Global solution for a coupled parabolic system with degenerate coefficients and time-weighted sources

TL;DR

This paper determines the Fujita exponent for a coupled degenerate parabolic system with time-weighted sources, providing a global solution framework under specific degeneracy and source conditions.

Contribution

It introduces a novel analysis of a coupled parabolic system with degenerate coefficients and time-weighted sources, establishing the Fujita exponent for such systems.

Findings

Derived the Fujita exponent for the system.

Established conditions for global solutions.

Analyzed effects of degeneracy on solution behavior.

Abstract

In this paper, we obtain the so-called Fujita exponent to the following parabolic system with time-weighted sources and degenerate coefficients $ u_{t}- \mbox{div} ( \omega(x)\nabla { u} )= t^{r} v^{p} $ and $ v_{t}- \mbox{div} ( \omega(x)\nabla {v} )= t^{s} u^{p}$ in $\mathbb{R}^{N} \times (0,T)$ with initial data belonging to $ \left[L^\infty(\mathbb{R}^N)\right]^2.$ Where $p,q > 0$ with $ pq > 1$; $r,s>-1 $; and either $\omega(x) = | x_1|^{a},$ or $\omega(x) = | x |^{b}$ with $a,b > 0$.