© H. El Ouardi et al., published by EDP Sciences, 2017

1 Introduction

Let (N ≥ 1) be a bounded Lipshitz domain and 0 < T < ∞. It will be assumed throughout this paper that p(x) is continuous function defined in with logarithmic module of continuity:$$2<p^{-}=\underset{\bar{\mathrm{\Omega }}}{\inf } p\left(x\right)\le p\left(x\right)\le p^{+}=\underset{\bar{\mathrm{\Omega }}}{\sup }p\left(x\right)<\mathrm{\infty },$$ $$\left|p\left(x\right)-p\left(y\right)\right|\le -\frac{C}{\log \left|x-y\right|},\hspace{1em}\mathrm{for} \mathrm{any} x,y\in \mathrm{\Omega } \mathrm{with} \left|x-y\right|<\frac{1}{2}.$$(1)

We set Q _T  = Ω × (0, T) and Σ _T  = ∂Ω × (0, T). Our aim is to prove the existence and uniqueness of solutions u = (u ₁, u ₂) to the nonlinear (p ₁(x), p ₂(x))-Laplacian system:$$\{\begin{array}{cc}{a}_1\left(x\right)\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }t}-{\Delta }_{p_1\left(x\right)}{u}_1=f_1\left(x,u_1,u_2\right)& \mathrm{in} Q_T,\\ a_2\left(x\right)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }t}-{\Delta }_{p_2\left(x\right)}{u}_2=f_2\left(x,u_1,u_2\right)& \mathrm{in} Q_T,\\ u_1=u_2=0,& \mathrm{in} {\mathrm{\Sigma }}_T,\\ \left(u_1\left(.,0\right),u_2\left(.,0\right)\right)=\left({\phi }_1,{\phi }_2\right)& \mathrm{on} \mathrm{\Omega } .\end{array}$$(2)where is a function, (i = 1, 2).

The operator is called p(x)-Laplacian, which will be reduced to the p-Laplacian when p(x) = p a constant.

The (p ₁(x), p ₂(x))-Laplacian system (2) can be viewed as a generalization of (p, q)-Laplacian system$$\{\begin{array}{cc}{u}_t-{\Delta }_{p}u=f\left(x,u,v\right)& \mathrm{in} Q_T,\\ v_t-{\Delta }_{q}v=g\left(x,u,v\right)& \mathrm{in} Q_T,\\ u=v=0& \mathrm{in} {\mathrm{\Sigma }}_T,\\ \left(u\left(.,0\right),v\left(.,0\right)\right)=\left({\phi }_1,{\phi }_2\right)& \mathrm{on} \mathrm{\Omega } .\end{array}$$(3)

For the case p _i (x) = p _i  > 2, and a _i (x) = 1, (i = 1, 2), system (2) models as non-Newtonian fluids [2, 21] and nonlinear filtration [2], etc. In the non-Newtonian fluids theory, (p ₁, p ₂) is a characteristic quantity of the fluids, there have been many results about the existence, uniqueness of the solutions. We refer the readers to the bibliography given in [7, 9, 10, 11, 28, 31] and the references therein.

In recent years, the research of nonlinear problems with variable exponent growth conditions has been an interesting topic. p(·)-growth problems can be regarded as a kind of nonstandard growth problems and these problems possess very complicated nonlinearities, for instance, the p(x)-Laplacian operator is inhomogeneous. And these problems have many important applications in nonlinear elastic, electrorheological fluids and image restoration. The reader can find in [14, 22] several models in mathematical physics where this class of problem appears.

The case of a single equation of the type (2) has been studied in [4–6, 20] and the authors established the existence and uniqueness results, in [20], the authors use the difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p _i (x).

The more interesting question concerning parabolic systems of (p ₁(x), p ₂(x))-Laplacian type is to understand the asymptotic behavior of solutions when time goes to infinity. The study of the asymptotic behavior of the system is giving us relevant information about the structure of the phenomenon described in the model.

Concerning the elliptic systems with variable exponents, the results about existence and non-existence are proved in [26, 27, 29, 30].

Note that system (2) has a more complicated nonlinearity than the classical (p,q)-Laplacian system since it is nonhomogenous.

