On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

Abstract

This paper is concerned with the study of the nonlinear damped wave equationwhere is a bounded domain of R 2 having a smooth boundary ∂ = . Assuming that g is a function which admits an exponential growth at the infinity and, in addition, that h is a monotonic continuous increasing function with polynomial growth at the infinity, we prove both: global existence as well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, optimal and uniform decay rates of the energy are proved for global solutions.

Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

“…Vitillaro in [43] combined the arguments in [12] and [21] to extend these results to situations where the damping is nonlinear and the solution has positive initial energy. See also the recent paper of Alves and Cavalcanti [4] where analogue results have been shown for the two-dimensional wave equation with exponential source. Similar results have also been established by Todorova [40], [41] and Levine and Park [19] for different Cauchy problems.…”

Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping

“…The papers [8,9] are also among the pioneers in establishing uniform decay rates estimates for the viscoelastic wave equation. In the absence of memory term, we can see the results regarding the wave equation subject to damping and source terms in the classical papers [10][11][12][13][14][15]. Finally, in what concerns the purely wave equation subject to a localized dissipation, namely, b(x) ≥ β > 0 in some proper subset ω ⊂ Ω, we may refer [16][17][18][19].…”

Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms

“…In the presence of the viscoelastic term (g ̸ = 0), the blow-up result of [1] was improved by the same author [6] to positive initial energy. For more related results, we refer the reader to [7][8][9][10][11][12]. For problem (1.2) in R n , Todorova [13] obtained results similar to [4] and also studied the following problem…”