×

Boundedness and blow-up behavior for reaction-diffusion systems in a bounded domain. (English) Zbl 0932.35125

The author studied the boundedness or the blow-up behaviour of the solution of the following reaction-diffusion system \[ \begin{cases} u_t-\Delta u= f(u,v),\;(x,t)\in\Omega\times (0,+\infty),\\ v_t-\Delta v= g(u,v),\;(x,t)\in\Omega\times (0,+\infty),\end{cases} \] subject to continuous and bounded initial data and homogeneous Dirichlet boundary conditions.
Similar results were obtained for nonlinear boundary conditions (including Neumann and Robin boundary conditions).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal., 92, 153-192 (1986) · Zbl 0596.35061
[2] Caristi, G.; Mitidieri, E., Blowup estimate of positive solutions of a parabolic system, J. Differential Equations, 113, 265-271 (1994) · Zbl 0807.35066
[3] Deuring, P., An initial boundary value problem for a certain density-dependent diffusion system, Math. Z., 194, 375-396 (1987) · Zbl 0622.35038
[4] M. Escobedo, M.A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata (IV), CLXV (1993) 315-336.; M. Escobedo, M.A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata (IV), CLXV (1993) 315-336. · Zbl 0806.35088
[5] Escobedo, M.; Levine, H. A., Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal., 129, 47-100 (1995) · Zbl 0822.35068
[6] Filo, J.; Kacur, J., Local existence of general nonlinear parabolic systems, Nonlinear Anal. TMA, 24, 11, 1597-1618 (1995) · Zbl 0830.35053
[7] Galaktionov, V. A.; Kurdyumov, S. P.; Samarskii, A. A., A parabolic system of quasilinear equations I, Differential Equations, 19, 1558-1571 (1983) · Zbl 0556.35071
[8] Galaktionov, V. A.; Kurdyumov, S. P.; Samarskii, A. A., A parabolic system of quasilinear equations II, Differential Equations, 21, 1049-1062 (1985) · Zbl 0599.35085
[9] Jong, U. K., Smooth solutions to a quasilinear system of diffusion equations for a certain population model, Nonlinear Anal. TMA, 8, 10, 1121-1144 (1984)
[10] Leung, A., Equilibria and stabilities for competing species of reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73, 204-218 (1980) · Zbl 0427.35011
[11] Lu, G.; Sleeman, B. D., Non-existence of global solutions to systems of semi-linear parabolic equations, J. Differential Equations, 104, 147-168 (1993) · Zbl 0816.35060
[12] Pao, C. V., On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87, 165-198 (1982) · Zbl 0488.35043
[13] Redlinger, R., Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics, J. Differential Equations, 118, 219-252 (1995) · Zbl 0826.35054
[14] J.D. Rossi, N. Wolanski, Blow-up vs. global existence for a semilinear reaction-diffusion system in a bounded domain, Commun. PDE 20 (11-12) (1995) 1991-2004.; J.D. Rossi, N. Wolanski, Blow-up vs. global existence for a semilinear reaction-diffusion system in a bounded domain, Commun. PDE 20 (11-12) (1995) 1991-2004. · Zbl 0851.35064
[15] Ruan, W., Bounded solutions for reaction-diffusion systems with nonlinear boundary conditions, Nonlinear Anal. TMA, 14, 12, 1051-1077 (1990) · Zbl 0733.35058
[16] Ruan, W. H.; Pao, C. V., Positive steady-state solutions of a competing reaction-diffusion system, J. Differential Equations, 117, 411-427 (1995) · Zbl 0923.35063
[17] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blowup in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995.; A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blowup in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995. · Zbl 1020.35001
[18] Senba, T., On the support properties of solutions for some degenerate quasilinear parabolic systems, Nonlinear Anal. TMA, 14, 9, 789-805 (1990) · Zbl 0707.35008
[19] Yagi, A., Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Anal. TMA, 21, 8, 603-630 (1993) · Zbl 0810.35046
[20] Yamada, Y., Stability of steady states for prey-predatordiffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21, 327-345 (1990) · Zbl 0702.35123
[21] Yamada, Y., Global solutions for quasilinear parabolic systems with cross-diffusion effects, Nonlinear Anal. TMA, 24, 9, 1395-1412 (1995) · Zbl 0863.35052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.