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Blow-up for quasilinear heat equations with critical Fujita's exponents

Published online by Cambridge University Press:  14 November 2011

Victor A. Galaktionov
Victor A. Galaktionov
Affiliation:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Sq. 4,125047 Moscow, Russia
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Abstract

We consider the Cauchy problem for the quasilinear heat equation

where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].

In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivity

with σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 517 - 525
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Aronson, D. G. and Vazquez, J. L.. The porous medium equation as a finite-speed approximation to a Hamilton–Jacobi equation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 4 (1987), 203–230.Google Scholar
2Aronson, D. G. and Weinberger, H. F.. Multidimensional nonlinear diffusion arising in popular genetics. Adv. Math. 30 (1978), 33–76.CrossRefGoogle Scholar
3Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice Hall, 1964).Google Scholar
4Fujita, H.. On the blowing up of solutions to the Cauchy problem for. J. Fac. Sci. Univ. Tokyo, Sect. IA 13 (1966), 109–124.Google Scholar
5Galaktionov, V. A.. On conditions of nonexistence of global solutions for a class of quasilinear parabolic equations. Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), 322–338.Google Scholar
6Galaktionov, V. A., Kurdyumov, S. P., Mikhailov, A. P. and Samarskii, A. A.. On unbounded solutions of the Cauchy problem for parabolic equation. Dokl. Akad. Nauk SSSR, Ser. Math. Phys. 252 (1980), 1362–1364 (in Russian).Google Scholar
7Galaktionov, V. A., Posashkov, S. A. and Vazquez, J. L.. Asymptotic convergence to dipole solutions in nonlinear parabolic equations (to appear).Google Scholar
8Galaktionov, V. A. and Vazquez, J. L.. Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach. J. Funct. Anal. 100 (1991), 435–462.CrossRefGoogle Scholar
9Hayakawa, K.. On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Japan. Acad. 49 (1973), 503–505.Google Scholar
10Kalashnikov, A. S.. Some questions of the qualitative theory of nonlinear degenerate parabolic second-order equations. Uspekhi Mat. Nauk 42 (1987), 135–176 (in Russian).Google Scholar
11Kobayashi, K., Sirao, T. and Tanaka, H.. On the growing up problems for semilinear heat equations. J. Math. Soc. Japan 29 (1977), 407–424.CrossRefGoogle Scholar
12Langlais, M. and Phillips, D.. Stabilization of solutions of nonlinear degenerate evolution equations. Nonlinear Anal. 9 (1985), 321–333.CrossRefGoogle Scholar
13Levine, H.. The role of critical exponents in blow-up problems. SI AM Rev. 32 (1990), 262–288.CrossRefGoogle Scholar
14Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P.. Blow-up in Problems for Quasilinear Parabolic Equations (Moscow: Nauka, 1987, in Russian; English translation: Berlin: Walter de Gruyter, to appear).Google Scholar