Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Lai, Ning-An^1,2  Schiavone, Nico Michele³  Takamura, Hiroyuki⁴ 

[1] Lishui Univ, Inst Nonlinear Anal, 1 Xueyuan Rd, Lishui City 323000, Zhejiang, Peoples R China

[2] Lishui Univ, Dept Math, 1 Xueyuan Rd, Lishui City 323000, Zhejiang, Peoples R China

[3] Sapienza Univ Rome, Dept Math Guido Castelnuovo, Piazzale Aldo Moro 5, I-00185 Rome, Italy

[4] Tohoku Univ, Math Inst, Aoba Ku, Sendai, Miyagi 9808578, Japan

关键词: Semilinear wave equation;    Scale-invariant damping;    Blow-up;    Lifespan;   

DOI  :  10.1016/j.jde.2020.08.020

来源: Elsevier

【 摘 要 】

In this work we consider several semilinear damped wave equations with subcritical nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between wave-like and heat-like behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the transition surfaces separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results. We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation. The main tool employed in the proof is a Kato's type lemma, established by iteration argument. (C) 2020 The Authors. Published by Elsevier Inc.

【 授权许可】

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