Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:470

Article

Ikeda, Masahiro^1,2  Sobajima, Motohiro³ 

[1] Keio Univ, Fac Sci & Technol, Dept Math, Kohoku Ku, 3-14-1 Hiyoshi, Yokohama, Kanagawa 2238522, Japan

[2] RIKEN, Ctr Adv Intelligence Project, Wako, Saitama, Japan

[3] Tokyo Univ Sci, Fac Sci & Technol, Dept Math, 2641 Yamazaki, Noda, Chiba, Japan

关键词: Evolution equations;    Two dimensional exterior domain;    Small data blow-up;    Upper bound of lifespan;    Fujita exponent;    Double exponential type;   

DOI  :  10.1016/j.jmaa.2018.10.004

来源: Elsevier

【 摘 要 】

In this paper we consider the following initial-boundary value problem with the power type nonlinearity vertical bar u vertical bar(p) with 1 < p <= 2 in a two-dimensional exterior domain {tau partial derivative(2)(t)u (x, t) - Delta u (x, t) + e(i zeta) partial derivative(t)u (x, t) = lambda vertical bar u (x, t)vertical bar(p), (x, t) is an element of Omega x (0, T), u (x, t) = 0 (x, t) is an element of partial derivative Omega x (0,T), u (x, 0) = epsilon f (x), x is an element of Omega, (0.1) partial derivative(t)u (x, 0) = epsilon g (x), x is an element of Omega, where Omega is given by Omega = {x is an element of R-2 ; vertical bar x vertical bar > 1}, zeta is an element of [-pi/2, pi/2], lambda is an element of C and tau is an element of{0, 1} switches the parabolicity, dispersivity and hyperbolicity. Remark that 2 = 1 + 2/N is well-known as the Fujita exponent. If p > 2, then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata [11]), and if p < 2, then global-in-time solutions cannot exist for any positive initial data (see Ogawa Takeda [22] and Lai Yin [14]). The result is that for given initial data (f, tau g) is an element of H-0(1) (Omega) x L-2 (Omega) satisfying (f + tau g) log vertical bar x vertical bar is an element of L-1 (Omega) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p = 2: LifeSpan (u) <= exp[exp(C epsilon(-1))]. The crucial idea is to use test functions which approximates the harmonic function loglx1 satisfying Dirichlet boundary condition and the technique modified from [9]. (C) 2018 Elsevier Inc. All rights reserved.

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