Critical counterexamples for linear wave equations with time-dependent propagation speed

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Ghisi, Marina¹  Gobbino, Massimo² 

[1] Univ Pisa, Dipartimento Matemat, Pisa, Italy

[2] Univ Pisa, Dipartimento Ingn Civile & Ind, Pisa, Italy

关键词: Wave equation;    Strong damping;    Derivative loss;    Gevrey spaces;    Baire category;    Residual set;   

DOI  :  10.1016/j.jde.2020.08.044

来源: Elsevier

【 摘 要 】

We investigate an abstract wave equation with a time-dependent propagation speed, and we consider both the non-dissipative case, and the case with a strong damping that depends on a power of the elastic operator. Previous results show that, depending on the values of the parameters and on the time regularity of the propagation speed, this equation exhibits either well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or ill-posedness with severe derivative loss. In this paper we examine some critical cases that were left open by the previous literature, and we show that they fall into the pathological regime. The construction of the counterexamples requires a redesign from scratch of the basic ingredients, and a suitable application of Baire category theorem in place of the usual iteration scheme. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free