Recently, [24] study the equation the p(x)-Laplacian equation$$a\left(x\right)\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathrm{div}\left(u^{m-1}{\left|\mathrm{Du}\right|}^{\lambda -1}\mathrm{Du}\right),$$where λ > 0, m + λ − 2 > 0 and a(x) is a positive continuous function. They examine under which conditions on behavior of a(x) corresponding nonnegative solutions of the Cauchy problems possess the finite speed of propagations or interface blow-up phenomena.

In this paper, we consider the existence and uniqueness for the problem of the type (2) under some assumptions. The proof consists of two steps. First, we prove that the approximating problem admits a global solution; then we do some uniform estimates for these solutions. We mainly use skills of inequality estimation and the method of approximation solutions. By a standard limiting process, we obtain the existence to problem of the type (2).

The outline of this paper is the following: In Section 2, we introduce some basic Lebesgue and Sobolev spaces and state our main theorems. In Section 3, we give the existence and uniqueness of weak solutions. In Section 4, the blow-up results will be proved. The asymptotic behaviour of solution is established in Section 5.

2 Preliminaries

To consider problems with variable exponents, one needs the basic theory of spaces L ^p(x)(Ω) and W ^1p(x)(Ω). For the convenience of readers, let us review them briefly here. The details and more properties of variable-exponent Lebesgue-Sobolev spaces can be found in [16, 17].

Let When p ⁻ > 1, one can introduce the variable-exponent Lebesgue space$$L^{p\left(x\right)}\left(\mathrm{\Omega }\right)=\left\{u:\mathrm{\Omega }\to \mathbb{R};\hspace{1em}u \mathrm{is} \mathrm{measurable} \mathrm{and}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|u\right|}^{p\left(x\right)}\mathrm{d}x<\mathrm{\infty }\right\},$$endowed with the Luxemburg norm.$${\Vert w\Vert }_{p\left(x\right)}=\inf \left\{\lambda >0:{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\frac{w}{\lambda }\right|}^{p\left(x\right)}\mathrm{d}x\le 1\right\}.$$

The conjugate space is L ^q(x)(Ω), with

As in the case of a constant exponent, set$$W^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)=\left\{u\left(x\right)\in L^{p\left(x\right)}\left(\mathrm{\Omega }\right): {\left|\nabla u\right|}^{p\left(x\right)}\in L^1\left(\mathrm{\Omega }\right)\right\},$$endowed with the norm$${\Vert u\Vert }_{1,p\left(x\right)}={\Vert u\Vert }_{p\left(x\right)}+{\Vert \nabla u\Vert }_{p\left(x\right)}.$$

Similarly we also denote by the closure of in and is the dual of with respect to the inner product in

In Propositions 2.1–2.3, we describe some results about the Luxembourg norm.

Proposition 2.1 [16, 17]

1. The space is a separable, uniformly convex Banach space, and its conjugate space is where For any and we have the following Hölder-type inequality: $$\left|{\int }_{\mathrm{\Omega }}\mathrm{}\mathrm{uv}\mathrm{d}x\right|\le \left(\frac{1}{p^{-}}+\frac{1}{\left(q^{-}\right)}\right){\Vert u\Vert }_{p\left(x\right)}{\Vert v\Vert }_{q\left(x\right)}\le 2{\Vert u\Vert }_{p\left(x\right)}{\Vert v\Vert }_{q\left(x\right)}.$$

2. If for any the imbedding is continuous, the norm of the imbedding does not exceed

Proposition 2.2 [16]

If we denote$$\rho \left(w\right)=\underset{\mathrm{\Omega }}{\int }\mathrm{}{\left|w\right|}^{r\left(x\right)}\mathrm{d}x,\forall w\in L^{r\left(x\right)}\left(\mathrm{\Omega }\right),$$then

1. |w|_(r(x)) < 1(= 1; > 1) ⇔ ρ(w) < 1 (= 1; > 1);

2. 

3. |w|_(r(x)) → 0 ⇔ ρ(w) → 0; |w|_(r(x)) → ∞ ⇔ ρ(w) → ∞.

Proposition 2.3 [16]

For there exists a constant such that:$${\Vert u\Vert }_{p\left(x\right)}\le C{\Vert \nabla u\Vert }_{p\left(x\right)},$$

This implies that and are equivalent norms of

System (2) does not admit classical solutions in general. So, we introduce weak solutions in the following sence.

