Existence and upper semicontinuity of global attractors for \(p\)-Laplacian systems
The authors consider the system \[ \left \{\begin{aligned} u_{t} - \text{div}\; (D_1 |\nabla u|^{p_1(x)-2} \nabla u) &\in F(u,v)\;\text{in}\;(0,\infty) \times \Omega, \\ v_{t} - \text{div}\; (D_1 |\nabla v|^{p_2(x)-2} \nabla v) &\in G(u,v)\;\text{in}\;(0,\infty) \times \Omega,\\ u(0,x)=u_0(x),\;v(0,x)&=v_{0}(x)\; \text{in}\;\Omega, \end{aligned}\right.\tag{S} \] under the assumptions that \(\Omega \subset \mathbb{R}^{N}\) with \(N \geq 1\) is a bounded domain with smooth boundary, the functions \(p_{i}\) are continuous on \(\overline{\Omega}\) with \(\text{ess inf}\; p_i >2\), \(u_0, v_0\) belong to \(L^{2}(\Omega)\) and \(D_1, D_2\) belong to \(L^{\infty}(\Omega)\) and satisfy the condition \(0<\sigma \leq D_{i}(x) \leq M < \infty\) a.e. in \(\Omega\). \(F,G\) are assumed to be bounded, upper semicontinuous and positively sublinear multivalued operators from \(L^2(\Omega) \times L^2(\Omega)\rightarrow P(L^{2}(\Omega))\). In the paper it is proved that the \(p(x)\)-Laplace operator is subdifferential of a convex, proper and lower semicontinuous non-negative map. The paper extends the results of \textit{J. Simsen} and \textit{C. B. Gentile} [J. Math. Anal. Appl. 368, No. 2, 525--537 (2010; Zbl 1191.35163)] where the constant exponent \(p\) case was considered. The proof of existence of a global compact invariant attractor for the system (S) follows the steps of the proof in [Simsen and Gentile, loc. cit.], while circumventing the major difficulty of \(p=p(x)\): proving that the generalized semiflow associated to the system (S) is \(B\)-dissipative. The study of dependence on parameters is also considered.
scientific article; zbMATH DE number 6005669
scientific article