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This paper concerns the global in time existence of solutions for asemilinear 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 initialfunction and $f\in C^1([0,\infty)) \cap C^2((0,\infty))$ denotes superlinearnonlinearity of the problem. We consider the global in time existence andnonexistence of solutions for problem~\eqref{eq:P}. The main purpose of thispaper is to determine the critical decay rate of initial functions for theglobal existence of solutions. In particular, we show that it is characterizedby 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 $q\ge 1$.

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