Lifespan of solutions to the damped wave equation with a critical nonlinearity

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Ikeda, Masahiro¹  Ogawa, Takayoshi² 

[1] Kyoto Univ, Grad Sch Sci, Dept Math, Sakyo Ku, Kitashirakawa Oiwakecho, Kyoto 6068502, Japan

[2] Tohoku Univ, Math Inst, Sendai, Miyagi 9808578, Japan

关键词: Damped wave equation;    Lifespan;    Upper bound;    Fujita exponent;    Higher dimensions;   

DOI  :  10.1016/j.jde.2016.04.016

来源: Elsevier

【 摘 要 】

In the present paper, we study a lifespan of solutions to the Cauchy problem for sernilinear damped wave equations {partial derivative(2)(t) - Delta u + partial derivative(t)u = f (u), (t, x) is an element of [0, T (epsilon)) x R-n, u(0, x) = epsilon u(0)(x), x is an element of R-n, (DW) partial derivative(t)u(0, x) = epsilon u(1)(x), x is an element of R-n, where n >= 1, f (u) = +/-vertical bar u vertical bar(p-1)u or vertical bar u vertical bar(p), p >= 1, epsilon >= 0 is a small parameter, and (u(0), u(1)) is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is f (u) = vertical bar u vertical bar(p) and the nonlinear power is the Fujita critical exponent p = p(F) = 1 + 2/n, then the upper estimate to the lifespan is estimated by T (epsilon) <= exp(C epsilon(-p)) for all epsilon is an element of (0, 1] and suitable data (u(0), u(1)), without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35]. We also prove a sharp lower estimate of the lifespan T (epsilon) to (DW) in the critical case p = p(F). (C) 2016 Elsevier Inc. All rights reserved.

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