Young measure solutions of a class of forward--backward diffusion equations
type: Item
instance of: scholarly article
title: Young measure solutions of a class of forward--backward diffusion equations
zbMATH Open document ID: 1029.35143
DOI: 10.1016/S0022-247X(03)00054-4
author: Jingxue Yin
author: Chunpeng Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: In this paper, the authors consider the problem \[ u_t= \nabla\cdot q(\nabla u),\quad (x,t)\in \Omega\times (0,T),\tag{1} \] where \(\Omega \subset {\mathbb R}^N\), is a bounded domain with smooth boundary, with Dirichlet boundary conditions on \(\partial \Omega\) and appropriate initial conditions. The assumptions on \(q\) are that \(q = \nabla \varphi\), where \(\varphi \in C^1(\mathbb R^N)\), \[ (\lambda |A|^{1+\delta} -1)_+ \leq \varphi(A) \leq \Lambda |A|^{1-\delta} + 1 \] and \(|q(A)|\leq \Lambda |A|^\delta\) for all \(A \in {\mathbb R}^N\), \(\delta \in [0,1]\). These assumptions cover a multitude of important cases and include backward-forward diffusion and equations of interest in the theory of phase transitions. The main results of the paper are contained in Theorems 3.1 and 4.1 and complement the results of \textit{S. Demoulini} [SIAM J. Math. Anal. 27, 376-403 (1996; Zbl 0851.35066)], which are only valid in the Hilbert space case \(\delta=1\). Theorem 3.1 deals with the regular case \(\delta \in (0,1)\) and the result is that if the initial data \(u_0 \in W_0^{1,1+\delta}(\Omega) \cap L^\infty (\Omega)\), equation (1) with Dirichlet boundary conditions admits at least one Young measure solution. Similarly, in the singular case \(\delta=0\), we have that if \(u_0\in BV(\Omega) \cap L^\infty (\Omega)\) and satisfies the boundary condition in the sense of trace, (1) admits at least one Young measure solution. There are many components in the proofs, but the main idea in Theorem 3.1 is [following \textit{D. Kinderlehrer} and \textit{P. Pedregal}, SIAM J. Math. Anal. 23, 1-19 (1992; Zbl 0757.49014)] to prove existence of Young measure solutions to time difference equations and then pass to the limit, while in Theorem 4.1 one takes a limit of solutions in the regular case as \(\delta\) goes to zero.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1915778
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: phase transitions
MaRDI profile type: Publication
cites work: Existence of Young measure solutions of a class of singular diffusion equations
cites work: Young Measure Solutions for a Nonlinear Parabolic Equation of Forward-Backward Type
cites work: Weak Convergence of Integrands and the Young Measure Representation
cites work: Gradient Young measures generated by sequences in Sobolev spaces
cites work: Young measure solutions for nonlinear evolutionary systems of mixed type
cites work: Remarks on Chacon's Biting Lemma
cites work: Direct methods in the calculus of variations
cites work: Q5567300
cites work: Q3878033
cites work: Q3320728
cites work: Q3726570
cites work: Q4547057
cites work: Q3995411
cites work: Q3972449
cites work: Dynamics of measured valued solutions to a backward-forward heat equation
cites work: Non-parametric mean curvature evolution with boundary conditions
cites work: Estimates for evolutionary surfaces of prescribed mean curvature
cites work: Evolutionary surfaces of prescribed mean curvature
cites work: Fine phase mixtures as minimizers of energy
cites work: Weak continuity and weak lower semicontinuity for some compensation operators
cites work: Equilibrium configurations of crystals
cites work: Q4547060
cites work: Lower semicontinuity of multiple integrals and the Biting Lemma
cites work: Intégrandes normales et mesures paramétrées en calcul des variations
full work available at URL: s0022-247x(03)00054-4
OpenAlex ID: W2049963569
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description: scientific article; zbMATH DE number 1915778