Life span of solutions for a semilinear parabolic problem with small diffusion (Q5946952)

This paper concerns the following initial boundary value problem for a semilinear parabolic equation NEWLINE\[NEWLINEu_t = \varepsilon \Delta u + |u|^{p-1} u\quad\text{in } \Omega \times (0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,t) =0 \quad \text{on } \partial \Omega \times (0,\infty), \qquad u(x,0)=\varphi(x)\quad \text{in } \Omega,NEWLINE\]NEWLINE where \(p>1\), \(\varepsilon >0\), \(\Omega \subset \mathbb{R}^N\) is a bounded domain and \(\varphi\) is a continuous function on \(\overline{\Omega}\) satisfying NEWLINE\[NEWLINE -\min_{x \in \overline{\Omega}}\varphi(x) < \max_{x \in \overline{\Omega}}\varphi(x),NEWLINE\]NEWLINE hence, not necessarily with constant sign. Denoted by \(T(\varepsilon)\) the blow-up time of the solution, the authors prove that \(T(\varepsilon) \rightarrow \frac{1}{p-1} |\varphi|_\infty^{1-p}\) as \(\varepsilon \rightarrow 0\). Such a result improves an analogous one obtained by \textit{A. Friedman} and \textit{A. A. Lacey} [SIAM J. Math. Anal. 18, 711-721 (1987; Zbl 0643.35013)], when the initial data (and hence also the solution) are positive. Moreover, the higher order term of \(T(\varepsilon)\) which reflects the pointedness of the peak of \(|\varphi|\) is determined, when the maximum of \(|\varphi(x)|\) is attained at only one point.