Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level

Boundary Value Problems

Yonghong Wu¹  Lishan Liu²  Fenglong Sun² 

[1] Department of Mathematics and Statistics, Curtin University;School of Mathematical Sciences, Qufu Normal University;

关键词: Petrovsky type equation;    Memory term;    Nonlinear damping;    Blow-up;   

DOI  :  10.1186/s13661-019-1136-x

来源: DOAJ

【 摘 要 】

Abstract In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, nonlinear weak damping, and a superlinear source: utt+Δ2u−∫0tg(t−τ)Δ2u(τ)dτ+|ut|m−2ut=|u|p−2u,in Ω×(0,T). $$ u_{tt}+\Delta ^{2} u- \int _{0}^{t} g(t-\tau )\Delta ^{2} u(\tau )\,\mathrm{d} \tau + \vert u_{t} \vert ^{m-2}u_{t}= \vert u \vert ^{p-2}u,\quad \text{in }\varOmega \times (0,T). $$ When the source is stronger than dissipations, we obtain the existence of certain weak solutions which blow up in finite time with initial energy E(0)=R $E(0)=R$ for any given R≥0 $R\geq 0$.

【 授权许可】

Unknown