# Reaction-diffusion systems with initial data of low regularity

2 MAMBA - Modelling and Analysis for Medical and Biological Applications
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
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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Preprints, Working Papers, ...

Cited literature [37 references]

https://hal.sorbonne-universite.fr/hal-02271103
Contributor : Benoît Perthame <>
Submitted on : Monday, August 26, 2019 - 3:22:51 PM
Last modification on : Wednesday, June 10, 2020 - 10:48:02 AM
Document(s) archivé(s) le : Friday, January 10, 2020 - 4:48:45 AM

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• HAL Id : hal-02271103, version 1
• ARXIV : 1908.09693

### Citation

El-Haj Laamri, Benoît Perthame. Reaction-diffusion systems with initial data of low regularity. 2019. ⟨hal-02271103⟩

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