THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

doi:10.1007/s10473-022-0107-y

Abstract

Abstract: In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents:\[u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(x,\tau){\rm d}\tau +\mu u_t=|u|^{p(x)-2}u,\] where $\mu$ is a nonnegative constant and the exponent of nonlinearity $p(\cdot)$ and $g$ are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function $g$, we prove a finite-time blow-up result. We also give some numerical applications to illustrate our theoretical results.

Key words: nonlinear damping, blow up, finite time, variable nonlinearity, arbitrary positive energy

CLC Number:

35A01

Ala A. TALAHMEH, Salim A. MESSAOUDI, Mohamed ALAHYANE. THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY[J].Acta mathematica scientia,Series B, 2022, 42(1): 141-154.

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