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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.

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
Full Text: DOI
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