Hadamard well-posedness for wave equations with \(p\)-Laplacian damping and supercritical sources. (English) Zbl 1262.35151

Authors’ abstract: “We study the global well-posedness of the nonlinear wave equation \(u_{tt}-\Delta u-\Delta_p u_t=f(u)\) in a bounded domain \(\Omega\subset\mathbb{R}^n\) with Dirichlet boundary conditions. The nonlinearity \(f(u)\) represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator \(f(u)\) is not locally Lipschitz from \(H^1_0(\Omega)\) into \(L^2(\Omega)\). The nonlinear term \(\Delta_p u_t\) is a strong damping, where \(\Delta_p\) denotes the \(p\)-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution depends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.”


35L20 Initial-boundary value problems for second-order hyperbolic equations
35L72 Second-order quasilinear hyperbolic equations