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

In this paper, the parabolic problem u_t- div(ω(x) ∇ u) = h(t) f(u) + l(t) g(u) with non-negative initial conditions pertaining to C_b(R^N) , will be studied, where the weight ω is an appropriate function that belongs to the Muckenhoupt class A1+2N and the functions f, g, h and l are non-negative and continuous. The main goal is to establish the global and non-global existence of non-negative solutions. In addition, will be obtained both the so-called Fujita’s exponent and the second critical exponent in the sense of Lee and Ni (Trans Am Math Soc 333(1):365–378, 1992), in the particular case when h(t)∼tr(r>-1), l(t)∼ts(s>-1), f(u) = u^p and g(u) = (1 + u) [ln (1 + u)] ^p. The results of this paper extend those obtained by Fujishima et al. (Calc Var Partial Differ Equ 58:62, 2019) that worked when h(t) = 1 , l(t) = 0 and f(u) = u^p.