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Abstract <p> We consider the Cauchy problem for a nonclassical third-order partial differential equation with gradient nonlinearity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|\nabla u(x,t)|^q\)</EquationSource> </InlineEquation>. A solution of this problem is understood in a certain weak sense. Using Schauder estimates, we show that a local-in-time weak solution of the Cauchy problem has a certain smoothness with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q&gt;N/(N-1)\)</EquationSource> </InlineEquation>. </p>

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On the smoothness of the solution of one nonlinear equation with gradient nonlinearity

  • 期刊论文

  • 发表日期：

    2025年12月23日

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  • A. K. Matveeva

摘要

Abstract

We consider the Cauchy problem for a nonclassical third-order partial differential equation with gradient nonlinearity \(|\nabla u(x,t)|^q\) . A solution of this problem is understood in a certain weak sense. Using Schauder estimates, we show that a local-in-time weak solution of the Cauchy problem has a certain smoothness with \(q>N/(N-1)\) .