Abstract
In this paper we study a class of nonlinear parabolic problems with \(p(x,t)\) growth conditions. We prove the existence and uniqueness of bounded solutions to such a problem, with less constraint to \(p(x,t)\). Our results are generalizations of the corresponding results in the constant exponent case.
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1 Introduction
Our main goal is to prove the existence and uniqueness of solutions to the following nonlinear parabolic problem:
where Ω is an open bounded subset of \(\mathbb{R}^{N}\) (\(N\ge1\)) with Lipschitz boundary ∂Ω, T is a positive constant, \(u_{0}\in L^{\infty}(\Omega )\), \(Q=\Omega\times(0, T)\) with the lateral boundary \(\partial\Omega\times(0, T)\).
Here, we make the following assumptions on a, ϕ, g, f, and F:
- (H1):
-
The function \(a:Q\times\mathbb{R}\times\mathbb {R}^{N}\) is a Carathéodory function and there exist a continuous function \(p:\bar{Q}\rightarrow(1,+\infty)\) and a positive constant α such that
$$a(x,t,s,\xi)\xi\geq\alpha|\xi|^{p(x,t)}, \quad\mbox{a.e. } (x,t)\in Q, \forall s\in\mathbb{R} \mbox{ and } \forall\xi\in\mathbb{R}^{N}. $$ - (H2):
-
There exist a continuous function b from \(\mathbb {R}^{+}\) into \(\mathbb{R}^{+}\) and a nonnegative function \(c\in L^{p'(x,t)}(Q)\) such that
$$\bigl|a(x,t,s,\xi)\bigr|\leq b\bigl(|s|\bigr) \bigl[|\xi|^{p(x,t)-1}+c(x,t) \bigr], \quad \mbox{a.e. } (x,t)\in Q, \forall s\in\mathbb{R} \mbox{ and } \forall\xi\in \mathbb{R}^{N}. $$ - (H3):
-
\((a(x,t,s,\xi)-a(x,t,s,\zeta))\cdot(\xi-\zeta )>0\) holds for almost every \((x,t)\in Q\) and for every \(\xi,\zeta\in\mathbb {R}^{N}\) with \(\xi\neq\zeta\).
- (H4):
-
\(g:Q\times\mathbb{R}\) is a Carathéodory function, satisfying \(\sup_{|s|\leq n} |g(\cdot,s)|=h_{n}(\cdot )\in L^{1}(Q)\) and
$$g(x,t,s)s\geq0, \quad\mbox{for a.e. } (x,t)\in Q \mbox{ and } \forall s\in \mathbb{R}. $$ - (H5):
-
The function ϕ is continuous on \(\mathbb{R}\) with values in \(\mathbb{R}^{N}\).
- (H6):
-
\(f\in L^{q(x,t)}(Q)\) and \(F\in(L^{q(x,t)(p^{-})'}(Q))^{N}\), where \(q^{-}>\max \{1+\frac{N}{p^{-}},2\}\).
As we have seen, problem (P) includes parabolic equation which is nonlinear with respect to the gradient of the solution, and with variable exponents of nonlinearity. Thus it is natural to solve problem (P) under the framework of Sobolev spaces with variable exponents. The problem we study here is closely related to the model of electro-rheological fluids (see [1–3]). For more applications, we refer the reader to [4–17].
In the case of p and q are two constants, the existence and regularity of the solutions to problem (P) have been intensively studied by many authors. We refer the reader to the bibliography [18] and references therein. Especially, it is well known that problem (P) have a weak solution belongs to \(L^{\infty}(Q)\), provided that \(q>\max \{1+\frac{N}{p},2\}\).
In the stationary case and \(p=p(x)\), there have been several results concerning the existence, uniqueness and regularity of entropy or renormalized solutions to such problems with \(q=1\) and \(F\equiv0\); see [7] and [8] for example. More precisely, in [7], it is assumed that \(p(x)\) belongs to \(W^{1,\infty}(\Omega )\); in [8], it is assumed that \(p(x)\in C(\bar{\Omega})\) satisfies the log-continuity condition. We also remark that the existence of bounded weak solutions to this type of problems have been studied in [9–11], assuming that \(p(x)\in C(\bar{\Omega})\).
Recently some papers appeared in the case of parabolic problems with non-standard growth. When \(p=p(x)\in C(\bar{\Omega})\) satisfies the log-continuity condition, the existence and uniqueness of an entropy solution to problem (P) without lower order terms were proved in [12], under the assumption \(f\in L^{1}(Q)\) and \(F\equiv0\). When \(p=p(x)\) only belongs to \(C(\bar{\Omega})\) with \(p^{-}>1\), M. Bendahmane et al. have also proved the existence and uniqueness of renormalized solutions, by the semigroup approach; see [13]. If \(p\in C(\bar{\Omega})\) and \(p^{-}>2\), the existence of weak solutions to problem (P) is proved in [14], for \(\phi =0\) and \(F\equiv0\).
