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The critical exponents of parabolic equations and blow-up in Rn

Published online by Cambridge University Press:  14 November 2011

Yuan-wei Qi
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong

Extract

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for ppc, where pcm + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Aguirre, J. and Escobedo, M.. On the blow-up of solutions of a convective reaction diffusion equation. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 433–60.CrossRefGoogle Scholar
2Aronson, D. G. and Weinberger, H. F.. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1978), 3376.CrossRefGoogle Scholar
3Bandle, C. and Levine, H.. On the existence and nonexistence of global solutions of reaction–diffusion equations in sectorial domains. Trans. Amer. Math. Soc. 655 (1989), 595624.CrossRefGoogle Scholar
4Chen, Y. Z.. Hölder continuity of the gradients of solutions of nonlinear degenerate parabolic systems. Acta Math. Sinica, NS 2 (1986), 309–31.Google Scholar
5Chen, Y. Z. and Benedetto, E. Di. On the local behaviour of solutions of singular parabolic equations. Arch. Rational Mech. Anal. 103 (1988), 319–45.Google Scholar
6Chen, Y. Z. and Benedetto, E. Di. Boundary estimates for solutions of nonlinear degenerate parabolic systems. J. Reine Angew. Math. 395 (1989), 102–31.Google Scholar
7Benedetto, E. Di. C1 + x local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(1983), 827–50.CrossRefGoogle Scholar
8Escobedo, M. and Herrero, M. A.. Boundedness and blow up for a semilinear reaction–diffusion system. J. Differential Equations 89 (1991), 176202.CrossRefGoogle Scholar
9Friedman, A. and McLeod, J. B.. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–77.CrossRefGoogle Scholar
10Fujita, H.. On the blowing up of solutions of the Cauchy problem for ut, = Δu + u1 + x. J. Fac. Sci. Tokyo Sect. IA, Math. 13 (1966), 102–24.Google Scholar
11Galaktionov, V. A.. Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 517–25.CrossRefGoogle Scholar
12Galaktionov, V. A., Kurdyumov, S. P., Mikhailov, A. P. and Samarskii, A. A.. Unbounded solutions of the Cauchy problem for the parabolic equation ut, = ∇(uσ∇u) + uβ. Soviet Phys. Dokl. 25 (1980), 458–9.Google Scholar
13Haraux, A. and Weissler, F. B.. Nonuniqueness for a semilinear initial problem. Indiana Univ. Math. J. 31 (1982), 167–89.CrossRefGoogle Scholar
14Hayakawa, K.. On nonexistence of global solutions of some semilinear parabolic equations. Proc. Japan Acad. 49 (1973), 503–25.Google Scholar
15Herrero, M. and Pierre, M.. The Cauchy problem for ut = Δum when 0 < m < 1. Trans. Amer. Math. Soc. 291 (1985), 145–58.Google Scholar
16Kobayashi, K., Sirao, T. and Tanaka, H.. On the blowing up problems for semilinear heat equation. J. Math. Soc. Japan 29 (1977), 407–24.CrossRefGoogle Scholar
17Levine, H. A.. A Fujita type global existence–global non-existence theorem for a weakly coupled system of reaction–diffusion equations. ZAMP 42 (1990), 408–30.Google Scholar
18Levine, H. A., Lieberman, G. and Meier, P.. On critical exponent for some quasilinear parabolic equations. Math. Meth. Appl. Sci. 109 (1992), 7380.Google Scholar
19Levine, H. A. and Meier, P.. A blowup result for the critical exponent in cones. Israel J. Math. 67 (1989), 18.CrossRefGoogle Scholar
20Levine, H. A. and Meier, P.. The value of the critical exponent for reaction–diffusion equations in cones. Arch. Rational Mech. Anal. 109 (1989), 7380.CrossRefGoogle Scholar
21Mochizuki, K. and Mukai, K.. Existence and nonexistence of global solutions to fast diffusions with source. Math. Appl. Anal. 2 (1994), 92102.CrossRefGoogle Scholar
22Qi, Y. W.. On the equation ut = Δuα + wβ. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 373–90.CrossRefGoogle Scholar
23Qi, Y. W.. The existence of moving boundary solution of a porous media equation with a source term. Adv. Math. Sci. Appl. (1994) (to appear).Google Scholar
24Qi, Y. W.. The critical exponents of degenerate parabolic equations. Sci. China Ser. A 38 (1994), 1153–62.Google Scholar
25Qi, Y. W. and Levine, H.. The critical exponent of degenerate parabolic systems. ZAMP 44 (1993), 249–65.Google Scholar
26Weissler, F. B.. Existence and nonexistence of global solutions for semilinear heat equations. Israel J. Math. 38(1981), 2940.CrossRefGoogle Scholar