Now, we prove the local existence of solutions for system (3.1).
Theorem 3.4
Under the assumptions of Theorem 1.2, system (3.1) has a unique solution for all \(t\in [0,T]\), where \(0<T<T^*\) if \(f=|u|^{p-2}u\) and \(0<T<\infty \) if f satisfies \((f_1)\).
Proof
First, problem (3.1) is equivalent to the problem
where \(\psi (t)=(a_{mj}(t))\in {\mathbb {R}}^m\),\(A=diag(\frac{\lambda _j}{1+\lambda _j})_{m\times m}, B= diag(\frac{1}{1+\lambda _j})_{m\times m},\) \(\xi =(b_{mj})\in {\mathbb {R}}^m\), \(\mu =(\lambda _j)\in {\mathbb {R}}^m\), \(R(\psi (t))_j=\lambda (|u_m|^{p-2}u_m,wj)\), \(G_j=(g,w_j)\). By Laplace transform or convoluting with \(g_\beta \), we transform (3.7) into the following Volterra type system
Therefore, we only need to prove that system (3.8) admits a unique continuous solution.
Notice
Then, \(\Vert u_m(t)\Vert _{L^2(\Omega )}=\Vert \psi (t)\Vert _{R^m}\), \(\Vert u_m(t)\Vert _{L^2(\Omega )}\le {\mathcal {S}} \Vert u_m(t)\Vert _{H_0^\alpha (\Omega )}\). Let
Then we obtain a prior estimate \(\Vert \psi (t)\Vert _{{\mathbb {R}}^m}\le R_0\).
Define the operator as
for all \(\psi \in E_T\), where
Set \(d(\psi _1,\psi _2)=\max _{t\in [0,T]}\Vert \psi _1(t)-\psi _2(t)\Vert _{{\mathbb {R}}^m}\). Since \(C(0,T;{\mathbb {R}}^m)\) is a Banach space, it follows that \((E_T,d)\) is a complete metric space. Then
Observe that
and
Thus, we have
Now we assume that T is small enough such that
Then \(\Vert \Psi \psi \Vert _{{\mathbb {R}}^m}\le 2R_0\), which means that \(\Phi \) maps from \(E_{T}\) to \(E_T\).
Next we show that \(\Phi \) is contractive in \(E_T\).
If \(2<p\le 2_\alpha ^*:=\frac{2N}{N-2\alpha }\), then by the Hölder inequality we have
thanks to the following basic inequality:
Then
Observe that
Thus,
Gathering (3.9) and (3.10), we arrive at
where
Consequently, we prove that \(\Phi \) is contractive on \(E_T\) provided T is small enough such that \({\mathcal {D}} T^{1-\beta }<1\). Thus, by the Banach contraction mapping theorem, we know that the map \(\Phi \) has a unique fixed point on some small interval \([0,T_0]\). Therefore, we prove that system (3.7) has a unique solution on \([0,T_0]\).
On the other hand, if f satisfies the Lipschitz condition, the existence of unique solution of system (3.7) on some small interval \([0,T_0]\) can be proved similarly as above.
Finally, we show that the local solution can be extended to (0, T]. Let \(T_0\) and \(u_m(T_0)\) be the initial data. Then repeating the same process as above, we can get a unique continuous solution on \([T_0,2T_0]\). Divide [0, T] into \([(k-1)T_0.kT_0]\) with \(k=1,2,\ldots K\) and \(T/K\le T_0\). Then we can obtain a unique continuous solution in [0, T]. In conclusion, we show that system (3.1) has a unique solution in \(C(0,T;{\mathbb {R}}^m)\).
\(\square \)
Proof of Theorem 1.2
Gathering Lemma 3.1 and Lemma 3.2, we get
and
By Proposition 2.12, we deduce that there exist a subsequence (still denoted by \(\{u_m\}\)) and \(u\in L^{\infty }(0,T;H_0^\alpha (\Omega )\cap L^{p}(\Omega ))\) such that
By the Vitali convergence theorem, one can show that
for all \(v\in C^1(0,T;H_0^\alpha (\Omega ))\).
Since \(\{\partial _t^\beta u_m\}\) is bounded in \(L^2(0,T;L^2(\Omega ))\), up to a subsequence we may assume that
Next we show that \(\chi =\partial _t^\beta u\). By Lemma 2.8, we obtain
It follows from (3.11) that
which means that \(\chi =\partial _t^\beta u\). Further, by (3.1) we conclude that
for any \(v\in L^2(0,T;H_0^\alpha (\Omega ))\).