Definition 2.4 A function u = (u ₁ , u ₂) is said to be a weak solution of equation (2) , if the following conditions are satisfied:

1. such that:

2. For any $$\underset{0}{\overset{T}{\int }}\mathrm{}\underset{\mathrm{\Omega }}{\int }\mathrm{}\left(a_i\right(x)u_i{\varphi }_{\mathrm{it}}-{\left|\nabla u_i\right|}^{p_i\left(x\right)-2}\nabla u_i\nabla {\varphi }_i-f_i(x,u_1,u_2\left){\varphi }_i\right)\mathrm{d}x\mathrm{d}t=0$$

3. 

In the study of the global existence of solutions, we need the following hypotheses (H):

• (H1)

• (H2)

• (H3)

3 Main results

Remark 3.1 In this paper, we shall denote by c _i , C _i differents constants, depending on p _i (x), T, Ω, but not on n, which may vary from line to line. Sometimes we shall refer to a constant depending on specific parameters C _i (T), etc.

Our main existence result is the following:

Theorem 3.2 Let (H1)–(H3) hold. Then system (2) admits a unique solution u = (u ₁ ,u ₂ ) ∈ (C((0, T); L ²(Ω)))² . Moreover, the mapping (φ ₁, φ ₂) → (u ₁(t), u ₂(t)) is continuous in L ²(Ω) × L ²(Ω).

Proof of the main results.

3.1 Existence

We will semi-discrete (2) in time and solve the corresponding elliptic problem. Based on the semi-discrete problem, we construct the corresponding approximate solutions. The key procedure is to establish necessary a priori estimates for finding the limit of the approximate solutions via a compactness argument.

We first consider the discrete scheme (4)$$\begin{array}{cc}{a}_i\left(x\right)\frac{u_i^n-u_i^{n-1}}{\tau }-{\Delta }_{p_i\left(x\right)}{u}_i^n=f_i\left(x,u_1^{n-1},u_2^{n-1}\right)& \mathrm{in} \mathrm{\Omega },\\ u_i^n=0& \mathrm{on} \mathrm{\partial }\mathrm{\Omega },\\ u_i^0={\phi }_i& \mathrm{in} \mathrm{\Omega },\end{array}$$(4)where Nτ = T and T is a fixed positive real, and 1 ≤ n ≤ N.

Lemma 3.3 For any fixed n, if Problem (4) admits a weak solution

Proof. On the space we consider the functional$$\mathrm{\Phi }\left(v\right)={\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla v\right|}^{p_i\left(x\right)}\mathrm{d}x+\frac{1}{2\tau }{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left|v\right|}^2\mathrm{d}x-{\int }_{\mathrm{\Omega }}\mathrm{}\mathrm{gv}\mathrm{d}x.$$where is a known function. Using Young’s inequality and Proposition 2.1, there exist constants C ₁, C ₂ > 0, such that:$$\mathrm{\Phi }\left(v\right)\ge \frac{1}{p_i^{+}}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla v\right|}^{p_i\left(x\right)}\mathrm{d}x-C_2{\Vert g\Vert }_{L^2}^2\ge \frac{1}{p_i^{+}}{\Vert v\Vert }_{1,p_1\left(x\right)}^{p_1^{-}}-C_2{\Vert g\Vert }_{L^2}^2,$$hence Φ(v) → ∞, as Since the norm is lower semi-continuous and is continuous functional, Φ(v) is weakly lower semi-continuous on and satisfy the coercive condition. From [14] we conclude that there exists such that:$$\mathrm{\Phi }\left(v^{\mathrm{*}}\right)=\underset{v\in W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)}{\inf }\mathrm{\Phi }\left(v\right),$$and v* is the weak solutions of the Euler equation corresponding to Φ(v),$$a_i\left(x\right)\frac{v}{\tau }-{\Delta }_{p_i\left(x\right)}v=g.$$

Choosing we obtain a weak solution of (4).$$a_i\left(x\right)\frac{u_i^1-u_i^0}{\tau }-{\Delta }_{p_i\left(x\right)}{u}_i^1=f_i\left(x,u_1^0,u_2^0\right).$$(5)

