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Damped Wave Equations of the p-Laplacian Type with Supercritical Sources
This dissertation focuses on a strongly damped wave equation of the p-Laplacian type on a bounded domain in dimension three under a generalized Robin boundary condition and in the presence of nonlinear source feedback terms on the boundary and in the interior. The strong damping term is often referred to as Kelvin-Voigt damping, emphasizing its role in describing viscoelastic materials. The source terms are allowed to be of supercritical order, in that their associated Nemytskii operators need not be even locally Lipschitz continuous. With suitable assumptions on the parameters a rigorous proof of the existence of local weak solutions to this problem obeying an energy inequality is given using a Galerkin scheme. The existence of global solutions is then demonstrated provided the source terms satisfy a suitable growth condition and, conversely, solutions are shown to blow-up in finite time provided the source terms and initial energy overcome the damping terms in an appropriate sense.
Kass, Nicholas J, "Damped Wave Equations of the p-Laplacian Type with Supercritical Sources" (2018). ETD collection for University of Nebraska-Lincoln. AAI10791760.