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

A. K. Matveeva

Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

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 $q>N/(N-1)$.

Keywords: Sobolev-type nonlinear equations, blow-up, local solvability, Schauder-type estimates, smoothness of solutions, weak solution.

PACS: 02.30.Jr, 02.30.Rz, 02.30.Em

MSC: 31B35, 35A05, 35B33,35B60,35B65,35M20

Received: 09.06.2025
Revised: 06.07.2025

DOI: 10.4213/tmf11024

 English version: Theoretical and Mathematical Physics, 2025, 225:3, 2089–2104

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