2022/08/24 by Yohei Fujishima, Fujishima, Yohei, Norisuke Ioku +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2208.11330
openalex publication_date 2022/08/24 · openalex created_date 2022/08/27 · openalex updated_date 2026/07/28
This paper concerns the global in time existence of solutions for a semilinear heat equation \begincases ∂t u = Δu + f(u), amp;x∈ ℝN, tgt;0,
u(x,0) = u0(x) ≥ 0, amp;x∈ ℝN, \endcases where N≥ 1, u0 is a nonnegative initial function and f∈ C1([0,∞)) ∩ C2((0,∞)) denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem~\eqrefeq:P. The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions W of ΔW + (y)/(2)⋅ ∇ W + f(W)F(W) + f(W) + (|∇ W|2)/(f(W)F(W)) [ q - f'(W)F(W) ] = 0, y ∈ ℝN, where F(s):=∫s∞\dfrac1f(η)dη and q≥ 1.