Life span of solutions with large initial data in a semilinear parabolic equation (Q2758113)

The paper deals with the Cauchy problem for the semilinear parabolic equation NEWLINE\[NEWLINE \begin{cases} u_t=\Delta u +|u|^{p-1}u\quad & \text{in} {\mathbb R}^N\times (0,\infty),\\ u(x,0)=\lambda \varphi(x)\quad & \text{in} {\mathbb R}^N,\end{cases} NEWLINE\]NEWLINE with \(p>1,\) \(\lambda>0\) and \(\varphi\) being a bounded continuous function. The authors show that the blowup time \(T(\lambda)\) of the solution satisfies NEWLINE\[NEWLINE T(\lambda)={{1}\over {p-1}} |\varphi|_\infty^{1-p} \lambda^{1-p} + o(\lambda^{1-p})\quad \text{ as} \lambda\to\infty. NEWLINE\]NEWLINE Moreover, when the maximum of \(|\varphi(x)|\) is attained at one point, the higher-order term of \(Y(\lambda)\) is determined which reflects the pointedness of the peak of \(|\varphi|.\)