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Journal Articles Journal of Differential Equations Year : 2020

Reaction-diffusion systems with initial data of low regularity

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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Dates and versions

hal-02271103 , version 1 (26-08-2019)

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El-Haj Laamri, Benoît Perthame. Reaction-diffusion systems with initial data of low regularity. Journal of Differential Equations, 2020, 269 (11), ⟨10.1016/j.jde.2020.06.042⟩. ⟨hal-02271103⟩
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