On solvability of some boundary value problems for a class of quasilinear parabolic equations
type: Item
instance of: scholarly article
title: On solvability of some boundary value problems for a class of quasilinear parabolic equations
zbMATH Open document ID: 0933.35102
DOI: 10.1007/BF02674727
published in: Siberian Mathematical Journal
publication date: 2000-01-31T00:00:00Z
review text: The article is devoted to studying the solvability of initial-boundary value problems for the nonlinear parabolic equation \[ u_t =a(t,x,u,u_x)u_{xx}+ F(t,x,u,u_x),\;\;(t,x)\in Q=(0,T)\times (-l,l)\;(l>0). \tag{1} \] Equation (1) is furnished with the boundary condition of the second, or third boundary value problem, or the problem when on one end of the interval \((-l,l)\) these conditions are replaced by the Dirichlet boundary condition. The first question considered in the article is a priori bounds for the maximum of the function \(u_x\) under the condition that a classical solution \(u\) is bounded. The second question are existence and uniqueness theorems of a classical solution to the above-mentioned problems. The peculiarities of the results are in the fact that the growth condition for the function \(F\) does not coincide with the celebrated Bernshteĭn condition. It is assumed that the function \(F\) is representable as \[ F(t,x,u,p)=b(t,x,u,p)+ c(t,x,p) \] or \[ F(t,x,u,p)=b(t,x,u,p)+ c(t,u,p), \] where the function \(b/a\) meets the Bernshteĭn condition (i.e., the growth in \(p\) as \(|p|\to \infty\) is at most 2), and \(c\) is of arbitrary growth in \(p\) and satisfies some simple additional conditions; in particular, there are no conditions at all if \(c=c(t,p)\).
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1396679
zbMATH Keywords: existence and uniqueness theorems of a classical solution
zbMATH Keywords: Bernshteĭn condition
author: A. S. Tersenov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1396679