Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

被引:16
|
作者
Guo, Yanqiu [1 ]
Rammaha, Mohammad A. [2 ]
Sakuntasathien, Sawanya [3 ]
机构
[1] Florida Int Univ, Dept Math & Stat, Miami, FL 33199 USA
[2] Univ Nebraska, Dept Math, Lincoln, NE 68588 USA
[3] Silpakorn Univ, Dept Math, Fac Sci, Nakhon Pathom 73000, Thailand
关键词
Viscoelasticity; Memory; Integro-differential; Damping; Source; Blow-up; GLOBAL NONEXISTENCE THEOREMS; HADAMARD WELL-POSEDNESS; WAVE-EQUATIONS; EVOLUTION-EQUATIONS; ASYMPTOTIC STABILITY; UNIFORM DECAY; EXISTENCE; BOUNDARY; SYSTEMS; ENERGY;
D O I
10.1016/j.jde.2016.10.037
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: {u(tt) - k(0)Delta u - integral(infinity)(0) k'(s)Delta u(t-s)ds + [u(t)](m-1) u(t) = vertical bar u vertical bar(P-1)u, in Omega x (0, T), u(x, t) = u(0)(x, t), in Omega x (-infinity, 0], where Omega is a bounded domain in R-3 with a Dirichlet boundary condition. The relaxation kernel k is mono-tone decreasing and k(infinity) = 1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p > max{m, root K(0)}, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to -infinity, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory. (C) 2016 Elsevier Inc. All rights reserved.
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页码:1956 / 1979
页数:24
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