When p is Lipschitz continuous with respect to the space variables and \(\frac{\beta}{2}\)-Hölder continuous with respect to time, Acerbi et al. [15] studied the regularity results for parabolic systems without lower order terms and \(f,F\equiv0\). As \(p=p(x,t)\in C(\bar{Q})\) satisfies the log-continuity condition and \(F\equiv0\), Antontsev and Shmarev [16] studied the existence of solutions of similar problems with anisotropic parabolic equation. Moreover, it is worth to mention that Alkhutov and Zhikov [17] obtained the existence results without any assumption on the regularity of the exponent, if the terms \(g,\phi,F\equiv0\).
The main idea of this paper relies on [13, 14, 16, 18, 19]. Using Galerkin’s approximation technique, we shall prove the existence and uniqueness of bounded solutions to problem (P) (Theorem 3.1 and Theorem 4.1), which generalizes the corresponding results in the constant exponents. In order to prove Theorem 3.1, a key result (Lemma 3.2) about an \(L^{\infty}\) estimate for solution to problem (P) is proved.
This paper is organized as follows: in Section 2 we recall some basic notations and properties of Sobolev spaces with variable exponents; in Section 3, we prove the existence of solutions to problem (P); in Section 4, we give the proof of uniqueness of solutions to problem (P).
2 Some preliminaries and notations
In what follows, we recall some definitions and basic properties of the generalized Lebesgue space \(L^{p(x)}(\Omega)\) and the generalized Sobolev spaces \(W^{1,p(x)}(\Omega)\) (see [20] and [21], etc.).
Set \(C_{+}(\bar{\Omega})=\{h\in C(\bar{\Omega}): \inf_{x\in \bar{\Omega}}h(x)>1\}\). For any \(h\in C_{+}(\bar{\Omega })\), we define
For any \(p\in C_{+}(\bar{\Omega})\), we define the variable exponent Lebesgue spaces \(L^{p(x)}(\Omega)\) to consist of all measurable functions such that the modular
is finite, endowed with the Luxemburg norm
Lemma 2.1
(1) The space \(L^{p(x)}(\Omega)\) is a separable and reflexive Banach space, and its dual space is isomorphic to \(L^{p'(\cdot)}(\Omega )\), where \(\frac{1}{p(x)}+\frac{1}{p'(x)}=1\). For any \(u\in L^{p(x)}(\Omega)\) and \(v\in L^{p'(\cdot)}(\Omega)\),
(2) If \(p_{1},p_{2}\in C_{+}(\bar{\Omega})\) with \(p_{1}(x)\leq p_{2}(x)\), for any \(x\in\Omega\), then there exists the continuous embedding \(L^{p_{2}(\cdot)}(\Omega)\hookrightarrow L^{p_{1}(\cdot )}(\Omega)\), whose norm does not exceed \(|\Omega|+1\).
(3) \(C_{0}^{\infty}(\Omega)\) is dense in \(L^{p(x)}(\Omega)\).
(4) For any \(u\in L^{p(x)}(\Omega)\), we have
(5) Let \(\{v_{n}\}\subseteq L^{p(x)}(\Omega)\) and \(v\in L^{p(x)}(\Omega )\), the following statements are equivalent:
-
(i)
\(\lim_{n\rightarrow\infty}\|v_{n}-v\| _{L^{p(x)}(\Omega)}=0\);
-
(ii)
\(\lim_{n\rightarrow\infty}\rho_{p}(v_{n}-v)=0\);
-
(iii)
\(v_{n}\) converges to v in measure and \(\lim_{n\rightarrow \infty}\rho_{p}(v_{n})=\rho_{p}(v)\).
Remark 2.1
Obviously, if p is a constant function, then the variable exponent Lebesgue space coincides with the usual Lebesgue space.
Set
where the norm is defined by
The space \(W^{1,p(x)}(\Omega)\) is called a generalized Sobolev space. By \(W_{0}^{1,p(x)}(\Omega)\) we denote the subspace of \(W^{1,p(x)}(\Omega)\) which is the closure of \(C_{0}^{\infty}(\Omega )\) with respect to the norm (2.2). We denote the dual space of \(W_{0}^{1,p(x)}(\Omega)\) by \((W_{0}^{1,p(x)}(\Omega))^{\star}\).
Lemma 2.2
The space \(W^{1,p(x)}(\Omega)\) and \(W_{0}^{1,p(x)}(\Omega)\) are reflexive Banach spaces. For any \(u\in W_{0}^{1,p(x)}(\Omega)\), the Poincaré inequality
holds true, where c is a constant depending on Ω, N, and p.
Lemma 2.3
(see [20])
Let \(p,d\in C_{+}(\bar{\Omega})\) with \(p^{+}< N\) and \(d(x)< p^{\ast}(x):=\frac {Np(x)}{N-p(x)}\) almost everywhere in Ω, then there is a continuous and compact imbedding \(W_{0}^{1,p(x)}(\Omega)\hookrightarrow \hookrightarrow L^{d(\cdot)}(\Omega)\), and
where \(\tilde{C}\) depends only on Ω, N, \(p^{+}\), and \(d^{+}\).
