In this paper we study the p-Laplacian reaction–diffusion equation

$$\begin{aligned} u_{t}-\text{ div }(|\nabla u|^{p-2}\nabla u)=k(t)f(u) \end{aligned}$$

subject to appropriate initial and boundary conditions. We show the positive solution \(u(\pmb {x},t )\) exists globally, under the conditions on f, k and the boundary conduction function. It is proved that the solution blows up at finite time, for some initial data and additional energy type conditions, by establishing accurate estimates and using the Sobolev inequality in multi-dimensional space.

中文翻译：

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多维空间中p-Laplacian反应-扩散方程的动力学性质

在本文中，我们研究了p -Laplacian 反应-扩散方程

$$\begin{aligned} u_{t}-\text{ div }(|\nabla u|^{p-2}\nabla u)=k(t)f(u) \end{aligned}$$

受适当的初始和边界条件的约束。我们展示了正解\(u(\pmb {x},t )\)在f , k和边界传导函数的条件下全局存在。通过建立准确的估计并在多维空间中使用 Sobolev 不等式，证明了对于一些初始数据和附加能量类型条件，该解在有限时间内会爆炸。