\(\varvec{L^1-L^1}\) estimates for a doubly dissipative semilinear wave equation

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• 作者：Marcello D’Abbicco
• 关键词：Dissipative wave equation ; Critical exponent ; Global small data solutions ; Power nonlinearity ; \(L^1 ; L^1\) ; estimates
• 刊名：Nonlinear Differential Equations and Applications NoDEA
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：24
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-9004
• 卷排序：24

文摘

In this paper, we derive energy estimates and \(L^1-L^1\) estimates, for the solution to the Cauchy problem for the doubly dissipative wave equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}-\Delta u+u_t-\Delta u_t=0,&{}\quad t\ge 0,\ x\in {\mathbb {R}}^n,\\ (u,u_t)(0,x)=(u_0,u_1)(x). \end{array}\right. } \end{aligned}$$The solution is influenced both by the diffusion phenomenon created by the damping term \(u_t\), and by the smoothing effect brought by the damping term \(-\Delta u_t\). Thanks to these two effects, we are able to obtain linear estimates which may be effectively applied to find global solutions in any space dimension \(n\ge 1\), to the problems with power nonlinearities \(|u|^p\), \(|u_t|^p\) and \(|\nabla u|^p\), in the supercritical cases, by only assuming small data in the energy space, and with \(L^1\) regularity. We also derive optimal energy estimates and \(L^1-L^1\) estimates, for the solution to the semilinear problems.KeywordsDissipative wave equationCritical exponentGlobal small data solutionsPower nonlinearity\(L^1-L^1\) estimatesThe author is supported by INdAM - GNAMPA Project 2016 and is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).