Remark 2.2
In general, the smooth functions are not dense in \(W^{1,p(x)}(\Omega)\) (see [21]). However, if the exponent \(p(x)\) is assumed to be log-Hölder continuous, i.e. there exists a positive constant C such that
then the smooth functions are dense in \(W^{1,p(x)}(\Omega)\) and \(W_{0}^{1,p(x)}(\Omega )=W^{1,p(x)}(\Omega)\cap W_{0}^{1,1}(\Omega)\) (see [20, 21]). Moreover, if \(p\in C_{+}(\bar{\Omega})\) satisfies (2.5) and \(p^{+}< N\) then the Sobolev embedding holds also for \(d(x)=p^{\ast}(x)\), i.e. \(W^{1,p(x)}(\Omega )\hookrightarrow L^{p^{\ast}(x)}(\Omega)\). As in [13, 23], we do not need these condition to prove our result and will most exclusively work with \(p\in C_{+}(\bar{\Omega})\). We also observe that \(W_{0}^{1,p(x)}(\Omega )\) is stable by composition with Lipschitz functions, even if for a function \(v\in W^{1,p(x)}(\Omega)\) having trace zero does not guarantee that \(v\in W_{0}^{1,p(x)}(\Omega )\). In other words, if \(L: \mathbb{R}\rightarrow \mathbb{R}\) is Lipschitz continuous such that \(L(0)=0\) and \(v\in W_{0}^{1,p(x)}(\Omega )\), then \(L(v)\in W_{0}^{1,p(x)}(\Omega )\). For more details, one can refer to [13, 23] for example.
Now, for any \(p\in C_{+}(\bar{Q})\), we define
We may also consider the generalized Lebesgue space
endowed with the norm
which obviously shares the same type of properties as \(L^{p(x)}(\Omega)\).
We will also use the standard notations for Bochner spaces, i.e., if \(q \geq1\) and X is a Banach space, then \(L^{q}(0, T ; X)\) denotes the space of strongly measurable functions \(u : (0, T )\rightarrow X\) for which \(t\mapsto\|u(t)\|_{X} \in L^{q}(0, T )\). Moreover, \(C([0, T ]; X)\) denotes the space of continuous functions \(u : [0, T]\rightarrow X\) endowed with the norm \(\|u\|_{C([0,T ];X)} := \max_{t\in[0,T ]} \|u(t)\|_{X}\).
For any given \(k>0\), the truncation function \(T_{k}\) is defined as follows:
We use \(C(\theta_{1},\theta_{2},\ldots,\theta_{m})\) to denote positive constants depending only on specified quantities \(\theta_{1}, \theta _{2},\ldots, \theta_{m}\). Throughout this paper, the notation \(X^{\star}\) denotes the dual space of a Banach space X.
3 Existence of weak solution to problem (P)
First of all, we shall give the definition of weak solution to problem (P). To do this, we need to introduce the following Banach space:
endowed with the norm
Remark 3.1
The space \(W(Q)\) is reflexive and separable. Moreover, there exists an equivalent norm of \(W(Q)\):
As in [13], we have the following result.
Lemma 3.1
(i) We have the following continuous dense embeddings:
In particular, \(\mathscr{D}(Q)\) is dense in \(W(Q)\) and
(ii) If \(T\in W^{\star}(Q)\), there exists \(f_{0}\in L^{p(x,t)}(Q)\), \(F=(f_{1},\ldots, f_{N})\in(L^{p(x,t)}(Q))^{N}\) such that \(T=f_{0}-\operatorname{div} F\), and
Furthermore, we have
(iii) If \(v\in W(Q)\) with \(v_{t}\in L^{1}(Q)+W^{\star}(Q)\), then we have \(v\in C([0,T]; L^{1}(\Omega ))\). Furthermore, if \(v\in W(Q)\cap L^{\infty}(Q)\) with \(v_{t}\in L^{1}(Q)+W^{\star}(Q)\), then \(v\in C([0,T]; L^{2}(\Omega))\).
(iv) If \(v\in W(Q)\cap L^{2}(0,T;L^{2}(\Omega ))\) with \(v_{t}\in W^{\star}(Q)\), then \(v\in C([0,T]; L^{2}(\Omega))\).
Proof of Lemma 3.1
The proofs of results (i) and (ii) are similar to [24], the proofs of (iii) and (iv) are similar to [25]. We omit the details here. □
Now we give the definition of weak solutions to problem (P).
Definition 3.1
A function \(u(x,t)\in W(Q)\) is called a weak solution of problem (P), if \(a(x,t,u, \nabla u)\in(L^{p'(x,t)}(Q))^{N}\), \(\phi(u)\in(L^{p'(x,t)}(Q))^{N}\), and \(g(x,t,u)\in L^{1}(Q)\) such that such that
with \(u|_{t=0}=u_{0}\).
Remark 3.2
Note that if u is a weak solution of problem (P), then \(u\in W(Q)\) and \(u_{t}\in L^{1}(Q)+W^{\star}(Q)\), so \(u\in C([0,T]; L^{1}(\Omega))\). Therefore the initial condition \(u|_{t=0}=u_{0}\) makes sense.
