Global existence for a class of quadratic reaction–diffusion systems with nonlinear diffusions and L1 initial data. (June 2016)

• Record Type: Journal Article
• Authors: Pierre, Michel; Rolland, Guillaume
• Is Part Of: Nonlinear analysis. Volume 138(2016)
• Page Start: 369
• Page End: 387
• Publication Date: 2016-06
• Subjects: 35K51 -- 35K57 -- 35K59; Reaction–diffusion -- Quasilinear systems -- Global existence; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work, existence of global weak solutions in any space dimension is proven for a class of reaction–diffusion systems with L 1 -initial data, nonlinear diffusions and at most quadratic reactions. The proof relies on a dimension-independent L 2 -estimate, based on a total mass control assumption. If initial data are in L 2, this estimate provides a control of the quadratic nonlinearities in L 1 up to t = 0 . In the case of L 1 -initial data, we prove that the L 2 -estimate can be localized in time, which allows to pass to the limit in an approximate system for t > 0 . We then prove the continuity of the solution in L 1 at t = 0 .