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On blow-up and degeneracy for the semilinear heat equation with source

Published online by Cambridge University Press:  14 November 2011

V. A. Galaktionov
V. A. Galaktionov
Affiliation:
Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow, U.S.S.R.
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Synopsis

The asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 19 - 24
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Berger, M. and Kohn, R.. A rescaling algorithm for the numerical calculation of blowing up solutions. Comm. Pure Appl. Math, (to appear).Google Scholar
2Dold, J. W.. Analysis of the early stage of thermal runaway. Quart. J. Mech. Appl. Math. 38 (1985), 361–387.CrossRefGoogle Scholar
3Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice Hall, 1964).Google Scholar
4Galaktionov, V. A.. On nonexistence of global solutions of Cauchy problems for quasilinear parabolic equations. J. Vichisl. Mat. i Mat. Phys. 23 (1983), 1072–1087 (in Russian).Google Scholar
5Galaktionov, V. A., Dorodnitzin, V. A., Elenin, G. G., Kurdjumov, S. P. and Samarskii, A. A.. Quasilinear heat equation with source: blow-up, localization, symmetry, explicit solutions, asymptotics, structures. In Modern Mathematical Problems, Vol. 28, pp. 96–206 (Moscow: Viniti an U.S.S.R., 1986) (in Russian).Google Scholar
6Galaktionov, V. A., Kurdjumov, S. P., Mikhailov, A. P. and Samarskii, A. A.. On unbounded solutions of semilinear parabolic equations (preprint, Keldysh Inst. Appl. Math. Acad. Sci. U.S.S.R., no. 161, 1979) (in Russian).Google Scholar
7Galaktionov, V. A., Kurdjumov, S. P. and Samarskii, A. A.. On approximate self-similar solutions for some quasilinear heat conduction equations with source. Mat. Sb. 124 (1984), 163–188 (in Russian).Google Scholar
8Galaktionov, V. A. and Posashkov, S. A.. Equation u t = u xx + u β. Localization, asymptotic behaviour of unbounded solutions (preprint, Keldysh Inst. Appl. Math. Acad. Sci. U.S.S.R., no. 97, 1985) (in Russian).Google Scholar
9Galaktionov, V. A. and Posashkov, S. A.. Applications of new comparison theorems for investigations of unbounded solutions of nonlinear parabolic equations. Differentsial'nye Uravneniya 22 (1986), 1165–1173 (in Russian).Google Scholar
10Galaktionov, V. A. and Posashkov, S. A.. Explicit solutions of parabolic equations with quadratic nonlinearities (preprint, Keldysh Inst. Appl. Math. Acad. Sci. U.S.S.R., no.-115, 1988) (in Russian).Google Scholar
11Galaktionov, V. A. and Posashkov, S. A.. On some properties of evolution of unbounded solutions of semilinear parabolic equations (preprint, Keldysh Inst. Appl. Math. Acad. Sci. U.S.S.R., no. 232, 1987) (in Russian).Google Scholar
12Galaktionov, V. A. and Posashkov, S. A.. On new explicit solutions of parabolic equations with quadratic nonlinearities. J. Vichisl. Mat. i Mat. Phys. 29 (1989), 497–506 (in Russian).Google Scholar
13Hocking, L. M., Stewartson, K. and Stuart, J. T.. A non-linear instability burst in plane parallel flow. J. Fluid Mech. 51 (1972), 705–735.CrossRefGoogle Scholar
14Kawohl, B. and Peletier, L. A.. Observations on blow up and dead cores for nonlinear parabolic equations. Math. Z. (to appear).Google Scholar
15Lacey, A.. Global blow-up of a nonlinear heat equation. Proc. Roy. Soc. Edinburgh. Sect. A 104 (1986), 161–167.CrossRefGoogle Scholar
16Samarskii, A. A., Galaktionov, V. A., Kurdjumov, S. P. and Mikhailov, A. P.. Blow-up in Problems for Quasilinear Parabolic Equations (Moscow: Nauka, 1987) (in Russian).Google Scholar
17Samarskii, A. A.. On new methods of investigations of asymptotic properties for parabolic equations. Trudy Mat. Inst. Acad. Sci. U.S.S.R. 158 (1981), 153–162.Google Scholar