Since , we may prove by induction that (4) has a solution in We put and for any integer k > 0, we may take as a test function in (5) to get$${\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{\tau }(w_i-\mathrm{M\tau }{)}_{+}^{k+1}\mathrm{d}x+k{\int }_{\mathrm{\Omega }}\mathrm{}\left|\nabla (w_i-\mathrm{M\tau }{)}_{+}^{p_i\left(x\right)}\right|(w_i-\mathrm{M\tau }{)}_{+}^{k-1}\mathrm{d}x={\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{\tau }(w_i-\mathrm{M\tau }{)}_{+}^k{u}_i^0\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}f(x,u_1^0,u_2^0\left)\right(w_i-\mathrm{M\tau }{)}_{+}^k\mathrm{d}x.$$

By using the Hölder inequality and , we have$${\int }_{\mathrm{\Omega }}\mathrm{}(w_i-\mathrm{M\tau }{)}_{+}^{k+1}\mathrm{d}x\le \left({\int }_{\mathrm{\Omega }}\mathrm{}(w_i-\mathrm{M\tau }{)}_{+}^{k+1}\left(u_i^0+\mathrm{M\tau }\right)\mathrm{d}x\right)\le {\left({\int }_{\mathrm{\Omega }}\mathrm{}(w_i-\mathrm{M\tau }{)}_{+}^{k+1}\mathrm{d}x\right)}^{\frac{k}{k+1}}{\left({\int }_{\mathrm{\Omega }}^{k+1}\mathrm{}{\left(u_i^0+\mathrm{M\tau }\right)}^{k+1}\mathrm{d}x\right)}^{\frac{1}{k+1}}.$$

We deduce

Letting k → ∞, we get Consider , we get easily that , i.e. and if we choose such that , we obtain

This completes the proof of lemma 3.3.

Now, we define the approximate solutions as set by: for all n ∈ {1, …, N}.$$\forall t\in \left[(n-1)\tau ,\mathrm{n\tau }\right] \{\begin{array}{l}{u}_{\mathrm{i\tau }}\left(t\right)=u_i^n, \\ \\ {\tilde{u}}_{\mathrm{i\tau }}\left(t\right)=\frac{\left(t-\left(n-1\right)\tau \right)}{\tau }\left(u_i^n-u_i^{n-1}\right)+u_i^{n-1},\end{array}$$are well defined and satisfied in addition$$a_i\left(x\right)\frac{\mathrm{\partial }{\tilde{u}}_{\mathrm{i\tau }}}{\mathrm{\partial }t}-{\Delta }_{p_{i\left(x\right)}}{u}_{\mathrm{i\tau }}=f_i\left(x,u_{1\tau }\left(.-\tau \right),u_{2\tau }\left(.-\tau \right)\right).$$(6)

We first establish some energy estimates of .

We need several lemmas to complete the proof of Theorem 3.2.

Lemma 3.4 There exists a positive constant C(T, u ₀) such that, for all n = 1, …, N $$u_i^n\in L^{\mathrm{\infty }}\left(0,T;L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\right),$$(7) $$u_{\mathrm{i\tau }},{\tilde{u}}_{\mathrm{i\tau }} \mathrm{are} \mathrm{bounded} \mathrm{in} L^{p_i\left(x\right)}\left(0,T;W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)\right)\cap L^{\mathrm{\infty }}\left(0,T;L^2\left(\mathrm{\Omega }\right)\right),$$(8) $$\frac{\mathrm{\partial }{\tilde{u}}_{\mathrm{i\tau }}}{\mathrm{\partial }t} \mathrm{is} \mathrm{bounded} \mathrm{in} L^2\left(Q_T\right),$$(9)and$$u_{\mathrm{i\tau }},{\tilde{u}}_{\mathrm{i\tau }} \mathrm{are} \mathrm{bounded} \mathrm{in} L^{\mathrm{\infty }}\left(0,T;W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)\right).$$(10)

Proof. (a) By lemma 3.3, for any n ∈ N, is bounded; whence (7)

(b) Multiplying (4) by , summing from n = 1 to N and integrating over Ω, we obtain$$\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\left(\frac{u_i^n-u_i^{n-1}}{\tau }\right)u_i^n\mathrm{d}x+\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_i^n\right|}^{p_i\left(x\right)}\mathrm{d}x=\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i\left(x,u_1^{n-1},u_2^{n-1}\right)u_i^n\mathrm{d}x.$$(11)

