In this work, we study the initial-value problem for an equation of evolution of wave fronts in chemical reactions. We show that the associated initial-value problem is locally and globally well-posed in Sobolev spaces \(H^s({\mathbb {R}})\), where \(s>1/2\). The well-posedness in critical space \(\dot{H}^{1/2}(\mathbb R)\) for small initial data is obtained. We also show that our result is sharp, in the sense that the flow-map data solution is not \(C^2\) at origin, for \(s<1/2\). Furthermore, we study the behavior of the solutions when \(\mu \downarrow 0\).

中文翻译：

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低规则Sobolev空间中化学反应中波前演化方程的IVP

在这项工作中，我们研究化学反应中波前演化方程的初值问题。我们证明了相关的初值问题在Sobolev空间\（H ^ s（{\ mathbb {R}}）\）中是局部和全局适当的，其中\（s> 1/2 \）。获得了临界空间\（\ dot {H} ^ {1/2}（\ mathbb R）\）中小的初始数据的适定性。我们还表明，我们的结果是尖锐，在这个意义上流动的地图数据的解决方案是不\（C ^ 2 \）在原点，对于\（S <1/2 \） 。此外，我们研究\（\ mu \ downarrow 0 \）时解的行为。