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

In this paper, the parabolic problem $$u_t - div(\omega (x) \nabla u)= h(t) f(u) + l(t) g(u)$$ u t - d i v ( ω ( x ) ∇ u ) = h ( t ) f ( u ) + l ( t ) g ( u ) with non-negative initial conditions pertaining to $$C_b({\mathbb {R}}^N)$$ C b ( R N ) , will be studied, where the weight $$\omega $$ ω is an appropriate function that belongs to the Muckenhoupt class $$A_{1 + \frac{2}{N}}$$ A 1 + 2 N 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 $$h(t)\sim t^r \,(r>-1)$$ h ( t ) ∼ t r ( r > - 1 ) , $$l(t)\sim t^s \, (s>-1)$$ l ( t ) ∼ t s ( s > - 1 ) , $$f(u)=u^p$$ f ( u ) = u p and $$g(u)=(1+u)[\ln (1+u)]^p$$ 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 $$h(t)=1$$ h ( t ) = 1 , $$l(t)=0$$ l ( t ) = 0 and $$f(u)=u^p $$ f ( u ) = u p .