Existence and non-existence of global solutions for a heat equation with degenerate coefficients

TL;DR

This paper investigates the conditions for existence and non-existence of solutions to a degenerate heat equation with weighted coefficients, extending classical results by identifying critical exponents for specific nonlinearities.

Contribution

It establishes new criteria for global solutions and non-solutions for a weighted parabolic problem, including the derivation of Fujita-type and second critical exponents.

Findings

Derived conditions for solution existence based on weight function properties.

Identified Fujita's exponent and second critical exponent for specific nonlinearities.

Extended previous results to more general weighted equations.

Abstract

In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function that belongs to the Munckenhoupt class $A_{1 + \frac{2}{N}}$ and the functions $f$, $g$, $h$ and $l$ are non-negative and continuous. The main goal is to establish of global and non-global existence of non-negative solutions. In addition, to present the particular case when $h(t) \sim t^r ~~ (r>-1)$, $l(t) \sim t^s ~~ (s>-1)$, $f(u) = u^p$ and $g(u)= (1+u)[\ln(1+u)]^p,$ we obtain both the so-called Fujita's exponent and the second critical exponent in the sense of Lee and Ni \cite{Lee-Ni}. Our results extend those obtained by Fujishima et al. \cite{Fujish} who worked when $h(t)=1$, $l(t)=0$ and $f(u)=u^p $.