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

Abstract

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0, &x\in \mathbb{R}^N, \end{cases} \end{equation} where $N\ge 1$, $u_0$ is a nonnegative initial function and $f\in C^1([0,\infty)) \cap C^2((0,\infty))$ denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem~\eqref{eq: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 \begin{equation} \notag \Delta W + \frac{y}{2}\cdot \nabla W + f(W)F(W) + f(W) + \frac{|\nabla W|^2}{f(W)F(W)} \Bigl[ q - f'(W)F(W) \Bigr] = 0, \quad y \in \mathbb{R}^N, \end{equation} where $F(s):=\displaystyle\int_s^{\infty}\dfrac{1}{f(\eta)}d\eta$ and $q\ge 1$.