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Life span of solutions with large initial data in a semilinear parabolic equation. (English) Zbl 0996.35006
The paper deals with the Cauchy problem for the semilinear parabolic equation $$ \cases u_t=\Delta u +|u|^{p-1}u\quad & \text{in} {\Bbb R}^N\times (0,\infty),\\ u(x,0)=\lambda \varphi(x)\quad & \text{in} {\Bbb R}^N,\endcases $$ 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 $$ T(\lambda)={{1}\over {p-1}} |\varphi|_\infty^{1-p} \lambda^{1-p} + o(\lambda^{1-p})\quad \text{ as} \lambda\to\infty. $$ 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 $|\varphi|.$

35B30Dependence of solutions of PDE on initial and boundary data, parameters
35K55Nonlinear parabolic equations
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