By using the Young inequality, for small, there exists such that$$\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i\left(x,u_1^{n-1},u_2^{n-1}\right)u_i^n\mathrm{d}x\le ϵ\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_i^n\right|}^{p_i\left(x\right)}\mathrm{d}x+C_{ϵ}\left(T\right).$$(12)

With the aid of the identity 2α(α − β) = α ² − β ² + (α − β)², we get$$\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\left(\frac{u_i^n-u_i^{n-1}}{\tau }\right)u_i^n\mathrm{d}x=\frac{1}{2}\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\left({\left|u_i^n\right|}^2-{\left|u_i^{n-1}\right|}^2+{\left|u_i^n-u_i^{n-1}\right|}^2\right)\mathrm{d}x=\frac{1}{2}\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\left({\left|u_i^n\right|}^2-{\left|u_i^{n-1}\right|}^2\right)\mathrm{d}x+\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left|u_i^N\right|}^2\mathrm{d}x-\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left|{\phi }_i\right|}^2\mathrm{d}x.$$

With the above estimates, we get (8).

(c) Multiplying the equation (4) by and summing from n = 1 to N, we get$$\tau \sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left(\frac{u_i^n-u_i^{n-1}}{\tau }\right)}^2\mathrm{d}x+\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_i^n\right|}^{p_i\left(x\right)-2}\nabla u_i^n.\nabla \left(u_i^n-u_i^{n-1}\right)\mathrm{d}x=\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i(x,u_1^{n-1},u_2^{n-1})\left(u_i^n-u_i^{n-1}\right)\mathrm{d}x.$$

By using the Young inequality, we get$$\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i\left(x,u_1^{n-1},u_2^{n-1}\right)\left(u_i^n-u_i^{n-1}\right)\mathrm{d}x\le C_{ϵ}\left(T\right)+\frac{\tau }{2}\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{\left(\frac{u_i^n-u_i^{n-1}}{\tau }\right)}^2\mathrm{d}x.$$(13)

From the convexity of the expression we get the following inequality:$${\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i^n\right|}^{p_i\left(x\right)}\mathrm{d}x-{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i^{n-1}\right|}^{p_i\left(x\right)}\mathrm{d}x\le {\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_i^n\right|}^{p_i\left(x\right)-2}\nabla u_i^n.\nabla \left(u_i^n-u_i^{n-1}\right)\mathrm{d}x,$$(14)

which imply with (12) and (13) that$$\frac{\tau }{2}\sum _{n=1}^N\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left(\frac{u_i^n-u_i^{n-1}}{\tau }\right)}^2\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i^N\right|}^{p_i\left(x\right)}\mathrm{d}x\le C_i.$$

By lemma 3.4, there exists M _i  > 0 independent of such that:$${\Vert u_{\mathrm{i\tau }}-{\tilde{u}}_{\mathrm{i\tau }}\Vert }_{L^{\mathrm{\infty }}\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}\le \underset{1\le n\le N}{\max }{\Vert u_i^n-u_i^{n-1}\Vert }_{L^2\left(\mathrm{\Omega }\right)}\le M_i\sqrt{\tau }.$$(15)

Therefore, taking and up to subsequence, we get that there exists such that , and as ,$$u_{\mathrm{i\tau }}\stackrel{\mathrm{*}}{\to }{u}_i\hspace{1em}\mathrm{in} L^{\mathrm{\infty }}\left(0,T;W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\right)\mathrm{and} {\tilde{u}}_{\mathrm{i\tau }}\stackrel{\mathrm{*}}{\to }{v}_i\hspace{1em}\mathrm{in} L^{\mathrm{\infty }}\left(0,T;W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\right),$$(16) $$\frac{\mathrm{\partial }{\tilde{u}}_{\mathrm{i\tau }}}{\mathrm{\partial }t}\to \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}\hspace{1em}\mathrm{in} L^2\left(Q_T\right).$$(17)

From (14), it follows that u _i  = v _i . From (15), we get that$$u_{\mathrm{i\tau }},{\tilde{u}}_{\mathrm{i\tau }}\to u_i\hspace{1em}\mathrm{in} L^{\mathrm{\infty }}\left(0,T;L^q\left(\mathrm{\Omega }\right)\right),\hspace{1em}\forall q>1.$$(18)