Next we show that
for any \(\varphi \in H_0^\alpha (\Omega )\).
For any \(t\in (0,T)\), let \(\chi _{(0,t)}\) denote the characteristic function in (0, t). Let \(\varphi \in H_0^\alpha (\Omega )\). Taking \(v=\varphi \chi _{(0,t)}\) in (3.12), we get
Since
we differentiate (3.14) with respect to t and get that
for any \(\varphi \in H_0^\alpha (\Omega )\) and a.e. \(t\in [0,T]\). Thus, (3.13) holds.
It follows that u is a weak solution of problem (1.1). \(\square \)
In the next lemma we shall show that under some assumptions the solution of problem (1.1) is nonnegative.
Lemma 3.5
If \(f(x,\xi )\le 0\) for any \(\xi \le 0\), \(g\ge 0\) and \(u_{0}\ge 0\) a.e. in \(\Omega \). Then the solutions of problem (1.1) are nonnegative.
Proof
Let u be a weak solution to problem (1.1). Clearly,
Taking \(v=-u^{-}\) in Definition 1.1, we obtain
Observe that for a.e. \(x,y\in \Omega \),
Then
Moreover, \(g(-u^-)\le 0\) and \(f(x,u)u^{-}=0\) a.e. \(x\in \Omega \).
To apply Lemma 2.2, we can use the same regularization discussion as above. For convenience, we omit this process. In view of Lemma 2.2, we have
and
Combining these facts with (3.15), it yields
This implies that
Since \(u_0\ge 0\) a.e. in \(\Omega \), it leads to
Then convoluting above inequality with \(g_\beta \), it leads to
Thus, we get \(u^{-} (t)=0\) a.e. in \(\Omega \) and for any \(t>0\). Hence, \(u(x,t)\ge 0\) a.e. in \(\Omega \) and for any \(t>0\). \(\square \)
At the end of this section, we study the uniqueness of solutions of problem (1.1).
Theorem 3.6
(Comparison theorem). Assume that f satisfies \((f_2)\). Let \({\underline{u}}\) be a subsolution of problem (1.1) and let \({\overline{u}}\) be a supsolution of problem (1.1). Then \({\underline{u}}\le {\overline{u}}\).
Proof
We deduce from Definition 1.1 that
for any \(0\le v\in H_0^\alpha (\Omega )\). Taking \(v=({\underline{u}}-{\overline{u}})^+=\max \{{\underline{u}}-{\overline{u}},0\}\) and applying Lemma 2.2 and \((f_2)\), it leads to
Set \(Y(t)=\Vert ({\underline{u}}(t)-{\overline{u}}(t))^+\Vert _2^2+[({\underline{u}}(t)-{\overline{u}}(t))^+]_\alpha ^2\). Then
Convoluting above inequality with \(g_\beta \), it follows that
By Lemma 2.6 and \(Y(0)=0\), one can get that \(Y(t)=0\). Thus, we get \({\underline{u}}\le {\overline{u}}\) a.e. in \(\Omega \times (0,T)\), which ends the proof. \(\square \)
Lemma 3.7
Assume that f satisfies \((f_2)\). Then the solution of problem (1.1) is unique.
Proof
Assume that \(u_1\) and \(u_2\) are two solutions of problem (1.1). Then we deduce from 1.1 that
for any \(v\in H_0^\alpha (\Omega )\). Taking \(v=u_1-u_2\), we get
Using a similar discussion as above and applying Lemma 2.2 and \((f_2)\), it leads to
Set \(Z(t)=\Vert (u_1(t)-u_2(t)\Vert _2^2+[u_1(t)-u_2(t)]_\alpha ^2\). Then
Convoluting above inequality with \(g_\beta \), it follows that
which together with Lemma 2.2 and \(Z(0)=0\) yields that \(Z(t)=0\). Thus, we get \(u_1=u_2\) a.e. in \(\Omega \times (0,T)\), which ends the proof. \(\square \)
4 Decay Estimates of Solutions
In this section, we give a decay estimate for problem (1.1) in which f satisfies \((f_2)\).