Remark 3.3
If u is a solution of problem (P), by Remark 3.2, the equality (3.4) reads
where \(\langle\cdot, \cdot\rangle\) denotes the duality pairing between \(W^{\star}(Q)+L^{1}(Q)\) and \(W(Q) \cap L^{\infty}(Q)\).
In order to find some estimates for weak solutions and also to get the uniqueness result, the following integration-by-parts-formula is needed (of which the proof will be given in the Appendix):
Lemma 3.2
Let \(\varphi: \mathbb{R}\mapsto \mathbb{R}\) be a continuous piecewise \(C^{1}\) function such that \(\varphi(0)=0\) and φ is zero outside a compact set of \(\mathbb{R}\). Let us denote \(\tilde{\varphi}(s)=\int_{0}^{s}\varphi (r)\,\mathrm{d}r\). If \(u\in W(Q)\) with \(\frac{\partial u}{\partial t}\in W^{\star}(Q)+L^{1}(Q)\) and if \(\psi\in C^{\infty}(\bar{Q})\), then we have, for any \(\tau\in(0,T]\),
From the proof of Lemma 3.2, it is easy to obtain the following conclusion.
Corollary 3.1
Let \(u\in W(Q)\) with \(\frac{\partial u}{\partial t}\in W^{\star}(Q)+L^{1}(Q)\). If \(\varphi: \mathbb{R}\mapsto \mathbb{R}\) is a continuous function such that \(\varphi(u)\in W(Q)\) and \(\tilde{\varphi}(u)\in C([0,T];L^{1}(\Omega ))\), then (3.6) holds true.
Theorem 3.1
Let \(p\in C_{+}(\bar{Q})\), assume that (H1)-(H6) hold, then problem (P) admits at least a weak solution \(u\in L^{\infty}(Q)\cap C(0,T;L^{2}(\Omega ))\).
Before giving the proof Theorem 3.1, we need an \(L^{\infty}\) estimate which is stated as follows.
Lemma 3.3
Let \(u\in L^{\infty}(Q)\cap C([0,T];L^{2}(\Omega ))\) be a weak solution to problem (P) and suppose that the assumptions of Theorem 3.1 hold true, then there exists a positive constant M such that
where M is a positive constant only depending on \(p^{-}\), \(p^{+}\), α, N, Ω, \(\|f\|_{L^{q^{-}}(Q)}\), \(\|u_{0}\|_{L^{\infty}(\Omega)}\) and \(\||F|^{\frac{p^{-}}{p^{-}-1}}\|_{L^{q^{-}}(Q)}\).
Remark 3.4
It is well known that if \(p>1\) is a constant function, then one may obtain an \(L^{\infty}\) estimates for u provided that \(q>\max \{1+\frac{N}{p},2\}\). The above result is a generalization of the corresponding result in the constant exponent case.
To prove Lemma 3.3, we need the following result, which can be viewed as a generalization of Lemma 4.1 in [18].
Lemma 3.4
Let \(Y_{n}\), \(n=0,1,2,\ldots\) , be a sequence of positive numbers, satisfying the inequalities
where j is a positive integer, \(c,b>1\) and \(\beta_{i}\) are given positive numbers with
Assuming that
then \(\lim_{n\rightarrow\infty}Y_{n}=0\).
Proof of Lemma 3.4
The proof is by induction as in [18]. However, the details of the proof are omitted. In order to be complete and self-contained, let us briefly explain the argument.
In view of (3.9), we get \(0< Y_{0}<1\). Thus, by (3.8) and (3.9), we get
where \(\mu=\beta-1\).
Obviously \(0< Y_{1}<1\), thus, using (3.8) again, we have
By induction, we easily find that
Letting n tend to infinity in the above inequality, we obtain the desired result immediately. □
Proof of Lemma 3.3
Set \(\bar{M}=\|u\|_{L^{\infty}(Q)}\). Without loss of generality, we may assume \(\bar{M}>2\|u_{0}\|_{L^{\infty}(\Omega)}\). For every k with \(\max \{\|u_{0}\|_{L^{\infty}(\Omega )},\bar {M}-1\}\leq k \leq\bar{M}\) and any given \(\tau\in(0, T )\), taking \(\eta(x,t)=\operatorname{sign}u(|u|-k)_{+}\chi_{(0,\tau)}(t)\) as a test function in problem (P), we obtain
For the first term of (3.10), by Lemma 3.2 (or Corollary 3.1) we have
Set
Concerning the second term of (3.10), we estimate as follows:
Since \(1< p^{-}\leq p(x,t)\), applying Hölder’s inequality we have
The above inequality and Young’s inequality show that
By (H5), we may assume that \(\phi=(\phi_{1},\phi_{2},\ldots ,\phi_{N})\), where \(\phi_{i}\in C(\mathbb{R})\) for \(1\leq i\leq N\). Let \(\tilde{\phi}_{i}(\theta)=\int_{0}^{\theta}\chi_{\{|\eta |\geq k\}}\phi_{i}(\eta)\,\mathrm{d}\eta\) and set \(\tilde{\phi }=(\tilde{\phi}_{1},\tilde{\phi}_{2},\ldots,\tilde{\phi}_{N})\), then it is easy to see that, after using the divergence theorem,
where \(\vec{n}\) is the outward pointing unit normal field of the boundary ∂Ω.
