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Comparison principles for strongly coupled reaction diffusion equations in unbounded domains. (English) Zbl 0712.35052

For strongly coupled systems of the form \[ u_ t=D\Delta u+f(x,t,u,\nabla u) \] comparison principle are studied.
P. C. Fife and M. M. Tang obtained a comparison principle in which D is a positive diagonal matrix. [See J. Differ. Equations 40, 168- 185 (1981; Zbl 0431.35008).] Terman obtained a comparison principle in an unbounded domain.
The author obtains a comparison principle in which D is a non-singular, \(n\times n\), constant diffusion matrix with nonnegtive eigenvalues and at least one nonnegative left eigenvector in an unbounded domain.
Applications to the study of threshold phenomena for the FitzHugh-Nagumo equations and to coupled nerve fibre equations are also given.
Reviewer: A.Tsutsumi

MSC:

35K57 Reaction-diffusion equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
92C05 Biophysics

Citations:

Zbl 0431.35008
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References:

[1] Friedman, Partial Differential Equations of Parabolic Type (1964) · Zbl 0144.34903
[2] DOI: 10.1016/0022-0396(81)90016-4 · Zbl 0431.35008 · doi:10.1016/0022-0396(81)90016-4
[3] DOI: 10.1512/iumj.1977.26.26029 · Zbl 0368.35040 · doi:10.1512/iumj.1977.26.26029
[4] DOI: 10.1016/0022-0396(81)90001-2 · Zbl 0431.92010 · doi:10.1016/0022-0396(81)90001-2
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[9] Terman, Comparison theorems for reaction-diffusion systems defined in an unbounded domain (1982)
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