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

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 \( \Omega \subset {\mathbb {R}}^3: u_{tt}-\nabla \cdot (|\nabla u|^{p(x, t)-2} \nabla u)-\lambda \Delta u- \Delta u_t+ |u|^{q(x, t)-2}u=g\) $\mathrm{\Omega }\subset {\mathbb{R}}^3:u_{\mathit{tt}}-\mathrm{\nabla }·{(|\mathrm{\nabla }u|}^{p(x,t)-2}\mathrm{\nabla }u)-\lambda \mathrm{\Delta }u-\mathrm{\Delta }{u}_t+{|u|}^{q(x,t)-2}u=g$ , together with the perturbed parameter \(\lambda \in [0,1]\) $\lambda \in [0,1]$ and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each \(\lambda _0\in (0,1]\) ${\lambda }_0\in (0,1]$ , the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback \({\mathscr {D}}\) $\mathcal{D}$ -exponential attractor \({\mathcal {E}}_\lambda =\{E_\lambda (t)\}_{t\in {\mathbb {R}}}\in {\mathscr {D}}\) ${\mathcal{E}}_{\lambda }={\{E_{\lambda }(t)\}}_{t\in \mathbb{R}}\in \mathcal{D}$ which is Hölder continuous w.r.t. \(\lambda \) $\lambda$ at \(\lambda _0\) ${\lambda }_0$ ; (iii) they have also a family of finite dimensional pullback \({\mathscr {D}}\) $\mathcal{D}$ -attractors \({\mathcal {A}}_\lambda =\{A_\lambda (t)\}_{t\in {\mathbb {R}}}\) ${\mathcal{A}}_{\lambda }={\{A_{\lambda }(t)\}}_{t\in \mathbb{R}}$ which are upper semicontinuous and residual continuous w.r.t. \(\lambda \in (0,1]\) $\lambda \in (0,1]$ . In particular, when \(\lambda \in (0,1]\) $\lambda \in (0,1]$ 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 \(H^1\times H^1\) $H^1\times H^1$ -norm, the pullback \({\mathscr {D}}\) $\mathcal{D}$ -attractor and pullback \({\mathscr {D}}\) $\mathcal{D}$ -exponential attractor in weaker \({\mathcal {Y}}_1\) ${\mathcal{Y}}_1$ -norm can be regularized to be those in stronger \(H^1\times H^1\) $H^1\times H^1$ -norm, which are also the standard ones in \({\mathcal {H}}_t\) ${\mathcal{H}}_t$ -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.