Recalling that \(\bar{M}-1 \leq k \leq\bar{M}\), it is straightforward that
For the second term of the right hand side of (3.10), we have
It follows from (3.10)-(3.16) that
Taking the supremum for \(\tau\in[0,T]\), we obtain
Now considering the sequence
replacing k by \(k_{n}\) in (3.17) and using Lemma 3.1 of [18], we find that
where \(C_{0}:=C(\alpha,p^{-},N,\|f\|_{L^{q^{-}}(Q)},\||F|^{\frac {p^{-}}{p^{-}-1}}\|_{L^{q^{-}}(Q)})\) and
The equality (3.18) is equivalent to
where \(b=2^{\frac{p^{-}(N+2)}{N}}\).
It follows from Lemma 3.4 that
provided
here \(\beta=\min_{1\leq i\leq4}\{\beta_{i}\}\).
In view of (3.20), we arrive that
which contradicts the definition of \(\bar{M}\).
Since
i.e.
we have a contradiction of (3.21) if \((\frac{2}{\bar {M}})^{p^{-}}\int_{0}^{T}\int_{\Omega}|u|^{p^{-}}\,\mathrm{d}x\,\mathrm{d}t\leq c^{\ast}\). Hence, we conclude that
Now taking \(\eta(x,t)=u\) as a test function in problem (P) and arguing as in (3.13), we find that
where we have used similar results to (3.14) and (3.16).
Taking the supremum for \(\tau\in[0,T]\), we have
where we have used the Poincaré inequality, (3.22), and (3.23).
Applying Young’s inequality and the Poincaré inequality in (3.24), we obtain
From (3.22) and (3.25), we obtain the desired result of Lemma 3.3. □
To prove Theorem 3.1, we have to consider approximating problems. We define a truncation \(\bar{a}\) of a by
where M is defined as in Lemma 3.3.
Similarly, the truncation \(\bar{\phi}\) of ϕ is defined as
For each \(\varepsilon>0\), we define
and the sequences \(\{f_{\varepsilon }\}\subseteq C_{c}^{\infty}(\Omega )\) and \(\{F_{\varepsilon }\}\subseteq C_{c}^{\infty}(\Omega )\) such that
Obviously, the function \(\bar{a}\) satisfies (H1), (H3) with a replaced by \(\bar{a}\). Moreover, due to (H2), there exist \(\bar{c}\in L^{p'(\cdot)}(\Omega)\) and a constant \(b_{M}>0\) such that
Now, we introduce a family of approximate problems:
In the following, we prove the existence of weak solutions of problem (\(P_{\varepsilon}\)). We will solve problem (\(P_{\varepsilon}\)) by Galerkin’s method.
For every fixed \(t\in[0, T]\), we introduce the Banach space
endowed with the norm
It is easy to see that \(V_{t}(\Omega)\) is reflexive and separable as a closed subspace of \(W_{0}^{1,p^{-}}(\Omega )\cap L^{2}(\Omega )\). Hence there exists a countable set of linearly independent functions \(\{\varphi_{i}\}_{i=1}^{\infty}\subseteq C_{0}^{\infty }(\Omega)\) consists a basis of \(V_{t}(\Omega)\). Without loss of generality, we may assume that \(\{\varphi_{i}\}_{i=1}^{\infty}\) also forms an orthonormal basis of \(L^{2}(\Omega)\). Fix now a positive integer m and let
One can check that, for any given \(v\in W(Q)\cap L^{2}(Q)\), there is a sequence \(v_{m}^{i}(t)\in C^{1}[0,T]\) such that
Now we consider the following approximate problem: find
where the coefficients \(u^{i}_{m}(t)\) satisfies
for \(i=1,2,\ldots,m\) and \(0\leq t\leq T\), where
The existence result of problem (3.28) and (3.29) is stated as follows.
Lemma 3.5
Fixed \(\varepsilon>0\), for each positive integer \(m=1,2,\ldots\) , there exists a function \(u_{m}\) of the form (3.28) satisfying (3.29).
Proof of Lemma 3.5
In order to prove our results, we introduce the following notations. For any element of \(v\in V\), we denote by
associated with the projection \(v^{m}\) of v on \(V_{m}\), i.e. \(v^{i}\in\mathbb{R}\) such that
Let \(G^{m}\) be the mapping from \(\mathbb{R}^{m}\) into itself whose ith component is
respectively.
Also we define \(F^{m}(t)\) to be the vector of \(\mathbb{R}^{m}\) whose ith component is
With the above notations, the problem (3.28) and (3.29) can be written as follows:
It is easy to check that \(G^{m}\) and \(F^{m}(t)\) are continuous. Hence, the ordinary system (3.30) has a local \(C^{1}\) solution \(\vec{u}_{m}(t)\) on some interval \([0,t_{m}]\), where \(t_{m}\) is a positive number.
