The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:251

Article

Takamura, Hiroyuki¹  Wakasa, Kyouhei² 

[1] Future Univ Hakodate, Fac Syst Informat Sci, Dept Complex & Intelligent Syst, Hakodate, Hokkaido 0418655, Japan

[2] Future Univ Hakodate, Sch Syst Informat Sci, Dept Complex Syst, Hakodate, Hokkaido 0418655, Japan

关键词: Lifespan;    Semilinear wave equation;    Critical exponent;    High dimensions;   

DOI  :  10.1016/j.jde.2011.03.024

来源: Elsevier

【 摘 要 】

The final open part of Strauss' conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) [18], or Zhou (2007) [21] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) [10], the lifespan T(epsilon) of solutions of u(tt) - Delta u = u(2) in R-4 x [0, infinity) with the initial data u(x, 0) = epsilon f(x), u(t)(x, 0) = epsilon g(x) of a small parameter epsilon > 0, compactly supported smooth functions f and g, has an estimate exp(c epsilon(-2)) <= T(epsilon) <= exp(C epsilon(-2)), where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free