Continuous dependence of renormalized solution for nonlinear degenerate parabolic problems in the whole space
type: Item
instance of: scholarly article
title: Continuous dependence of renormalized solution for nonlinear degenerate parabolic problems in the whole space
zbMATH Open document ID: 1323.35099
author: Adama Ouédraogo
author: Jean De Dieu Zabsonré
published in: Mediterranean Journal of Mathematics
publication date: 2014-10-17T00:00:00Z
review text: The authors study the continuous dependence of the so-called renormalized solution for the general degenerate parabolic equation \[ u_t - \Delta\,b(u) + \operatorname{div}F(b(U)) = f \qquad \text{in } Q = (0,T)\times\mathbb R^N,\, T > 0, \] under the initial condition \[ u(0,\cdot) = u_0 \qquad \text{on }\mathbb R^N, \] where \(F:\mathbb R \to\mathbb R^N\) is a Lipschitz continuous function and \(b\) is a nondecreasing continuous function on \(\mathbb R\). For normalization, the values \(b(0) = 0\) and \(F(0) = 0\) are fixed. The well-posedness of the renormanized solution for this problem with \(L^1\) initial data was studied in [\textit{M. Maliki} and \textit{A. Ouedraogo}, Ann. Fac. Sci. Toulouse, Math. (6) 17, No. 3, 597--611 (2008; Zbl 1173.35565)]. The main aim of this paper is to establish the continuous dependence of the renormalized solution with respect to the data. The novelty is that the authors work in the whole space \(\mathbb R^N\) with unbounded data \((u_0,f)\).
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 6358076
zbMATH Keywords: parabolic equation
zbMATH Keywords: continuous dependence
zbMATH Keywords: weak solution
zbMATH Keywords: entropy solution
zbMATH Keywords: renormalized solution
zbMATH Keywords: Kato's inequality
zbMATH Keywords: structure condition
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
full work available at URL: s00009-013-0328-3
OpenAlex ID: W2035741731
cites work: Quasilinear elliptic-parabolic differential equations
cites work: \(L^1\)-theory of scalar conservation law with continuous flux function
cites work: Renormalized Entropy Solutions for Quasi-linear Anisotropic Degenerate Parabolic Equations
cites work: Q4851005
cites work: On the general equation \(u_ t = a(.,u,\varphi (.,u)_ x)_ x + v\) in \(L^ 1\). II: The evolution problem
cites work: On mild and weak solutions of elliptic-parabolic problems
cites work: Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness
cites work: Q4318931
cites work: On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes
cites work: Q4705705
cites work: Entropy solutions for nonlinear degenerate problems
cites work: Q3740471
cites work: On a Nonlinear Parabolic Problem Arising in Some Models Related to Turbulent Flows
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: Q3353669
cites work: A Kinetic Formulation of Multidimensional Scalar Conservation Laws and Related Equations
cites work: Continuous dependence of the entropy solution of general parabolic equation
cites work: Renormalized solution for nonlinear degenerate problems in the whole space
cites work: Continuous dependence of generalized local solutions
cites work: Uniqueness of entropy solutions for nonlinear degenerate parabolic problems
cites work: \(L^ 1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations
cites work: Q4881719
cites work: Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure
cites work: Q4792090
cites work: Q3350007
DOI: 10.1007/S00009-013-0328-3
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description: scientific article; zbMATH DE number 6358076