Now we prove that \(t_{m}=T\). We still need some a priori estimates for the sequence of \(\{u_{m}\}\).
Multiplication of the first equality of (3.30) by \(\vec {u}_{m}(t)\) and integration over \((0,t)\), we obtain
Note that \(\int_{0}^{t}\int_{\Omega}\bar{\phi}(u_{m})\nabla u_{m} \,\mathrm{d}x\,\mathrm{d}t=0\), by Young’s inequality, assumption (H1) and (H4) we get
which implies that \(\vec{u}_{m}(t)\) remains bounded as t tends to \(T_{m}\), where C is a positive constant independent of m.
Since \(\vec{u}_{m}(t)\) does not blow up whenever t tends to \(t_{m}\), the system (3.30) admits a global solution on \([0,T]\). Thus, we have finished the proof. □
Proof of Theorem 3.1
The proof is divided into three steps.
Step 1: we prove the existence of solutions to problem (\(P_{\varepsilon }\)).
In view of (3.32) and Lemma 2.1, we infer that the solution \(u_{m}\) obtained in Lemma 3.5 is bounded in \(W(Q)\cap L^{\infty}(0,T;L^{2}(\Omega ))\) with respect to m. Hence, there exists a subsequence of \(\{u_{m}\}\) (still denoted by \(\{ u_{m}\}\)) such that as \(m\rightarrow\infty\),
and
Moreover, we have
By the above results, Lemma 3.1, it is easy to see that
Then using an Aubin’s type lemma (see Corollary 4 of [26]), we conclude that \(\{u_{m}\}\) contains a subsequence strongly convergent in \(L^{\tilde{s}}(Q)\), where \(\tilde{s}=\min \{ p^{-},s\}\). Thus, we can also draw a subsequence of \(\{u_{m}\}\) (still denoted by \(\{u_{m}\} \)) such that
Since \(u_{m}\) satisfies (3.28) and (3.29), it is easy to see that for all \(\varphi\in C^{1}([0,T];V_{m})\) and \(\tau\in(0,T]\),
Since, for any given \(\varphi\in C^{1}(0,T;C^{\infty}_{0}(\Omega ))\), there exists a sequence \(\varphi_{m}\in C^{1}([0,T];V_{m})\) such that \(\varphi _{m}\rightarrow\varphi\) in \(C^{1}(0,T;C^{\infty}_{0}(\Omega ))\), we have
As a consequence, it follows from (3.33)-(3.36) that
To identity the term \(\zeta_{\varepsilon }\), we shall prove the following result:
For this purpose, we need to choose an appropriate test function φ in (3.38). We will use the regularization method of Landes [27]. We define the regularization in time of the function \(u_{\varepsilon }\) by
where \(\bar{u}_{\varepsilon }(x,\theta)=u_{\varepsilon }(x,\theta)\) if \(\theta>0\); \(\bar{u}_{\varepsilon }(x,\theta)=0\) if \(\theta\leq0\).
As in [27], the function \((u_{\varepsilon})_{\nu}\in W^{1,p^{-}}(0,T;W_{0}^{1,p(x,t)}(\Omega ))\cap W(Q)\cap L^{2}(Q)\) satisfies
and
In order to deal with a nonzero initial datum \(u_{0}\), we now define
where \(\{u_{0j}\}\subseteq C_{0}^{\infty}(\Omega )\) such that \(u_{0j}\rightarrow u_{0}\) strongly in \(L^{\sigma}(\Omega )\) (for any \(\sigma \geq1\)) and weakly∗ in \(L^{\infty}(\Omega )\).
Obviously, this function satisfies the following problems:
Moreover, the \((u_{\varepsilon})_{\nu,j} \in W^{1,p^{-}}(0,T;W_{0}^{1,p(x,t)}(\Omega ))\cap W(Q)\cap L^{2}(Q)\) enjoys the property
Choosing \(\varphi=(u_{\varepsilon})_{\nu,j}\) as a test function in (3.38) and taking \(\varphi=u_{m}\) in (3.37), we get
where
In the following, we pass to the limit in (3.41) as \(m\rightarrow\infty\), \(\nu\rightarrow\infty\), and then \(j\rightarrow\infty\).
The limit of \(I_{1}(m,\nu,j)\): we rewrite \(I_{1}(m,\nu,j)\) as follows:
For \(I_{11}\), we have
which yields
Using the properties of \((u_{\varepsilon})_{\nu,j}\) and (3.36), we get the following estimate for \(I_{12}\):
Substituting (3.43) and (3.44) into (3.42),
The limit of \(I_{2}(m,\nu,j)\) , \(I_{3}(m,\nu,j)\) , and \(I_{4}(m,\nu,j)\): by (3.33), (3.34), and (3.36), it is easy to see that
As a consequence of (3.45)-(3.48), we find that
i.e. (3.39) holds true.
