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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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