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Coupled Systems of Reaction Diffusion Equations

In: Nonlinear Parabolic and Elliptic Equations

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  • C. V. Pao

    (North Carolina State University)

Abstract

The discussions in the previous chapters are concerned with scalar boundary-value problems where only one unknown function is involved in the equation. In this chapter the method of upper and lower solutions is extended to coupled systems of parabolic and elliptic equations. Special attention is given to coupled systems of two equations where the reaction function is quasimonotone. Three basic types of quasimonotone functions are treated, and two monotone sequences for each type are constructed. These monotone sequences lead to some existence-comparison theorems for both parabolic and elliptic boundary-value problems. These theorems are used to obtain invariant regions for time-dependent systems through suitable construction of upper and lower solutions. Using the same idea as for mixed quasimonotone functions the monotone method is extended to coupled systems of arbitrary finite numbers of parabolic equations, elliptic equations, parabolic-ordinary equations, and systems with nonlocal reaction functions. Existence-comparison results are also given for finite coupled systems of parabolic and elliptic equations where the reaction function is not necessarily quasimonotone. Here generalized coupled upper and lower solutions are introduced.

Suggested Citation

  • C. V. Pao, 1992. "Coupled Systems of Reaction Diffusion Equations," Springer Books, in: Nonlinear Parabolic and Elliptic Equations, chapter 0, pages 381-458, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-3034-3_8
    DOI: 10.1007/978-1-4615-3034-3_8
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