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.

中文翻译：

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具有低规律性初始数据的反应扩散系统

来自生态学、化学反应和其他几个应用领域的模型导致具有超线性增长的半线性抛物线方程。即使在一般情况下会发生爆炸，这些模型也具有质量控制必不可少的特性。在很多情况下，已知这个$L^1$控制足以证明弱解的全局存在。该理论基于由 M. Pierre 及其合作者发起的基本估计，他们引入了证明 $L^2$ 解的先验估计的方法。在这里，我们使用 $L^1$ 中的初始数据建立这样一个关键估计，而通常的理论使用 $L^2$。这使我们能够大大简化某些结果的证明。我们还建立了新的半线性存在结果，它们在复杂的化学反应中是超二次的。