Differential and Integral Equations

Blow-up of positive solutions to wave equations in high space dimensions

Mohammad Rammaha, Hiroyuk Takamura, Hiroshi Uesaka, and Kyouhei Wakasa

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Abstract

This paper is concerned with the Cauchy problem for the semilinear wave equation: $$ u_{tt}-\Delta u=F(u) \quad \mbox{in}\quad \mathbb R^n\times[0,\infty), $$ where the space dimension $n\geq 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p > 1$. Here, the Cauchy data are non-zero and non-compactly supported. Our results on the blow-up of positive radial solutions (not necessarily radial in low dimensions $n=2, 3$) generalize and extend the results of Takamura [19] for zero initial position and Takamura, Uesaka and Wakasa [21] for zero initial velocity. The main technical difficulty in the paper lies in obtaining the lower bounds for the free solution when both initial position and initial velocity are non-identically zero in even space dimensions.

Article information

Source
Differential Integral Equations Volume 29, Number 1/2 (2016), 1-18.

Dates
First available in Project Euclid: 24 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1448323250

Mathematical Reviews number (MathSciNet)
MR3450746

Zentralblatt MATH identifier
06562164

Subjects
Primary: 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations 58J45: Hyperbolic equations [See also 35Lxx]

Citation

Rammaha, Mohammad; Takamura, Hiroyuk; Uesaka, Hiroshi; Wakasa, Kyouhei. Blow-up of positive solutions to wave equations in high space dimensions. Differential Integral Equations 29 (2016), no. 1/2, 1--18. https://projecteuclid.org/euclid.die/1448323250.


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