Let \(\varphi _1\) be the corresponding eigenfunction to the first eigenvalue \(\lambda _1\) of the fractional Laplacian. Clearly, \(\varphi _1>0\) and \(\varphi _1\in L^\infty (\Omega )\). Let \(0\le u_0(x)\le \eta _0\varphi _1(x)\), where \(\eta _0>0\). Assume that there exists \(h_0>0\) such that \(f(x,\xi )\le h_0\xi \) for all \(x\in \Omega \) and \(\xi \ge 0\).
Set \(v(x,t)=\varphi _1(x)\eta (t)\), where \(\eta \) satisfies
where \(0<\lambda <\frac{\lambda _1}{h_0}\).
By [30, Theorem 7.1], there exist \(c_1,c_2>0\) such that
A simple calculation gives that
Since \(v(x,0)=\varphi _1(x)\eta _0\) and \(\varphi _1(x)\eta _0\ge u_0(x)\), we know that v(x, t) is a supersolution of problem (1.1). Then by Theorem 3.6, we obtain that the unique solution u of problem (1.1) satisfies
Further, (4.1) implies that
5 Blow-Up of Solutions
In this section, we consider the blow-up property of solutions of problem (1.1). Some techniques are inspired from [23] and [39]. Let \(u_0\in H_0^\alpha (\Omega )\) satisfy \(u_0\ge 0\). If \(2<p\) and \(\int _\Omega u_0\varphi _1 dx>(\frac{\lambda _1}{\lambda })^{1/(p-2)}\), then the solution of problem (1.1) blows up in finite time.
Proof of Theorem 1.4
Taking \(\varphi =\varphi _1(x)\varphi _2(t)\) with \(\varphi _2\in C^1(0,T)\) in Definition 1.1 and integrating over (0, T), we get
which gives that
Set \(H(t)=\int _\Omega u(x,t)\varphi _1(x)dx\). Then
It follows from Jensen’s inequality that
thanks to \(p>2\). By Lemma 2.9, we have
where
On one hand, choose \(\varphi _2=I_{T-}^\beta {\bar{\varphi }}(t)\) with \({\bar{\varphi }}\in C_0^1(0,T)\) and \({\bar{\varphi }}\ge 0\). Then we deduce from [13, Lemma 2.21 ] that
Applying Lemma 2.8 to the left of (5.2), it leads to
The arbitrary of \({\bar{\varphi }}\) gives that
Since \(H(0)>(\frac{\lambda _1}{\lambda })^{1/(p-2)}\), we have \(H(t)>(\frac{\lambda _1}{\lambda })^{1/(p-2)}\) as t small enough. Then we have
On the other hand, choose \(\varphi _2=(1-\frac{t}{T})_+^k\), \(t\in [0,T]\), \(k>\max \{1,(p-1)\beta /(p-2)\}\), in (5.1). By a direct calculation (see [13]), one can show that
Then
By the Hölder inequality, one has
Further, using the Young inequality, for any \(\varepsilon >0\) we get
We know that there exists a constant \(C>0\) such that
Choosing \(\varepsilon \) small enough such that \(H(0)>(\frac{\lambda _1}{\lambda -\varepsilon })^{-1/(p-2)}\), we get \(H(0)\le CT^{\beta -(p-1)\beta /(p-2)}\) for some \(C>0\). If problem (1.1) has a global weak solution, we obtain \(H(0)=0\) by letting \(T\rightarrow \infty \), which contradicts \(H(0)>(\frac{\lambda _1}{\lambda })^{-1/(p-2)}\). Therefore we show that the global nonexistence of problem (1.1). \(\square \)
References
Bekkai, A., Rebiai, B., Kirane, M.: On local existence and blowup of solutions for a time-space fractional diffusion equation with exponential nonlinearity. Math. Methods Appl. Sci. 42, 1819–1830 (2019)
Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)
Bisci, G.M., Rădulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)
Caffarelli, L.: Some nonlinear problems involving non-local diffusions, ICIAM 07–6th International Congress on Industrial and Applied Mathematics, pp. 43–56. Zürich, Eur. Math. Soc. (2009)
Caffarelli, L.: Non-local diffusions, drifts and games. Nonlinear Partial Differ. Equ. Abel Symposia 7, 37–52 (2012)
Can, N.H., Kumar, D., Viet, T.V., Nguyen, A.T.: On time fractional pseudo-parabolic equations with nonlocal integral conditions. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7196
Chaoui, A., Rezgui, N.: Solution to fractional pseudoparabolic equation with fractional integral condition. Rend. Circ. Mat. Palermo, II. Ser. 67, 205–213 (2018)
Colombo, F., Gantner, J.: An application of the S-functional calculus to fractional diffusion processes. Milan J. Math. 86(2), 225–303 (2018)
del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional diffusion in plasma turbulance. Phys. Plasmas 11, 3854–3864 (2004)
do Ó, J.M., Giacomoni, J., Mishra, P.K.: Nehari manifold for fractional Kirchhoff systems with critical nonlinearity. Milan J. Math. 87(2), 201–231 (2019)
