Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Abstract

We consider a non-homogeneous parabolic equation with degenerate coefficients of the form $u_t-L_{\omega } u=u^p$, where $L_{\omega }=\omega ^{-1}\mathrm div(\omega \nabla )$. This paper establishes the existence/non-existence of global-in-time mild solutions based on a critical exponent, known as the Fujita exponent. Similar topics for a semilinear heat equation with degenerate coefficients are treated in Fujishima (Calc Var Partial Differ Equ 58:25, 2019). They considered an equation $u_t-\textrm{div}(\omega \nabla u) =u^p$, which is not self-adjoint, with two types of homogeneous weights: $\omega (x) = |x_1|^a$ and $\omega (x) = |x|^b$ where $a,b>0$. In this paper we consider the case of a self-adjoint operator, and extend to more general weights that meet certain restrictions such as being in the Muckenhoupt class $A_2$, non-decreasing, and where the limits $\alpha :=\lim _{|x'|\rightarrow \infty }(\log \omega (x))/(\log |x'|)$ and $\beta :=\lim _{|x'|\rightarrow 0}(\log \omega (x))/(\log |x'|)$ exist, where $x' = (x_1, \dots , x_n)$ and $1\le n\le N$. The main result establishes that the Fujita exponent is given by $p_F = 1+2/(N+\alpha )$. This means that the asymptotic behavior of the weight at infinity affects global existence of solutions and the one at the origin does not.