A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities
involved the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.
The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global a
ttractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.
We present a new method of investigating the so-called quasi-linear strongly damped wave equations $$ partial_t^2u-gammapartial_tDelta_x u-Delta_x u+f(u)= abla_xcdot phi( abla_x u)+g $$ in bounded 3D domains. This method allows us to establish the e
xistence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity $phi$ is less than 6 and $f$ may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case $phiequiv0$ which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction $|f(u)|leq C(1+ |u|^5)$.
