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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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[1] Friedman, A.; Lacey, A., The blow-up time of solutions of nonlinear heat equations with small diffusion, SIAM J. math. anal., 18, 711-721, (1987) · Zbl 0643.35013
[2] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \operatorname{\Delta} u + u^{1 + \alpha}\), J. fac. sci. univ. Tokyo sect. IA math., 16, 105-113, (1966)
[3] Gui, C.; Wang, X., Life span of solutions of the Cauchy problem for a semilinear heat equation, J. differential equations, 115, 166-172, (1995) · Zbl 0813.35034
[4] Lee, T.-Y.; Ni, W.-M., Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. amer. math. soc., 333, 1434-1446, (1992)
[5] Mizoguchi, N.; Yanagida, E., Life span of solutions with large initial data in a semilinear parabolic equation, Indiana univ. math. J., 50, 1, 591-610, (2001) · Zbl 0996.35006
[6] Mizoguchi, N.; Yanagida, E., Life span of solutions for a semilinear parabolic problem with small diffusion, J. math. anal. appl., 261, 350-368, (2001) · Zbl 0993.35011
[7] Payne, L.; Schaefer, P., Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. math. anal. appl., 328, 2, 1196-1205, (2007) · Zbl 1110.35031
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