On one-dimensional forward-backward diffusion equations with linear convection and reaction

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Kim, Seonghak¹  Yan, Baisheng² 

[1] Kyungpook Natl Univ, Dept Math, Daegu 41566, South Korea

[2] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

关键词: Forward-backward diffusion;    Linear convection and reaction;    Convex integration;    Baire's category;    Nonlocal differential inclusion;    Transition gauge;   

DOI  :  10.1016/j.jde.2018.08.004

来源: Elsevier

【 摘 要 】

We study the initial-Neumann boundary value problem for a class of one-dimensional forward-backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data 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. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward-backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free