The author considers the strongly damped wave equation
Here is a smooth, bounded domain, and . The emphasis of the paper is on the growth order of the nonlinearity which may be 5. A functional setting is imposed on (1), i.e. one sets on and then one defines the phase space of (1) in terms of the fractional power spaces , i.e. one sets . The assumptions on the nonlinearity are as follows:
One assumes that there is a decomposition , such that
some and one also assumes as , where . Based on these assumptions, Theorem 1 asserts the existence of a unique global solution to (1), what entails the existence of a strongly continuous solution semigroup , , which has suitable Lipschitz properties by Theorem 2. Theorem 3 then asserts the existence of an absorbing set. Based on this fact, the existence of a universal attractor follows (Theorem 4). The author then considers a subcritical case which arises if is -independent and where the in (2) satisfy:
Under assumptions (4), further properties of the global attractor can be deduced. In addition, the existence of exponential attractors can be proved, as is shown in the last part of the paper.