Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients
Year: 2013
Author: Wei Han
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 2 : pp. 138–150
Abstract
This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ where $u=u(x,t)$ is a real-valued scalar unknown function in $[0,+∞)×[0,+∞)$, here a(x) is a smooth real-valued function of the variable $x∈(0,+∞)$. The exponents p satisfies $1 < p < +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g., [1]). We will establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v26.n2.4
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 2 : pp. 138–150
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Semilinear wave equation