A shift in the Strauss exponent for semilinear wave equations with a not effective damping

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

D'Abbicco, Marcello¹  Lucente, Sandra²  Reissig, Michael³ 

[1] Univ Sao Paulo, Dept Comp & Matemat, BR-14040901 Ribeirao Preto, SP, Brazil

[2] Univ Bari, Dept Math, I-70125 Bari, Italy

[3] Tech Univ Bergakad Freiberg, Fac Math & Comp Sci, D-09596 Freiberg, Germany

关键词: Semilinear damped wave equation;    Not effective damping;    Small data global existence;    Strauss exponent;   

DOI  :  10.1016/j.jde.2015.06.018

来源: Elsevier

【 摘 要 】

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namely v(tt) - Delta v +2/1+t v(t) = broken vertical bar v broken vertical bar(P), v(0, x) = v(0)(x), v(t) (0, = v(1)(x), where p > 1, n > 2. We prove blow-up in finite time in the subcritical range p is an element of (1, p(2)(n)] and existence theorems for p > p2(n), n = 2, 3. In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture p2(n) = p(0)(n +2) for n > 2, where p0(n) is the Strauss exponent for the classical semilinear wave equation with power nonlinearity. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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