Long-term analysis of strongly damped nonlinear wave equations. (English) Zbl 1230.35019

Summary: We consider the strongly damped nonlinear wave equation \[ u_{tt} - \Delta u_t - \Delta u + f(u_t) + g(u) = h \] with Dirichlet boundary conditions, which serves as a model in the description of thermal evolution within the theory of type III heat conduction. In particular, the nonlinearity \(f\) acting on \(u_{t}\) is allowed to be nonmonotone and to exhibit a critical growth of polynomial order 5. The main focus is the long-term analysis of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space.


35B40 Asymptotic behavior of solutions to PDEs
35B33 Critical exponents in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35M10 PDEs of mixed type
35B41 Attractors
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