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Extinction properties of semilinear heat equations with strong absorption. (English) Zbl 0655.35050
Summary: Consider the initial-boundary value problem for $u\sb t=\Delta u-\lambda u\quad q$ with $\lambda >0$, $0<q<1$; the initial data are nonnegative and the boundary data vanish. It is well known that the solution becomes extinct in finite time T, i.e., u(x,t) becomes identically zero for $t\ge T$, where T is some positive number. In this paper we study the profile of $x\to u(x,t)$ as $t\to T$.

##### MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions of PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35K20 Second order parabolic equations, initial boundary value problems
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##### References:
 [1] Aronson, D. G.: Regularity properties of flows through porous media. SIAM J. Appl. math. 17, 461-467 (1969) · Zbl 0187.03401 [2] Brezis, H.; Friedman, A.: Estimates on the support of solutions of parabolic variational inequalities. Illinois J. Math. 20, 82-98 (1976) [3] Brezis, H.; Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions. J. math. Pures appl. 62, 73-97 (1983) · Zbl 0527.35043 [4] L. A. Caffarelli and A. Friedman, Blow-up of solutions of nonlinear heat equations, J. Math. Anal. Appl., in press. · Zbl 0653.35038 [5] Evans, L. C.; Knerr, B. F.: Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities. Illinois J. Math. 23, 153-166 (1979) · Zbl 0403.35052 [6] Friedman, A.: Partial differential equations of parabolic type. (1983) [7] Friedman, A.; Mcleod, B.: Blow-up of positive solutions of semilinear heat equations. Indiana univ. Math. J. 34, 425-447 (1985) · Zbl 0576.35068 [8] Fujita, J.: On the blowing up of solutions of the Cauchy problem for $ut = {\delta}u + u1 + {\alpha}$. J. fac. Sci. univ Tokyo sect. J 13, 109-124 (1966) · Zbl 0163.34002 [9] Giga, Y.; Kohn, R.: Asymptotically self-similar blow-up of semilinear heat equations. Comm. pure appl. Math. 38, 297-319 (1985) · Zbl 0585.35051 [10] Gmira, A.; Veron, L.: Large time behavior of solutions of a semilinear problem in rn. J. differential equations 53, 258-276 (1984) · Zbl 0529.35041 [11] A. S. Kalashnikov, The propagation of disturbances in problems of nonlinear heat conduction with absorption, USSR Comp. Math Math. Phys 14, 70--85. [12] Kalashnikov, A. S.: On the differential properties of generalized solutions of non-stationary filtration type. Vestnick Moscow univ. Math 29, 62-68 (1974) · Zbl 0272.35016 [13] Kamin, S.; Peletier, L. A.: Large time behavior of the heat equation with absorption. Ann. scuola norm. Sup. Pisa 12, 393-408 (1985) · Zbl 0598.35050 [14] Martinson, L. L.: The finite velocity of propagation of thermal perturbations in media with constant thermal conductivity. Zh. vychisl. Mat. i mat. Fiz 16, 1233-1241 (1976) [15] Weissler, F. B.: Single point blow-up for a semilinear initial value problem. J. differential equations 55, 204-224 (1984) · Zbl 0555.35061