Global in time solvability for a semilinear heat equation without the self-similar structure

This paper is devoted to the global in time solvability for a superlinear parabolic equation $$\begin{aligned} \partial _t u = \Delta u + f(u), \quad x\in {\mathbb {R}}^N, \quad t>0, \quad u(x,0) = u_0(x), \quad x\in {\mathbb {R}}^N,\quad \hbox {(P)} \end{aligned}$$ ∂ t u = Δ u + f ( u ) , x ∈ R N , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ R N , (P) where f(u) denotes superlinear nonlinearity of problem (P) and $$u_0$$ u 0 is a nonnegative initial function. As a continuation of the paper in 2018 by the authors of this paper, we consider the global in time existence and nonexistence of nonnegative solutions for problem (P). We prove the existence of global in time solutions based on a quasi-scaling property for (P). We also discuss the nonexistence of nontrivial nonnegative global in time solutions by focusing on the behavior of f(u) as $$u\rightarrow +0$$ u → + 0 . These results enable us to generalize the Fujita exponent, which is known as the critical exponent classifying the global in time solvability for a power-type semilinear heat equation.