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arXiv · 1908.09693

Reaction-diffusion systems with initial data of low regularity

El-Haj Laamri·Benoît Perthame
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Abstract

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this $L^1$ control is enough to prove the global existence of weak solutions. The theory is based on basic estimates initiated by M. Pierre and collaborators, who have introduced methods to prove $L^2$ a priori estimates for the solution. Here, we establish such a key estimate with initial data in $L^1$ while the usual theory uses $L^2$. This allows us to greatly simplify the proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations.

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El-Haj Laamri, Benoît Perthame. 2019-08-26. Reaction-diffusion systems with initial data of low regularity. https://arxiv.org/abs/1908.09693

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