arXiv 1904.071862019-04-15math.AP

一类非自治半线性heat equations方程组解在无穷远处的爆破

Blow-up at space infinity for solutions of a system of non-autonomous semilinear heat equations

Gabriela de Jesús Cabral-García, José Villa-Morales

备注：: 19 pages

摘要：

在本文中，我们将看到\begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t) Δu_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial u_{2}}{\partial t} & = & \mathit{k}_{2} (t) Δu_{2} + h_{2}(t) u_{2}^{p_{22}} u_{1}^{p_{21}}, \end{eqnarray*}解的整体或局部存在性依赖于初始数据，以及\begin{eqnarray*} \dfrac{dy_{1}}{dt} & = & h_{1}(t) y_{1}^{p_{11}}(t) y_{2}^{p_{12}}(t),\\ \dfrac{dy_{2}}{dt} & = & h_{2}(t) y_{2}^{p_{22}}(t) y_{1}^{p_{21}}(t). \end{eqnarray*}解的整体或局部存在性，并给出了偏微分系统解的最大存在时间的一些界。此外，如果这样的存在时间是有限的和$\min\{p_{11} + p_{12},p_{22}+p_{21}\} > 1$，那么我们将证明的偏微分系统的解，爆破在空间无限。

英文摘要：

In this paper we will see that the global or local existence of solutions to \begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t) Δu_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial u_{2}}{\partial t} & = & \mathit{k}_{2} (t) Δu_{2} + h_{2}(t) u_{2}^{p_{22}} u_{1}^{p_{21}}, \end{eqnarray*} depends on the initial datums and the global or local existence of solutions to \begin{eqnarray*} \dfrac{dy_{1}}{dt} & = & h_{1}(t) y_{1}^{p_{11}}(t) y_{2}^{p_{12}}(t),\\ \dfrac{dy_{2}}{dt} & = & h_{2}(t) y_{2}^{p_{22}}(t) y_{1}^{p_{21}}(t). \end{eqnarray*} We also give some bounds for the maximal existence time of the partial differential system. Moreover, if such existence time is finite and $\min\{p_{11} + p_{12},p_{22}+p_{21}\} > 1$ then we will prove the partial differential system has solutions that blows-up at space infinite.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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