Blow up for Initial-Boundary Value Problem of WaveEquation with a Nonlinear Memory in 1-D

Citation:

Ning-An LAI,Jianli LIU,Jinglei ZHAO.Blow up for Initial-Boundary Value Problem of WaveEquation with a Nonlinear Memory in 1-D[J].Chinese Annals of Mathematics B,2017,38(3):827~838
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Authors:

Ning-An LAI; Jianli LIU;Jinglei ZHAO

Foundation:

This work was supported by the National Natural Sicence Foundation of China (Nos.11301489, 11401367,11501273), the Natural Science Foundation of Zhejiang Province (Nos.LQ13A010013, LY14A010010), and the Doctoral Fund of Ministry of Education of China (No.20133108120002).
Abstract: The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: $$u_{tt}-u_{xx}=\frac{1}{\Gamma(1-\gamma)}\int_0^t(t-s)^{-\gamma}|u(s)|^p\rmd s.$$ The blow up result will be established when $p>1$ and $0<\gamma<1$, no matter how small the initial data are, by introducing two test functions and a new functional.

Keywords:

Blow up, Wave equation, Nonlinear memory, Initial-boundary value problem

Classification:

35L05, 35L70
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