Step 2: In this step, we identify the quantities \(\zeta_{\varepsilon }\), and prove that \(u_{\varepsilon}\) is a weak solution of problem (\(P_{\varepsilon}\)).
Equation (3.39) implies that, as m tends to infinity,
As a consequence of (3.49) and (H3), we have, for any \(0<\tau<T\),
At the possible expense of extending the functions of \(u_{m}\), \(a(x,t,s,\xi)\), \(g(x,t,s)\), f, and F on a time interval \((0,\bar {T})\) with \(\bar{T}>T\), in such a way such that all the assumptions (H1)-(H4) hold true and \(u_{m}\) is still a solution of problem (3.28) and (3.29) with T in place of \(\bar{T}\), we conclude that the previous convergence result (3.50) holds true in \(L^{1}(Q)\), i.e.
Using (3.33), (3.34), (3.36), (3.51), and arguing as in [28], we see that as \(m\rightarrow\infty\),
By (3.52), using the Vitali convergence theorem, we get, as \(m\rightarrow\infty\),
Moreover, it is easy to see that
and
It follows from (3.53)-(3.55) and (3.37) that
Applying now the Aubin type lemma, by the fact that the sequence \(\{ u_{m}\}\) is bounded in \(L^{\infty}(0,T;L^{2}(\Omega ))\), we get for \(s>\max \{\frac {2N}{N+2},p^{+}\}\)
which implies that \(u_{\varepsilon }|_{t=0}=u_{0}\).
Recalling that \(u_{\varepsilon }\in W(Q)\) and using the result (iv) of Lemma 3.1, we deduce that \(u_{\varepsilon }\in C([0,T];L^{2}(\Omega ))\). Combining this fact with the above convergence results and (3.38), we see that \(u_{\varepsilon }\) is a weak solution of problem (\(P_{\varepsilon}\)).
In the following, we prove that \(u_{\varepsilon }\) belongs to \(L^{\infty}(Q)\). Let k be chosen so that \(k\geq\|u_{0}\|_{L^{\infty}(\Omega )}\), and take \(\eta _{\varepsilon }(x,t)=\operatorname{sign}u_{\varepsilon }(|u_{\varepsilon }|-k)_{+}\chi_{(0,\tau)}(t)\) as a test function in problem (\(P_{\varepsilon }\)), we obtain
Firstly, by Young’s inequality and the Poincaré inequality, we get for any \(\delta>0\)
where \(A_{k\varepsilon }(t)=\{x\in\Omega:|u(x,t)|>k\}\) and \(\psi_{\varepsilon }(k)=\int_{0}^{T} \operatorname{meas}A_{k\varepsilon }(t)\,\mathrm{d}t\).
Secondly, it is easy to see that estimates (3.11)-(3.14) still hold with \(u_{\varepsilon }\) instead of u, \(A_{k\varepsilon }(t)\) instead of \(A_{k}(t)\), and \(\psi_{\varepsilon }(k)\) instead of \(\psi(k)\). Hence, taking δ small enough in (3.59) and then applying all these results in (3.58), we get
where \(C_{\varepsilon }=C(\alpha,p^{-},\Omega ,N,\|f_{\varepsilon }+\operatorname{div}F_{\varepsilon }\| _{L^{\infty}(Q)})\). Therefore, for \(l\geq k\), by Proposition 3.1 of [18] we have
where \(C_{\varepsilon 1}\) is a positive constant. Taking \(k=\|u_{0}\|_{L^{\infty}(\Omega )}\), we obtain
Then it follows that there exists a constant \(\sigma_{\varepsilon }>1\) such that
Let us consider the sequence \(k_{n}=M_{\varepsilon }(2-2^{-n})\), where \(M_{\varepsilon }= \max \{\sigma_{\varepsilon }+\|u_{0}\|_{L^{\infty}(\Omega )},C_{\varepsilon 1}\}\). Replacing l, k by \(k_{n+1}\), \(k_{n}\) in (3.60), respectively, it then follows that
which implies that
Obviously, (3.61) holds for \(l=k_{0}\). Hence, by Lemma 3.4 and (3.62), we have \(u_{\varepsilon }\in L^{\infty}(Q)\) such that \(\|u_{\varepsilon }\| _{L^{\infty}(Q)}\leq M_{\varepsilon }\).
Since \(u_{\varepsilon }\in L^{\infty}(Q)\cap C([0,T];L^{2}(\Omega ))\) is a weak solution of (\(P_{\varepsilon}\)), using the same argument of Lemma 3.3 we get
where M is defined as before.
Step 3: In view of (3.63), we have
Choosing \(u_{\varepsilon }\) as a test function in (\(P_{\varepsilon}\)), it follows from (H1)-(H6) that
where C is a positive constant independent of ε.