By Aubin-Simon’s compactness results [22], we have$${\tilde{u}}_{\mathrm{i\tau }}\to u_i \in C\left(0,T;L^2\left(\mathrm{\Omega }\right)\right).$$(19)

Now, multiplying (4) by and using (14) and (15), we get by straightforward calculations:$${\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\left(\frac{\mathrm{\partial }{\tilde{u}}_{\mathrm{i\tau }}}{\mathrm{\partial }t}-\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}\right)\left({\tilde{u}}_{\mathrm{i\tau }}-u_i\right)\mathrm{d}x\mathrm{d}t-{\int }_0^T\mathrm{}\langle {\Delta }_{p_i\left(x\right)}{u}_{\mathrm{i\tau }},u_{\mathrm{i\tau }}-u_i\rangle \mathrm{d}t={\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i\left(x,u_{1\tau }\left(.-\tau \right),u_{2\tau }\left(.-\tau \right)\right)\mathrm{d}x\mathrm{d}t+o_{\tau }\left(1\right),$$where as .

Thus, we get that$$\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left|{\tilde{u}}_{\mathrm{i\tau }}\left(T\right)-u_i\left(T\right)\right|}^2\mathrm{d}x-{\int }_0^T\mathrm{}\langle {\Delta }_{p_i\left(x\right)}{u}_{\mathrm{i\tau }}-{\Delta }_{p_i\left(x\right)}{u}_i,u_{\mathrm{i\tau }}-u_i\rangle \mathrm{d}t\le {\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_i\left(x,u_{1\tau }\left(.-\tau \right),u_{2\tau }\left(.-\tau \right)\right)\mathrm{d}x\mathrm{d}t+o_{\tau }\left(1\right),$$(20)and from (16) we have thus,$$u_{\mathrm{i\tau }}\to u_i\hspace{1em}\mathrm{in}\hspace{1em}{L}^{p_i\left(x\right)}\left(0,T;W_0^{1,p_i\left(x\right)}\left(\mathrm{\Omega }\right)\right), \hspace{1em}\mathrm{as} \tau \to 0^{+},$$and consequently by the same as that in [24]$${\Delta }_{p_i\left(x\right)}{u}_{\mathrm{i\tau }}\to {\Delta }_{p_i\left(x\right)}{u}_i\hspace{1em}\mathrm{in}\hspace{1em}{L}^{p_i(x{)}^{\mathrm{\prime }}}(0,T;W^{-1,p_i(x{)}^{\mathrm{\prime }}}(\mathrm{\Omega }\left)\right).$$

Therefore, u _i satisfies (3).

3.2 Uniqueness

Let (H1)–(H3) be satisfied. Then system (2) has a unique solution u = (u ₁, u ₂) in Q _T.

Proof. Let u = (u ₁, u ₂) and v = (v ₁, v ₂) be solutions of (2), we have:$${\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)\frac{\mathrm{\partial }\left(u_i-v_i\right)}{\mathrm{\partial }t}\left(u_i-v_i\right)\mathrm{d}x\mathrm{d}t-{\int }_0^T\mathrm{}\langle {\Delta }_{p_i\left(x\right)}{u}_i-{\Delta }_{p_1\left(x\right)}{v}_i,u_i-v_i\rangle \mathrm{d}t={\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}\left(f_i(x,u)-f(x,v)\right)\left(u_i-v_i\right)\mathrm{d}x\mathrm{d}t.$$

Since f _i (x,.,.) is locally Lipschitz uniformly in Ω, the difference w _i  = u _i  − v _i satisfies$$\frac{C}{2}\stackrel{2}{\sum \mathrm{}_{i=1}}{\left|w_i\right|}_{L^2\left(\mathrm{\Omega }\right)}^2+\stackrel{2}{\sum \mathrm{}_{i=1}}{\int }_0^T\mathrm{}\langle {\Delta }_{p_i\left(x\right)}{u}_i-{\Delta }_{p_i\left(x\right)}{v}_i,w_i\rangle \mathrm{d}t\le c\stackrel{2}{\sum \mathrm{}_{i=1}}{\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|w_i\right|}^2\mathrm{d}t,$$we observe that w → −Δ _p(x) w is monotone from to $$\stackrel{2}{\sum \mathrm{}_{i=1}}{\left|w_i\right|}^2\le 2c\stackrel{2}{\sum \mathrm{}_{i=1}}{\int }_0^T\mathrm{}{\left|w_i\right|}^2\mathrm{d}t.$$(21)

We finally deduce from Gronwall’s lemma,$$\stackrel{2}{\sum \mathrm{}_{i=1}}{\left|w_i\right|}^2\le \stackrel{2}{\sum \mathrm{}_{i=1}}{\left|w_i\left(0\right)\right|}^2\exp \left(2\mathrm{cT}\right),\hspace{1em}\forall t\in (0,T).$$

Thus, we deduce that u _i  = v _i .

