Global Existence and Non-existence Theorems for Nonlinear Wave Equations

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 |ut|m sgn(ut) and a source term of the form |u|p-1u, where k, p ≥ 1 and 0 \u3c m \u3c 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 \u3e 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

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

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.Comment: The 2nd version includes a new proof of the energy identit

Global Existence and Non-existence Theorems for Nonlinear Wave Equations

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 |ut|m sgn(ut) and a source term of the form |u|p-1u, where k, p ≥ 1 and 0 \u3c m \u3c 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 \u3e 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

The influence of damping and source terms on solutions of nonlinear wave equations

We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces.One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions

Weak solutions and blow-up for wave equations of \u3ci\u3ep\u3c/i\u3e-Laplacian type with supercritical sources

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f (u), in a bounded domain Ω ⊂ R3 and subject to Dirichlét boundary conditions. The operator Δp, 2 \u3c p \u3c 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 W01, p (Ω) into L2(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

Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms

This article is concerned with the blow-up of generalized solutions to the wave equation utt - Δu + |u|k j\u27 (ut = |u|p-1 u in Ω × (0, T), where p \u3e 1 and j\u27 denotes the derivative of a C1 convex and real valued function j. We prove that every generalized solution to the equation that enjoys an additional regularity blows-up in finite time; whenever the exponent p is greater than the critical value k + m, and the initial energy is negative. Indiana University Mathematics Journal ©