Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities

Abstract

The paper investigates the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional p(x, t)-Laplacian and q(x, t)-growth source term on a bounded domain \( \Omega \subset {\mathbb {R}}^3: u_{tt}-\nabla \cdot (|\nabla u|^{p(x, t)-2} \nabla u)-\lambda \Delta u- \Delta u_t+ |u|^{q(x, t)-2}u=g\), together with the perturbed parameter \(\lambda \in [0,1]\) and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each \(\lambda _0\in (0,1]\), the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback \({\mathscr {D}}\)-exponential attractor \({\mathcal {E}}_\lambda =\{E_\lambda (t)\}_{t\in {\mathbb {R}}}\in {\mathscr {D}}\) which is Hölder continuous w.r.t. \(\lambda \) at \(\lambda _0\); (iii) they have also a family of finite dimensional pullback \({\mathscr {D}}\)-attractors \({\mathcal {A}}_\lambda =\{A_\lambda (t)\}_{t\in {\mathbb {R}}}\) which are upper semicontinuous and residual continuous w.r.t. \(\lambda \in (0,1]\). In particular, when \(\lambda \in (0,1]\) and without the p(x, t)-Laplacian, the above mentioned results can be greatly improved, in the concrete; (iv) the weak solutions of the corresponding model possess additionally partial regularity and the Hölder stability in stronger \(H^1\times H^1\)-norm, the pullback \({\mathscr {D}}\)-attractor and pullback \({\mathscr {D}}\)-exponential attractor in weaker \({\mathcal {Y}}_1\)-norm can be regularized to be those in stronger \(H^1\times H^1\)-norm, which are also the standard ones in \({\mathcal {H}}_t\)-norm. The method provided here allows overcoming the difficulties arising from variable exponent nonlinearities and extending the analysis and the results for these type of models with constant exponent nonlinearities.