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

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)div(omega del). Ths paper establishes the existence/non-existence of global-n-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) - div(omega del u) = u(p), which is not self-adjoint, with two types of homogeneous weights: omega x) = vertical bar x(1)vertical bar(a) and omega x) = vertical bar x vertical bar(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(vertical bar x'vertical bar ->infinity)(log omega x))/(log vertical bar x'vertical bar) and beta := lim(vertical bar x'vertical bar -> 0) (log omega(x))/(log vertical bar x'vertical bar) exist, where x' = (x(')1,..., x(n)) and 1 <= n <= 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.