On one-dimensional forward-backward diffusion equations with linear convection and reaction
type: Item
instance of: scholarly article
title: On one-dimensional forward-backward diffusion equations with linear convection and reaction
zbMATH Open document ID: 1406.35165
author: Seonghak Kim
author: Baisheng Yan
published in: Journal of Differential Equations
publication date: 2018-12-03T00:00:00Z
review text: In the present work, for arbitrary \(T>0\), the authors study an initial-Neumann boundary value problem for a class of one-dimensional forward-backward diffusion equations with linear convection and reaction of the form: \[ u_t=(\sigma (u_x))_x+b(x,t)u_x+c(x,t)u+f(x,t) \] in \(\Omega \times (0,T)\), \(\Omega =(0,1)\), with the following initial and Neumann boundary conditions \(u=u_0\) on \(\Omega \times\{t=0\}\) and \(u_x=0\) on \(\partial \Omega \times (0,T)\). It is assumed that the initial function \(u_0(x)\in C^{2+\alpha}(\bar{\Omega})\) for some \(\alpha \in (0,1)\) and satisfy the compatibility condition \(u'_0=0\) on \(\partial \Omega\). The diffusion flux function \(\sigma (s)\), \(s\in \mathbb R\) is assumed to contain two forward-diffusion phases, or more specifically, it is supposed that there exist numbers \(s_2>s_1>0\) and \(\Lambda \geq \lambda >0\) such that \(\sigma (0)=0\), \(\sigma (s_1)>\sigma (s_2)\geq 0\), \(\sigma \in C^{1+\alpha}(J_1)\), \(\sigma'\in L_{\mathrm{loc}}^1(J_1,(0,\infty ))\bigcap L_{\mathrm{loc}}^1(J_2,[\lambda ,\Lambda ])\) where \(J_k=\{(-\infty ,\frac{s_2}{k})\cup(\frac{s_1}{k},\infty )\}\), \(k=1,2\). It is proved that for all smooth initial functions with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. In addition, the authors introduce the notion of transition gauge for this kind of solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant.
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35M13
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 49K21
zbMATH DE Number: 6987671
zbMATH Keywords: forward-backward diffusion
zbMATH Keywords: linear convection and reaction
zbMATH Keywords: convex integration
zbMATH Keywords: Baire's category
zbMATH Keywords: nonlocal differential inclusion
zbMATH Keywords: transition gauge
MaRDI profile type: Publication
full work available at URL: j.jde.2018.08.004
OpenAlex ID: W2887724453
Wikidata QID: Q59163403
cites work: A Degenerate Pseudoparabolic Regularization of a Nonlinear Forward-Backward Heat Equation Arising in the Theory of Heat and Mass Exchange in Stably Stratified Turbulent Shear Flow
cites work: Direct methods in the calculus of variations
cites work: Implicit partial differential equations
cites work: Existence of Infinitely Many Solutions for a Forward Backward Heat Equation
cites work: Convex Integration and Infinitely Many Weak Solutions to the Perona--Malik Equation in All Dimensions
cites work: On Lipschitz solutions for some forward-backward parabolic equations
cites work: On Lipschitz solutions for some forward-backward parabolic equations. II: The case against Fourier
cites work: Q4354424
cites work: Two-phase entropy solutions of a forward-backward parabolic equation
cites work: Optimal existence theorems for nonhomogeneous differential inclusions
cites work: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation
cites work: Convex integration for a class of active scalar equations
cites work: On existence of weak solutions for one-dimensional forward-backward diffusion equations
DOI: 10.1016/J.JDE.2018.08.004
description: scientific article; zbMATH DE number 6987671