On the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution
type: Item
instance of: scholarly article
title: On the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution
zbMATH Open document ID: 0674.35060
author: Takahiro Motai
published in: Tsukuba Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: We study the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution: \[ \partial^ 2_ tw(t)-\Delta w(t)+w(t)+\{V_{\gamma}*f(w(t))\}w(t)=0;\quad w(0)=\phi (x),\quad \partial_ tw(0)=\psi (x), \] where \(f(w)=w^ 2\), \(V_{\gamma}(x)=| x|^{-\gamma}\) in \((x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}.\) We prove the existence of weak solutions for \(0<\gamma <n\). We also prove that for \(0<\gamma <Min\{4,n\}\) the weak solution is unique and there exists a regular solution.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 4103606
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonlinear Klein-Gordon equation
zbMATH Keywords: cubic convolution
zbMATH Keywords: existence
zbMATH Keywords: weak solutions
zbMATH Keywords: unique
zbMATH Keywords: regular solution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160835
OpenAlex ID: W577189275
DOI: 10.21099/TKBJM/1496160835
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description: scientific article; zbMATH DE number 4103606