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A strongly damped wave equation including the displacement dependingnonlinear damping term and nonlinear interaction function is considered. Themain aim of the note is to show that under the standard dissipativityrestrictions on the nonlinearities involved the initial boundary value problemfor the considered equation is globally well-posed in the class of sufficientlyregular solutions and the semigroup generated by the problem possesses a globalattractor in the corresponding phase space. These results are obtained for thenonlinearities of an arbitrary polynomial growth and without the assumptionthat the considered problem has a global Lyapunov function.

A note on a strongly damped wave equation with fast growing nonlinearities