Fu, Y.Q., Zhang, X.J.: Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete Contin. Dyn.Syst. B (2021). https://doi.org/10.3934/dcdsb.2021091
Han, Y.Z.: Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity. Appl. Math. Lett. 99, 105986 (2020)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Ltd, Amsterdam (2006)
Korpusov, M.O., Sveshnikov, A.G.: Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics. Comput. Math. Math. Phys. 43, 1765–1797 (2003)
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)
Li, L., Liu, J.G.: Some compactness criteria for weak solutions of time fractional PDEs. SIAM J. Math. Anal. 50, 3963–3995 (2018)
Li, L., Liu, J.G., Wang, L.Z.: Cauchy problems for Keller-Segel type time-space fractional diffusion equation. J. Differ. Equ. 265, 1044–1096 (2018)
Long, Q.F., Chen, J.Q.: Blow-up phenomena for a nonlinear pseudo-parabolic equation with nonlocal source. Appl. Math. Lett. 74, 181–186 (2017)
Medveď, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214, 349–366 (1997)
Mingqi, X., Rădulescu, V.D., Zhang, B.L.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31, 3228–3250 (2018)
Mingqi, X., Rădulescu, V.D., Zhang, B.L.: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differ. Equ. 57, 1–27 (2019)
Molica Bisci, G., Rădulescu, V.D.: Ground state solutions of scalar field fractional for Schrödinger equations. Calc. Var. Partial Differ. Equ. 54, 2985–3008 (2015)
Nabti, A.: Life span of blowing-up solutions to the Cauchy problem for a time-space fractional diffusion equation. Comput. Math. Appl. 78, 1302–1316 (2019)
Nguyen, A.T., Hammouch, Z., Karapinar, E., Tuan, N.H.: On a nonlocal problem for a Caputo time-fractional pseudoparabolic equation. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7743
Padron, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation. Trans. Am. Math. Soc. 356, 2739–2756 (2004)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)
Ting, T.W.: Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal. 14, 1–26 (1963)
Tuan, N.H., Au, V.V., Xu, R.Z.: Semilinear Caputo time-fractional pseudo-parabolic equations. Commun. Pure Appl. Anal. 20, 583–621 (2021)
Vázquez, J.L.: Nonlinear diffusion with fractional Laplacian operators. Nonlinear Partial Differ. Equ. 7, 271–298 (2012). (Abel Symp. Springer, Heidelberg)
Vergara, V., Zacher, R.: Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods. SIAM J. Math. Anal. 47, 210–239 (2015)
Wang, X.C., Xu, R.Z.: Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. 10, 261–288 (2021)
Wang, F.L., Hu, D., Xiang, M.Q.: Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems. Adv. Nonlinear Anal. 10, 636–658 (2021)
Xiang, M., Rădulescu, V.D., Zhang, B.L.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. 84(1), 915–954 (2021)
Xiang, M., Hu, D., Zhang, B.L., Wang, Y.: Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth. J. Math. Anal. Appl. 501, 124269 (2021)
Xu, R.Z., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. 264, 2732–2763 (2013)
Xu, Y.Q., Tan, Z.: Blow-up of solutions for a time-space fractional evolution system. Acta Math. Sin. 29, 1067–1074 (2013)
Zacher, R.: Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients. J. Math. Anal. Appl. 348, 137–149 (2008)
Zacher, R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkc. Ekv. 52, 1–18 (2009)
Zhang, Q.G., Sun, H.R.: The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation, Topological Methods. Nonlinear Anal. 46, 69–92 (2015)
Acknowledgements
Mingqi Xiang was supported by the Natural Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program. The research of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, Project Number PCE 137/2021, within PNCDI III.
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Shen, R., Xiang, M. & Rădulescu, V.D. Time-Space Fractional Diffusion Problems: Existence, Decay Estimates and Blow-Up of Solutions. Milan J. Math. 90, 103–129 (2022). https://doi.org/10.1007/s00032-021-00348-5
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DOI: https://doi.org/10.1007/s00032-021-00348-5