Geodesic
Comparison principle for weakly-coupled non-cooperative elliptic and parabolic systems
Serdica Mathematical Journal, Tome 45 (2019) no. 2, pp. 143-156.

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In this review article is considered the comparison principle for linear and quasi-linear weakly coupled systems of elliptic and of parabolic PDE. It is demonstrated that a cooperativeness is a kind of a watershed quality for the comparison principle. Roughly speaking comparison principle holds for cooperative systems, while it does not hold for every noncooperative one. Considering a cooperative system one can apply the theory of a positive operator in a positive cone and prove the validity of the comparison principle. One particularly important result for cooperative systems is the existence of positive first eigenvalue and positive first eigenvector. Investigation of the validity of the comparison principle for non-cooperative system is more complicated. In this paper is mentioned the idea of division of the non-cooperative system in a cooperative and competitive part. Then the spectral properties of the cooperative part are employed in order to derive conditions for validity of comparison principle for the non-cooperative system. Some applications of comparison principle are given as well.
Keywords: Comparison principle, elliptic systems, parabolic systems, cooperative and noncooperative systems, 35J47, 35K40 
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     author = {Boyadzhiev Georgi, Kutev Nikolay},
     title = {Comparison principle for weakly-coupled non-cooperative elliptic and parabolic systems},
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     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SMJ2_2019_45_2_a3/}
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Boyadzhiev Georgi, Kutev Nikolay. Comparison principle for weakly-coupled non-cooperative elliptic and parabolic systems. Serdica Mathematical Journal, Tome 45 (2019) no. 2, pp. 143-156. http://geodesic.mathdoc.fr/item/SMJ2_2019_45_2_a3/