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

## TL;DR

This paper investigates conditions under which solutions to a non-autonomous semilinear heat system blow up at space infinity, relating solution existence to initial data and parameters, and providing bounds on maximal existence time.

## Contribution

It establishes criteria for solution blow-up at space infinity and links the existence of solutions to initial data and system parameters, including bounds on maximal existence time.

## Key findings

- Solutions can blow up at space infinity under certain conditions.
- Existence depends on initial data and system parameters.
- Bounds for the maximal existence time are provided.

## Abstract

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) \Delta u_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial u_{2}}{\partial t} & = & \mathit{k}_{2} (t) \Delta u_{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.