Thus the solution is unique. The continuity of the the mapping can be obtained similarly.

Remark 3.5 From Theorem 3.2, the solution of system (2) generates a semigroup in L ²(Ω) × L ²(Ω).

Remark 3.6 If we assume that f _i (x, s ₁, s ₂) = g _i (x, s ₁, s ₂) − h _i (x, s _i ) where is Carathéodory mapping and that there exist positive constants L _j , c _j , m _j , C _j such that:

1. for any x ∈ Ω and

2. for any x ∈ Ω and where with and

By the same argument as that in [12, 32], One can show in the same way that the semigroup associated with system (2) admits an absorbing set in there is a bounded set such that, for any bounded set B in L ²(Ω) × L ²(Ω), there exists a T ₀ > 0 such that for any t ≥ T ₀. Where T ₀ depends only on B.

4 Blow-up of solutions

In this section, we shall investigate the blow-up properties of solutions to system (2) using energy methods. To this end, we consider the following hypotheses on the data.

(H4) such that:$${\int }_{\mathrm{\Omega }}\mathrm{}F\left({\phi }_1\right(x),{\phi }_2(x\left)\right)\mathrm{d}x-\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla {\phi }_i\right|}^{p_i\left(x\right)}\mathrm{d}x\ge 0.$$

(H5) and H is such that:$$\underset{i=1}{\sum \mathrm{}^2}{\left|u_i\right|}^{\alpha }\le \alpha H\left(u_1,u_2\right)\le \underset{i=1}{\sum \mathrm{}^2}{u}_i\frac{\mathrm{\partial }F}{\mathrm{\partial }{u}_i},\hspace{0.5em}\alpha >\max (p_2^{+},2).$$

Throughout this section, we define for t ≥ 0$$E\left(t\right)=\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i(x,t)\right|}^{p_i\left(x\right)}\mathrm{d}x-{\int }_{\mathrm{\Omega }}\mathrm{}H\left(u_1\right(x,t),u_2(x,t\left)\right)\mathrm{d}x$$

Theorem 4.1 Let (H1)–(H5) be satisfied, then the solutions of system (2) blow up in finite time, namely, there exists a T* < ∞ such that as t → T*.

Proof. We define .

Multiplying the first equation of (2) by , the second by , integrating by parts, we have$$E\mathrm{\prime }\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}\left\{\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i\right|}^{p_i\left(x\right)}\mathrm{d}x-{\int }_{\mathrm{\Omega }}\mathrm{}H\left(u_1\left(x,t\right),u_2\left(x,t\right)\right)\mathrm{d}x\right\}=-\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right){\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}\right)}^2\mathrm{d}x\le 0,$$(22)which implies that E(t) ≤ E(0).

Next define

Multiplying the first equation of (2) by u ₁, the second by u ₂, integrating by parts, we have$$g\mathrm{\prime }\left(t\right)=\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)u_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}\mathrm{d}x=-\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i\right|}^{p_i\left(x\right)}\mathrm{d}x+\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{u}_i\frac{\mathrm{\partial }H}{\mathrm{\partial }{u}_i}\mathrm{d}x=-\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{p}_i\left(x\right)\frac{1}{p_i\left(x\right)}{\left|\nabla u_i\right|}^{p_i\left(x\right)}\mathrm{d}x+\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{u}_i\frac{\mathrm{\partial }H}{\mathrm{\partial }{u}_i}\mathrm{d}x\ge -\underset{i=1}{\sum \mathrm{}^2}{p}_i^{+}{\int }_{\mathrm{\Omega }}\mathrm{}\frac{1}{p_i\left(x\right)}{\left|\nabla u_i\right|}^{p_i\left(x\right)}\mathrm{d}x+\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{u}_i\frac{\mathrm{\partial }H}{\mathrm{\partial }{u}_i}\mathrm{d}x\ge -p_2^{+}\left(E\left(t\right)+{\int }_{\mathrm{\Omega }}\mathrm{}H\left(u_1\left(x,t\right),u_2\left(x,t\right)\right)\mathrm{d}x\right)+\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{u}_i\frac{\mathrm{\partial }H}{\mathrm{\partial }{u}_i}\mathrm{d}x\ge \underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{u}_i\frac{\mathrm{\partial }H}{\mathrm{\partial }{u}_i}\mathrm{d}x-p_2^{+}{\int }_{\mathrm{\Omega }}\mathrm{}H\left(u_1\left(x,t\right),u_2\left(x,t\right)\right)\mathrm{d}x\ge \left(\frac{\alpha -p_2^{+}}{\alpha }\right)\underset{i=1}{\sum \mathrm{}^2}{\int }_{\mathrm{\Omega }}\mathrm{}{\left|u_i\right|}^{\alpha }\mathrm{d}x.$$(23)

