Abstract <p> We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(|D_xu|^q\)</EquationSource> </InlineEquation>. Two exponents, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">\(q_1=N/(N-1)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">\(q_2=(N+1)/(N-1)\)</EquationSource> </InlineEquation>, are found such that for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(1<q\leq q_1\)</EquationSource> </InlineEquation>, there is no weak solution local in time for any <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(T>0\)</EquationSource> </InlineEquation>; for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2700_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">\(q_1<q\leq q_2\)</EquationSource> </InlineEquation>, there is a unique weak solution local in time; however, there is no weak solution global in time, i.e., independently of the “value” of the initial function, the solution to the Cauchy problem blows up in a finite time. </p>

Blow-up of the solution to the Cauchy problem for one \((N+1)\)-dimensional composite-type equation with gradient nonlinearity

期刊论文
发表日期：

2025年10月27日

查看全文

M. O. Korpusov,
A. A. Panin,
A. K. Matveeva

摘要

Abstract

We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity \(|D_xu|^q\) . Two exponents, \(q_1=N/(N-1)\) and \(q_2=(N+1)/(N-1)\) , are found such that for \(1<q\leq q_1\) , there is no weak solution local in time for any \(T>0\) ; for \(q_1<q\leq q_2\) , there is a unique weak solution local in time; however, there is no weak solution global in time, i.e., independently of the “value” of the initial function, the solution to the Cauchy problem blows up in a finite time.