The aim of this paper is to show some applications of Sobolev inequalities in partial differential equations. With the aid of some well-known inequalities, we derive the existence of global solution for the quasilinear parabolic equations. When the blow-up occurs, we derive the lower bound of the blow-up solution.

中文翻译：

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Sobolev不等式在证明拟线性抛物方程解的存在性中的应用

本文的目的是展示Sobolev不等式在偏微分方程中的一些应用。借助一些众所周知的不等式，我们得出了拟线性抛物方程的整体解的存在性。当爆炸发生时，我们得出爆炸解决方案的下限。