A `super-critical' nonlinear wave equation in 2 space dimensions
type: Item
instance of: scholarly article
title: A `super-critical' nonlinear wave equation in 2 space dimensions
zbMATH Open document ID: 1233.35144
author: Michael Struwe
published in: Milan Journal of Mathematics
publication date: 2011-10-19T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/39225
review text: This work is concerned with the initial value problem of a wave equation with an exponential nonlinearity in two space dimensions. Initial values for the displacement and the velocity fields are prescribed smooth functions. The form of the nonlinearity allows the author to derive a conservation law leading to a global energy identity evaluated over the entire plane and implying that the global energy is time-independent, so that it is determined by the initial values. The global energy is assumed to be larger than a particular number. Therefore, the so-called supercritical case is investigated. The principal aim of the work is to show that this nonlinear wave equation possesses a smooth radially symmetric unique solution for all time for all radially symmetric smooth Cauchy data. To this end, the author employs extensively techniques of the functional analysis and analyses carefully the blow-up conditions at the origin characterised by the concentration of energy so that they can be eliminated to reach a smooth solution.
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5960503
zbMATH Keywords: ritical growth
zbMATH Keywords: exponential nonlinearity
zbMATH Keywords: global energy identity
zbMATH Keywords: smooth radially symmetric solution
zbMATH Keywords: concentration of energy
reviewed by: Erdoğan S. Şuhubi
MaRDI profile type: Publication
OpenAlex ID: W2045387728
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DOI: 10.1007/S00032-011-0146-7
description: scientific article; zbMATH DE number 5960503