Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T11:26:43.548Z Has data issue: false hasContentIssue false
  • English
  • Français

On the blow-up of solutions of a convective reaction diffusion equation

Published online by Cambridge University Press:  14 November 2011

J. Aguirre  and
M. Escobedo
J. Aguirre
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, E 48080 Bilbao, Spain
M. Escobedo
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, E 48080 Bilbao, Spain
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equation

where u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 3 , 1993 , pp. 433 - 460
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aguirre, J., Escobedo, M. and Zuazua, E.. Self-similar solutions of a convection diffusion equation and related semilinear elliptic problems. Comm. Partial Differential Equations 15 (1990), 139–157.CrossRefGoogle Scholar
2Alfonsi, L. and Weissler, F. B.. Blow up in ℝN for a parabolic equation with a damping nonlinear gradient term. In Nonlinear Diffusion Equations and Their Equilibrium States (Boston: Birkhauser, to appear).Google Scholar
3Avrin, J. D.. The generalized Burgers' equation and the Navier-Stokes equation in ℝN with singular initial data. Proc. Amer. Math. Soc. 101 (1987), 29–40.Google Scholar
4Ball, J. M.. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford 28 (1977), 473–486.CrossRefGoogle Scholar
5Bandle, C. and Levine, H.. Blow up phenomena in reaction diffusion equations with convection terms (preprint).Google Scholar
6Brezis, H. and Friedman, A.. Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. 63 (1983), 73–97.Google Scholar
7Chen, T. F., Levine, H. A. and Sacks, P. E.. Analysis of a convective reaction-diffusion equation. Nonlinear Anal. 2(1989), 1349–1370.Google Scholar
8Chen Zhimin, T. F.. Long time small solutions to non linear parabolic equations. Ark. Mat. 28 (1990), 371–381.CrossRefGoogle Scholar
9Chipot, M. and Weissler, F. B.. Some blow up results for a nonlinear parabolic equation with gradient term. SIAM J. Math. Anal. 20 (1989), 886–907.CrossRefGoogle Scholar
10Escobedo, M., Vazquez, J. L. and Zuazua, E.. Asymptotic behavior and source type solutions for a diffusion-convection equation (preprint).Google Scholar
11Escobedo, M., Vazquez, J. L. and Zuazua, E. (in preparation).Google Scholar
12Escobedo, M. and Zuazua, E.. Large time behaviour for convection-diffusion equations in ℝN. J. Funct. Anal. 100 (1991), 119–161.CrossRefGoogle Scholar
13Fila, M.. Remarks on blow up for a nonlinear parabolic equation with a gradient term. Proc. Amer. Math. Soc. 111 (1991), 795–801.CrossRefGoogle Scholar
14Friedman, A.. Blow-up of solutions of nonlinear parabolic equations. In Nonlinear Diffusion Equations and Their Equilibrium States I pp. 301–318 (New York: Springer, 1988).CrossRefGoogle Scholar
15Friedman, A. and Kamin, S.. The asymptotic behaviour of gas in an n-dimensional porous medium. Trans. Amer. Math. Soc. 262 (1980), 551–563.Google Scholar
16Friedman, A. and Lacey, A. A.. Blow up of solutions of semilinear parabolic equations. J. Math. Anal. Appl. 132 (1988), 171–186.CrossRefGoogle Scholar
17Fujita, H.. On the blowing up of solutions of the Cauchy problem for u t = ∆u1+ α. J. Fac. Sci. Univ. Tokyo, Sect. I 13 (1966), 109–124.Google Scholar
18Jones, B. F., Jr., Singular integrals and parabolic equations. Bull. Amer. Math Soc. 69 (1963), 501–503.CrossRefGoogle Scholar
19Kamin, S. and Peletier, L. A.. Source type solutions of degenerate diffusion equations with absorption. Israel J. Math. 50 (1985), 219–230.CrossRefGoogle Scholar
20Kavian, O.. Remarks on the large time behaviour of a nonlinear diffusion equation. Ann. Inst. H. Poincaré, Anal. Nonlinéaire 4 (1987), 423–452.CrossRefGoogle Scholar
21Kawohl, B. and Peletier, L. A.. Observations on blow up and dead cores for nonlinear parabolic equations. Math. Z. 202 (1989), 207–217.CrossRefGoogle Scholar
22Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural, N. N.'ceva. Linear and quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs 23 American Mathematical, Society, Providence, R.I.: 1968).Google Scholar
23Levine, H. A.. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu t = Au + F(u). Arch. Rational Mech. Anal. 51 (1973), 371–386.CrossRefGoogle Scholar
24Levine, H. A., Payne, L. E., Sacks, P. E. and Straughan, B.. Analysis of a convective reaction-diffusion equation II. SIAM J. Math. Anal. 20 (1989), 133–147.CrossRefGoogle Scholar
25Liu, W.. Singular solutions for a convection diffusion equation with absorption (preprint).Google Scholar
26Liu, T.-P. and Pierre, M.. Source solutions and asymptotic behaviour in conservation laws. J. Differential Equations 51 (1984), 419–441.CrossRefGoogle Scholar
27Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (New York: Springer, 1984).CrossRefGoogle Scholar
28Weissler, F. B.. Local existence and nonexistence for semilinear parabolic equations in L p. Indiana Univ. Math. J. 29 (1980), 79–101.CrossRefGoogle Scholar
29Weissler, F. B.. Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38 (1981), 29–40.CrossRefGoogle Scholar
30Ding, Xiaxi. Superlinear conservation law with viscosity (preprint).Google Scholar