Global and unique weak solutions of nonlinear wave equations
type: Item
instance of: scholarly article
title: Global and unique weak solutions of nonlinear wave equations
zbMATH Open document ID: 0841.35067
author: Lev Kapitanski
published in: Mathematical Research Letters
publication date: 1995-11-16T00:00:00Z
review text: The existence and uniqueness of global weak solutions of the problem \({\mathcal L} u+ f(u)= 0\), \(u(0)= u_0\), \(u_t(0)= u_1\) is studied, where \(\mathcal L\) is a linear wave operator and the nonlinearity \(f\) has the so-called critical growth at infinity, typically, \[ f(u)= |u|^\sigma u,\quad \sigma= {4\over N- 2}, \] where \(N\) is the spatial dimension. The main result of the paper states that all weak solutions \((u, u_t)\) of the problem are continuous in time with values in the energy space \(H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)\). In particular, the energy is a continuous function of time.
reviewed by: Eduard Feireisl
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 817689
zbMATH Keywords: nonlinear wave equations
zbMATH Keywords: critical growth at infinity
MaRDI profile type: Publication
full work available at URL: mrl.1994.v1.n2.a9
OpenAlex ID: W2060831697
DOI: 10.4310/MRL.1994.V1.N2.A9
description: scientific article; zbMATH DE number 817689