Systems of Parabolic Equations with Continuous and Discrete Delays

Abstract In this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear “reaction function” may depend on both continuous (infinite or finite) and discrete delays. It is shown that if the reaction function is locally Lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function. For systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system. This approach yields a (global) attractor of the parabolic system, and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system. Application to three model problems in population dynamics and chemical reactions is given.

[1]  Philip Korman,et al.  On the existence and uniqueness of positive steady states in the volterra-lotka ecological models with diffusion , 1987 .

[2]  Yoshio Yamada.  Asymptotic behavior of solutions for semilinear Volterra diffusion equations , 1993 .

[3]  John J. Tyson.  Chemistry of the Belousov-Zhabotinskii Reaction , 1976 .

[4]  C. V. Pao,et al.  Quasisolutions and global attractor of reaction-diffusion systems , 1996 .

[5]  Hal L. Smith,et al.  Abstract functional-differential equations and reaction-diffusion systems , 1990 .

[6]  R. L. Pitliya,et al.  Oscillations in Chemical Systems , 1986 .

[7]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[8]  K. J. Brown.  Spatially inhomogeneous steady state solutions for systems of equations describing interacting populations , 1983 .

[9]  C. C. Travis,et al.  Existence, stability, and compactness in the -norm for partial functional differential equations , 1978 .

[10]  Xin Lu.  Monotone method and convergence acceleration for finite‐difference solutions of parabolic problems with time delays , 1995 .

[11]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[12]  S. Luckhaus.  Global boundedness for a delay differential equation , 1986 .

[13]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[14]  A. Schiaffino.  On a diffusion volterra equation , 1979 .

[15]  John J. Tyson,et al.  The Belousov–Zhabotinskii reaction’ , 1976, Nature.

[16]  C. V. Pao,et al.  Coupled nonlinear parabolic systems with time delays , 1995 .

[17]  Reinhard Redlinger,et al.  Existence theorems for semilinear parabolic systems with functionals , 1984 .

[18]  A. Schiaffino,et al.  Monotone methods and attractivity results for Volterra integro-partial differential equations , 1981, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  C. V. Pao,et al.  Dynamics of Nonlinear Parabolic Systems with Time Delays , 1996 .

[20]  R. M. Noyes,et al.  Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction , 1974 .