By using Hölder’s inequality, we have$$\frac{1}{2} {\int }_{\mathrm{\Omega }}\mathrm{}{a}_i\left(x\right)u_i^2\mathrm{d}x\le c_0\left(\frac{1}{2}\right){\left|\mathrm{\Omega }\right|}^{\frac{\alpha -2}{2}}{\left({\int }_{\mathrm{\Omega }}\mathrm{}{\left|u_i\right|}^{\alpha }\mathrm{d}x\right)}^{\frac{2}{\alpha }},$$(24)where

By the formula we have by combining (23), (24)$$g\mathrm{\prime }\left(t\right)\ge k{g}^{\frac{\alpha }{2}}\left(t\right),$$where

A direct integration of the above inequality over (0, t) then yields$$g^{\frac{\alpha }{2}-1}\left(t\right)\ge \frac{1}{g^{1-\frac{\alpha }{2}}\left(0\right)-k\left(\frac{\alpha }{2}-1\right)t},$$which implies that g(t) bows up at a finite time and so does u _i .

5 Asymptotic behaviour

This section is devoted to the asymptotic behaviour of solutions. In order to prove the asymptotic behaviour, we assume

(H6)

Theorem 5.1 The weak solution u = (u ₁(t), u ₂(t)) obtained in Theorem 3.2, satifies: where C _i  > 0 (i = 1, 2, 3), or or .

Proof. Let u _i be solution of (2).

Multiplying the first equation in (2) by u ₁ and integrating over Q _T ,$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_1\left(x\right){\left|u_1\right|}^2\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_1\right|}^{p_1\left(x\right)}\mathrm{d}x={\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_1\left(x,u_1,u_2\right)u_1\mathrm{d}x.$$(25)

Multiplying the second equation in (2) by u ₂ and integrating over Q _T ,$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_2\left(x\right){\left|u_2\right|}^2\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_2\right|}^{p_2\left(x\right)}\mathrm{d}x={\int }_0^T\mathrm{}{\int }_{\mathrm{\Omega }}\mathrm{}{f}_2\left(x,u_1,u_2\right)u_2\mathrm{d}x.$$(26)

Summing up (25) and (26), we have from hypothesis (H5) that$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_1\left(x\right){\left|u_1\right|}^2\mathrm{d}x+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}\mathrm{}{a}_2\left(x\right){\left|u_2\right|}^2\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_1\right|}^{p_1\left(x\right)}\mathrm{d}x+{\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_2\right|}^{p_2\left(x\right)}\mathrm{d}x\le 0.$$(27)

By u _i using Poincaré’s inequality, we obtain$${\Vert u_i\Vert }_{L^2}^2\le c{\Vert \nabla u_i\Vert }_{L^2}^2\le c{\Vert \nabla u_1\Vert }_{p_i\left(x\right)}^2.$$(28)

If and by Proposition 2.2,$${\left|\nabla u_1\right|}_{p_1\left(x\right)}^{p_1^{-}}\le {\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_1\right|}^{p_1\left(x\right)}\mathrm{d}x \mathrm{and} {\left|\nabla u_2\right|}_{p_2\left(x\right)}^{p_2^{-}}\le {\int }_{\mathrm{\Omega }}\mathrm{}{\left|\nabla u_2\right|}^{p_2\left(x\right)}\mathrm{d}x.$$(29)