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This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H 1 (Ω) × L2 (∂Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H 1 (Ω) into L2 (Ω), or L2 (∂Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.

In this article we focus on the global well-posedness of an initial-boundary value problem for a nonlinear wave equation in all space dimensions. The nonlinearity in the equation features the damping term |u| k |u t | m sgn(u t) and a source term of the form |u| p−1 u, where k, p ≥ 1 and 0 < m < 1. In addition, if the space dimension n ≥ 3, then the parameters k, m and p satisfy p, k/(1−m) ≤ n/(n−2). We show that whenever k + m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p > k + m and the initial energy is negative, then local weak solutions blow-up in finite time, regardless of the size of the initial data.

Articles you may be interested inBlow up of a solution for a system of nonlinear higher-orderwave equations with strong damping terms AIP Conf.Discrete rotating waves in a ring of coupled mechanical oscillators with strong damping Uniqueness of solutions to the helically reduced wave equation with Sommerfeld boundary conditionsThis paper investigates a quasilinear wave equation with Kelvin-Voigt damping,, in a bounded domain Ω ⊂ R 3 and subject to Dirichlét boundary conditions. The operator ∆ p , 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f (u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W 1, p 0 (Ω) into L 2 (Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy. C 2015 AIP Publishing LLC.

Abstract. Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

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We consider the local and global well-posedness of the coupled nonlinear wave equations u tt − Δu + g 1 (u t) = f 1 (u, v), v tt − Δv + g 2 (v t) = f 2 (u, v) in a bounded domain Ω ⊂ R n with Robin and Dirichlét boundary conditions on u and v respectively. The nonlinearities f 1 (u, v) and f 2 (u, v) have supercritical exponents representing strong sources, while g 1 (u t) and g 2 (v t) act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.