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Life span of solutions with large initial data for a superlinear heat equation. (English) Zbl 1154.35060
The author studies the parabolic semilinear initial-boundary value problem \begin{aligned} u_t=\Delta u+ f(u) &\quad\text{in }\Omega\times (0,\infty),\\ u= 0 &\quad\text{on }\partial\Omega\times (0,\infty),\\ u(x,0)= \rho\varphi(x) &\quad\text{on }\overline\Omega, \end{aligned} where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ with a smooth boundary $$\partial\Omega$$, $$\rho$$ is a positive parameter, $$\varphi(x)$$ is a nonnegative continuous function on the closure $$\overline\Omega$$ of $$\Omega$$, and $$f(u)$$ is a nonnegative superlinear continuous function on $$[0,\infty)$$. Let $$T(\rho)$$ denote the life span (or the blow-up time) of the solution $$u$$. It is shown that
$T(\rho)= \int_{\rho\|\varphi\|_\infty} {du\over f(u)}+ \text{h.o.t as }\rho\to\infty.$ When the maximum of $$\varphi$$ is attained at a finite number of points in $$\Omega$$, the higher-order term which depends on the minimal value of $$|\Delta\varphi|$$ at the maximal points of $$\varphi$$ can be determined. The proof is based on constructing a supersolution and a subsolution. It is remarked that the method works also for the Cauchy problem on the whole space.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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