Hence, arguing as before, up to subsequences (still denoted by \(\{u_{\varepsilon }\}\)), we infer that
and
Moreover, we have
which implies that
Then the same argument of (3.36) shows that for subsequences of \(\{u_{\varepsilon }\}\) (still denoted by \(\{u_{\varepsilon }\}\)),
Proceeding as in the proof of (3.53), we get
Obviously, we also obtain \(u\in W(Q)\cap L^{\infty}(Q)\) and \(\frac {\partial u}{\partial t}\in W^{\star}(Q)+L^{1}(Q)\), thus \(u\in C([0,T];L^{2}(\Omega ))\). Furthermore, we get \(u|_{t=0}=u_{0}\). Thus, let \(\varepsilon \rightarrow0\) in (\(P_{\varepsilon }\)), with the help of (3.63)- (3.68) and assumptions (H1)-(H6), we deduce that u is a solution to problem (P). □
4 Uniqueness of weak solutions to problem (P)
In order to get the uniqueness result, we need the following assumptions:
- (H7):
-
The function ϕ is locally Lipschitz continuous.
- (H8):
-
For every \(k>0\), there exist \(\bar{c}_{k}\in L^{p'(x,t)}(Q)\) and a constant \(\beta_{k}>0\) such that
$$\begin{aligned} &\bigl|a(x,t,s_{1},\xi)-a(x,t,s_{2},\xi)\bigr| \\ &\quad\leq|s_{1}-s_{2}| \bigl[\beta_{k}|\xi|^{p(x,t)-1}+\bar{c}_{k}(x,t) \bigr], \quad\mbox{a.e. } (x,t)\in Q, \end{aligned}$$(4.1)for every \(\forall\xi\in\mathbb{R}^{N}\) and every \(|s_{1}|\leq k\) and \(|s_{2}|\leq k\).
- (H9):
-
\(g:\Omega\times[0,T]\times\mathbb{R}\) is monotone with respect to the third variable.
Theorem 4.1
Assume \(p\in C_{+}(\bar{\Omega})\) and the conditions (H1)-(H9) hold, then problem (P) admits a unique weak solution \(u(x,t)\in W \cap C([0,T];L^{2}(\Omega ))\cap L^{\infty}(Q)\).
Proof of Theorem 4.1
The existence result is proved by Theorem 3.1. In the following, we prove the uniqueness result. Assume that \(u,v\in W \cap C([0,T];L^{2}(\Omega ))\cap L^{\infty}(Q)\) are two weak solutions of (P), then taking \(\eta=\frac{1}{\varepsilon }T_{\varepsilon }(u-v)\chi_{(0,\tau)(t)}\), the following equality holds:
Subtracting equality (4.3) from (4.2) and using Lemma 3.2, we obtain
where \(\tilde{T}_{\varepsilon }(r)=\int_{0}^{r}T_{\varepsilon }(s)\,\mathrm{d}s\).
Denote the four terms on the left hand side by \(L_{1}(\varepsilon )\), \(L_{2}(\varepsilon )\), \(L_{3}(\varepsilon )\), and \(L_{4}(\varepsilon )\), respectively.
Concerning the first term \(L_{1}(\varepsilon )\), we have
For the second term \(L_{2}(\varepsilon )\), in view of (H3) we get
By (H8), we obtain
where M is defined as Lemma 3.3.
Note that
using Lebesgue’s dominated convergence theorem, we find that
Combining (4.6) with (4.8), we obtain
Similarly to the proof of (4.8), using (H7) we have
Condition (H9) implies that
Let \(\varepsilon \rightarrow0\) in (4.4), it follows from (4.5) and (4.9)-(4.11) that
Hence, we have \(u=v\) a.e. in Q. □
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Acknowledgements
The authors wish to thank the referees for careful reading of the early version of this manuscript and providing valuable corrections, suggestions and comments. This work is supported by the NSFC of China (no. 11461048), the Natural Science Foundation of JiangXi Province of China (no. 20132BAB211006), the foundation of Jiangxi Educational Committee (no. GJJ14546), and NSFC of China (no. 11126338).
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Appendix
Appendix
Proof of Lemma 3.2
The proof is similar to the proof of Lemma 7.1 in [29], but we use another approximation here. For the sake of clarity and readability, we give the details below.
First of all, by Remark 3.2, we note \(u\in C([0,T]; L^{1}(\Omega))\). We take the Steklov average of the function u by
Appropriately extending the functions u outside \((0, T )\), we still get \(u_{h}\in W(Q)\) with \(\frac{\partial u}{\partial t}\in W(Q)\) and convergence, as \(h\rightarrow0\) strongly to u in \(W(Q)\) and a.e. in Q. By the properties of φ and Remark 3.2, we also get \(\varphi(u_{h})\in W(Q)\cap C([0,T]; L^{1}(\Omega))\) and \(\varphi (u_{h})\rightarrow\varphi(u)\) strongly in \(W(Q)\). Hence we have
which yields
Thus the assertion of the lemma follows. □
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Zou, W., Li, J. Existence and uniqueness of bounded weak solutions for some nonlinear parabolic problems. Bound Value Probl 2015, 69 (2015). https://doi.org/10.1186/s13661-015-0332-6
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DOI: https://doi.org/10.1186/s13661-015-0332-6