Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation

This paper concerns the global in time existence of solutions for a semilinear heat equation{partial derivative(t)u=triangle u+f(u),x is an element of RN, t >0,u(x,0) =u0(x)>= 0, x is an element of RN,(P)where N >= 1,u 0 is a nonnegative initial function andf is an element of C1([0,infinity))boolean AND C2((0,infinity))denotes superlinear nonlinearity of the problem. We consider the global in timeexistence and nonexistence of solutions for problem (P). The main purpose of thispaper is to determine the critical decay rate of initial functions for the globalexistence of solutions. In particular, we show that it is characterized by quasiself-similar solutions which are solutionsWof triangle W+y2<middle dot>del W+f(W)F(W) +f(W) +|del W|2f(W)F(W)[q-f '(W)F(W)]= 0iny is an element of RN, whereF(s) :=integral infinity s1f(eta)d eta andq >= 1.(c) 2023 Elsevier Ltd. All rights reserved.