Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities
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- 作者：Yanqiu Guo
- 关键词：35L10 ; 35L20 ; 35L70 ; 35B44
- 刊名：Journal of Differential Equations
- 出版年：2017
- 出版时间：5 February 2017
- 年：2017
- 卷：262
- 期：3
- 页码：1956-1979
- 全文大小：394 K
- 卷排序：262
文摘
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:{utt−k(0)Δu−∫0∞k′(s)Δu(t−s)ds+|ut|m−1ut=|u|p−1u, in Ω×(0,T),u(x,t)=u0(x,t), in Ω×(−∞,0], where Ω is a bounded domain in R3R3 with a Dirichlét boundary condition. The relaxation kernel k is monotone decreasing and k(∞)=1k(∞)=1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p>max{m,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 −∞, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.