Abstract

We study global existence, uniqueness and positivity of weak solutions of a class of reaction-diffusion systems of chemical kinetics type, under the assumptions of logarithmic Sobolev inequality and appropriate exponential integrability of the initial data.

Highlights

  • In this paper we consider chemical reactions between q 2 species Ai, i = 1, . . . , q, as follows q q αiAi i=1βiAi, i=1 where αi, βi ∈ N

  • C is of positive symmetric part and the nonlinearity must satisfy some moderate growth bound involving the dimension n to ensure global existence

  • In the two-by-two case, we assume these three conditions: (1) C1 = C3 and C2 = C4 and otherwise they are different, (2) the linear diffusion term satisfies logarithmic Sobolev inequality, (3) the initial datum f is nonnegative and satisfies some exponential integrability properties. Under these assumptions we prove that there exists a unique weak solution of the system of reaction-diffusion equation (1.2) which is nonnegative

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Summary

Introduction

Geometric characteristics and approximations of global and exponential attractors of general reaction-diffusion systems may be found in [19], [44], [45] (and references therein) in terms of precise estimates of their Kolmogorov ε-entropy In these papers, C is of positive symmetric part and the nonlinearity must satisfy some moderate growth bound involving the dimension n to ensure global existence. Logarithmic Sobolev inequality plays a key role to study existence results in a finite or an infinite dimensional setting, by a constructive approximation approach. (3) the initial datum f is nonnegative and satisfies some exponential integrability properties (made more precise later) Under these assumptions we prove that there exists a unique weak solution of the system of reaction-diffusion equation (1.2) which is nonnegative. We recall or detail tools used in the proof in three appendices: the entropic inequality, basics on Orlicz spaces, and some further topics on Markov semigroups and Orlicz spaces

Framework and main result
Iterative procedure
Preliminaries
A mollified problem
Uniqueness
Existence for the cornerstone linear problem
Non-negativity
Extension to the general case
Contraction property
Bochner measurability
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