×

zbMATH — the first resource for mathematics

Quasisolutions and global attractor of reaction-diffusion systems. (English) Zbl 0853.35056
The author considers a reaction-diffusion system \[ {\partial u_i\over \partial t}= L_i u_i+ f_i(x, u)\quad \text{in } \Omega\times (0, \infty), \] \[ B_i u_i= h_i\quad \text{on } \partial \Omega\times (0, \infty),\quad u_i(x, 0)= u_{i, 0}(x)\quad \text{in } \Omega, \] where \(u= (u_1,\dots, u_n)\). The nonlinear term \(f\) is called mixed quasimonotone if and only if for each \(i\) there exist \(a_i\) and \(b_i\) such that \(a_i+ b_i= n- 1\) and \(f\) is monotone nondecreasing in \(a_i\) components of \((u_1, u_2,\dots, u_{i- 1}, u_{i+ 1},\dots, u_n)\) and monotone nonincreasing in other \(b_i\) components of that. For this special case the global attractor of the dynamics is characterized as a cone defined in the ordered function space.
Reviewer: S.Jimbo (Sapporo)

MSC:
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brown, P.N., Decay to uniform states in ecological interactions, SIAM J. appl. math., 38, 22-37, (1980) · Zbl 0511.92019
[2] Casten, R.G.; Holland, C.J., Stability properties of solutions to systems of reaction-diffusion equations, J. diff. eqns, 27, 266-273, (1978) · Zbl 0338.35055
[3] Conway, E.D.; Hoff, D.; Smoller, J.A., Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. math. analysis, 35, 1-16, (1978) · Zbl 0383.35035
[4] Feng, W., Coupled system of reaction-diffusion equations and application to carrier facilitated diffusion, Nonlinear analysis, 17, 285-311, (1991) · Zbl 0772.35030
[5] FENG W. & RUAN W. H., Coexistence and stability in a three-species competition model, preprint. · Zbl 0897.34044
[6] Fife, P.C.; Tang, M.M., Comparison principles for reaction-diffusion systems, J. diff. eqns, 40, 168-185, (1981) · Zbl 0431.35008
[7] Gardner, R.A., Comparison and stability theorems for reaction-diffusion systems, J. diff. eqns, 37, 60-69, (1980) · Zbl 0413.34058
[8] Lakshmikantham, V.; Vatsala, A.S., Stability results for solutions of reaction-diffusion equations by the method of quasisolutions, Applic. analysis, 12, 229-235, (1981) · Zbl 0443.34024
[9] Leung, A.W.; Clark, D., Bifurcation and large time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. diff. eqns, 25, 113-127, (1980) · Zbl 0427.35014
[10] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044
[11] Pao, C.V., On nonlinear reaction-diffusion equations, J. math. analysis applic., 87, 165-198, (1982) · Zbl 0488.35043
[12] Ruan, W.H., Bounded solutions for reaction-diffusion systems with nonlinear boundary conditions, Nonlinear analysis, 169, 157-178, (1992) · Zbl 0799.35122
[13] Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary-value problems, Indiana univ. math. J., 21, 979-1000, (1972) · Zbl 0223.35038
[14] Zhou, L.; Pao, C.V., Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear analysis, 6, 1163-1184, (1982) · Zbl 0522.92017
[15] Brown, K.J., Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations, J. math. analysis applic, 95, 251-264, (1983) · Zbl 0518.92017
[16] Freedman, H.I.; Waltman, P., Persistence in a model of three competitive populations, Math. biosci., 73, 89-101, (1985) · Zbl 0584.92018
[17] Hallman, T.G.; Svoboda, L.J.; Gard, T.C., Persistence and extinction in the three species Lotka-Volterra competitive systems, Math biosci., 46, 117-124, (1979) · Zbl 0413.92013
[18] Korman, P.; Leung, A., On the existence and uniqueness of positive steady-states in the Volterra-Lotka ecological models with diffusion, Applic. analysis, 26, 145-160, (1987) · Zbl 0639.35026
[19] Lu, X., Persistence and extinction in a competition-diffusion system with time delays, Can. appl. math. quart., 2, 231-246, (1994) · Zbl 0817.35043
[20] Ladde, G.S.; Lakshmikantham, V.; Vatsala, A.S., Monotone iterative techniques for nonlinear differential equations, (1985), Pittman New York · Zbl 0658.35003
[21] Ali, S.W.; Cosner, C., On the uniqueness of the positive steady-state Lotka-Volterra models with diffusion, J. math. analysis applic., 163, 329-341, (1992) · Zbl 0799.35115
[22] Varga, R.S., Matrix iterative analysis, (1962), Prentice Hall Boston · Zbl 0133.08602
[23] Blat, J.; Brown, K.J., Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. math. analysis, 17, 1339-1353, (1986) · Zbl 0613.35008
[24] Gierer, A., Generation of biological patterns and form: some physical, mathematical, and logical aspects, Prog. biophys. molec. biol., 37, 1-47, (1981)
[25] Meinhardt, H., Models of biological pattern formulation, (1982), Academic Press Englewood, NJ
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.