Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations
type: Item
instance of: scholarly article
title: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations
zbMATH Open document ID: 0921.35017
author: Gustaf Gripenberg
author: Bernardo Cockburn
published in: Journal of Differential Equations
publication date: 1999-09-29T00:00:00Z
full work available at URL: http://purl.umn.edu/3094
review text: The paper deals with the problem in what way the solution to the parabolic equation \[ \partial_t v=\nabla\bigl(\Phi (v)\bigr)+\Delta \bigl( \varphi(v)\bigr) \] with the initial condition \(v(0,\cdot)=h\) in the domain \((0, \infty)\times\mathbb{R}^d\) depends on functions \(\Phi\) and \(\varphi\). These functions are uniformly differentiable, moreover, \(\varphi\) is nondecreasing. Basing on the properties of nonlinear semigroups, the authors prove that the solution to the initial problem uniformly depends on the initial conditions \(h\) and the functions \(\Phi\) and \(\varphi\). To be more precise, if \(v_j\), \(j=1,2\), satisfy \[ \partial_tv_j= \nabla\bigl(\Phi_j (v_j)\bigr)+ \Delta\bigl(\varphi_j (v_j) \bigr), \quad v_j(0,\cdot) =k_j, \] then \[ \begin{multlined}\bigl\| v_1(t,\cdot)-v_2 (t,\cdot)\bigr\|_{L^1(\mathbb{R}^d)} \leq\| h_1-h_2\|_{L^2 (\mathbb{R}^d)}+\| h_1 \|_{TV(\mathbb{R}^d)} \times\\ \times\Bigl(t\sup_{s\in I(h_1)}\bigl\|\Phi_1'(s)-\Phi_2'(s)\bigr\|_\infty +4\sqrt{td}\sup_{s\in I(h_1)}\bigl| \sqrt {\varphi_1'(s)}- \sqrt{\varphi_2'(s)} \bigr|\Bigr). \end{multlined} \] where \(I(h) {\overset{def} =}-\|h^-\|_{L^\infty},\bigl(\|h^+\|_{L^\infty}\bigr)\).
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1271347
zbMATH Keywords: nonlinear semigroups
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
OpenAlex ID: W2081783669
cites work: Q3940162
cites work: On mild and weak solutions of elliptic-parabolic problems
cites work: Kruzkov’s estimates for scalar conservation laws revisited
cites work: Uniqueness of solutions of the initial-value problem for \(u_t- \Delta\varphi (u)=0\)
cites work: The semigroup approach to first order quasilinear equations in several space variables
cites work: Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
cites work: Q5520948
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation
cites work: Q5890172
cites work: A Moving Mesh Numerical Method for Hyperbolic Conservation Laws
DOI: 10.1006/JDEQ.1998.3499
description: scientific article; zbMATH DE number 1271347