\(L^ 1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations
type: Item
instance of: scholarly article
title: \(L^ 1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations
zbMATH Open document ID: 0862.35078
author: Felix Otto
published in: Journal of Differential Equations
publication date: 1996-11-18T00:00:00Z
review text: The following quasilinear elliptic-parabolic equation: \[ \partial_t[b(u)]- \text{div}[a(\nabla u,b(u))]+ f(b(u))=0\quad\text{in }(0,T)\times\Omega \] in \(u\) is considered, where \(b:\mathbb{R}\to\mathbb{R}\) is monotone nondecreasing and continuous and \(a:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n\) s.t. \(u\in H^{1,r}(\Omega)\mapsto-\text{div}[a(\nabla u,w)]\in(H^{1,r}(\Omega))\) is strictly monotone for any \(w\in\mathbb{R}\) and some \(r\in(1,\infty)\). The problem is completed by some boundary conditions for the function \(u\) and some initial values for \(b(u)\). It is known that the natural solution space for this equation is given by all \(u\) of finite energy, i.e. \[ \sup_{t\in(0,T)} \int_\Omega\Psi(b(u(t)))+ \int_{(0,T)\times\Omega}|\nabla u|^r<+\infty,\text{ where }\Psi(w):=\sup_{z\in\mathbb{R}}\Biggl(zw- \int^z_0b(\xi)d\xi\Biggr), \] is the Legendre transform of the primitive of \(b\). Under these assumptions the \(L_1\)-contraction principle and uniqueness of solutions for this quasilinear elliptic-parabolic equation are proved.
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35G30
zbMATH DE Number: 946923
zbMATH Keywords: Legendre transform
zbMATH Keywords: \(L_ 1\)-contraction principle
zbMATH Keywords: uniqueness
reviewed by: Elena Gavrilova
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0155
OpenAlex ID: W2008683313
DOI: 10.1006/JDEQ.1996.0155
description: scientific article; zbMATH DE number 946923