On the Dirichlet problem for fully nonlinear parabolic equations
type: Item
instance of: scholarly article
title: On the Dirichlet problem for fully nonlinear parabolic equations
zbMATH Open document ID: 0932.35122
DOI: 10.1007/BF02366847
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-10-12T00:00:00Z
full work available at URL: https://eudml.org/doc/68749
review text: Let \(Q=\Omega\times (0,T)\) with \(\Omega\) a bounded domain of \(\mathbb{R}^n\). The paper concerns the solvability of the first initial-boundary value problem: \(G[u]=g\) in \(Q\), \(u(x,0)=\psi(x)\), and \(u(x,t)=\Phi\) on \(\partial\Omega \times [0,T]\), where \(G\) is a second order fully nonlinear parabolic operator defined as follows. Let \(G(s,S)\) be a function defined on \(D_0\subset \mathbb{R}^1 \times S^n\), where \(S^n\) is the set of \(n\times n\) symmetric matrices. Assume there exists a nonempty convex domain \(D\) where \(G(s,S)\) is convex and monotone in a suitable way. Indeed, two types of functions are considered. One type is: \(G(s,S)=-s+F(s)\), \(D=\mathbb{R}^1\times K(F)\), where \(K(F)\) is a component of the concavity and monotonicity of \(F\). One typical example is: \(F(s)=(\text{tr}_mS/\text{tr}_lS)^{1/(m-l)}\), \(0\leq l < m\leq\), \(K(F)=\{s\in S^n: \text{tr}_p S>0\), \(p=1,\ldots, m\}\). The second type discussed is: \(G(s,S)=(s \text{tr}_m S)^{-1}\), \(D=(-\infty,0)\times F(K)\). The function \(G(s,S)\) determines fully nonlinear parabolic operators in the following way: \(G[u]=G(a_0u_t+w,A(u_x)u_{xx}A(u_x)-a_1I +W)\), where \(a_0, a_1\geq 0\), \(w\) is a smooth function, \(W\) is a matrix, \(A(u_x)\) is a smooth positive matrix, and \(u_{xx}\) is the Hessian matrix of \(u\). The peculiarity of the equation considered is the nonlinear dependence both on the first order time derivative and the second order spatial ones. The solvability of the problems depend on the coefficients \(a_0\) and \(a_1\). They are classified as: evolution of open type if \(a_0>0, \;a_1=0\); evolution of closed type if \(a_0=0, \;a_1>0\); evolution of mixed type if \(a_0>0, \;a_1>0\). In order to have admissible solutions, the compatibility of \(\psi, \Phi\) are discussed. In contrast to the linear case, the initial velocity \(v(x)=u_t(x,0)\) must satisfy a nonlinear equation. The solvability of the general problem is investigated by using a suitable continu…
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1348498
zbMATH Keywords: second-order fully nonlinear parabolic operator
zbMATH Keywords: compatibility
zbMATH Keywords: continuity method
zbMATH Keywords: a priori estimates
author: N. M. Ivochkina
reviewed by: Giovanni Porru
MaRDI profile type: Publication
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OpenAlex ID: W2055402311
description: scientific article; zbMATH DE number 1348498