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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.

MSC:

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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