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 (P)∂tu=Δu+f(u),x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where N≥1, u0 is nonnegative initial function and f∈C1([0,∞))∩C2((0,∞)) denotes superlinear nonlinearity problem. We consider nonexistence problem (P). The main purpose this to determine critical decay rate functions solutions. In particular, we show that it characterized by quasi self-similar which are W ΔW+y2⋅∇W+f(W)F(W)+f(W)+|∇W|2f(W)F(W)[q−f′(W)F(W)]=0in y∈RN, where F(s)≔∫s∞1f(η)dη q≥1.