Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth
- 93 Downloads
We consider the Neumann and Cauchy problems for positivity preserving reaction–diffusion systems of m equations enjoying the mass and entropy dissipation properties. We show global classical existence in any space dimension, under the assumption that the nonlinearities have at most quadratic growth. This extends previously known results which, in dimensions \(n\ge 3\), required mass conservation and were restricted to the Cauchy problem. Our proof is also simpler.
KeywordsReaction–diffusion systems Mass dissipation Entropy Global existence
Unable to display preview. Download preview PDF.
- 3.M.C. Caputo, T. Goudon, A. Vasseur, Solutions of the \(4\)-species quadratic reaction–diffusion system are bounded and \(C^\infty \)-smooth, in any space dimension, Preprint arXiv:1709.05694 (2017).
- 11.T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 117–142.Google Scholar
- 12.J.I. Kanel, The Cauchy problem for a system of semilinear parabolic equations with balance conditions, Differentsial’nye Uravneniya 20 (1984), 1753–1760 (English translation: Differential Equations 20 (1984), 1260–1266).Google Scholar
- 13.J.I. Kanel, Solvability in the large of a system of reaction–diffusion equations with the balance condition, Differentsial’nye Uravneniya 26 (1990), 448–458 (English translation: Differential Equations 26 (1990), 331–339.Google Scholar
- 17.M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction diffusion systems with quadratic growth, Indiana Univ. Math. J. (2018), to appear (Preprint hal: 01671797).Google Scholar
- 19.P. Quittner, Ph. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, 2007, 584 p.+xi.Google Scholar