Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with arbitrary functionals. (English) Zbl 0737.35135

The paper looks for a new object for which an analogue of the maximum principle for solutions of parabolic equations is valid. The case of noncylindrical domains and nonlocal parabolic inequalities of the type \[ u^ i_ t(x,t)\leq f^ i(x,t,u(x,t), u^ i_ x(x,t), u^ i_{xx}(x,t);[u])\hbox { for a.e. } (x,t) \] \(i=1,\ldots,m\); \(u=(u^ 1,\ldots,u^ m)\), with some additional nonlocal assumptions is discussed. Here \(f^ i(\cdots;[u])\) are functionals with respect to \(u\).
Reviewer: U.Raitums (Riga)


35R10 Partial functional-differential equations
35B50 Maximum principles in context of PDEs
Full Text: DOI


[1] Byszewski, L., Strong maximum principle for implicit nonlinear parabolic functional-differential inequalities in arbitrary domains, Univ. Iagel. Acta. Math., 24, 327-339 (1984) · Zbl 0555.35130
[2] Byszewski, L., Strong maximum and minimum principles for parabolic functional-differential problems with initial inequalities \(u(t_0, x)_{(⩾)}^⩽K\), Ann. Polon. Math., 52, 79-85 (1990)
[3] Byszewski, L., Strong maximum and minimum principles for parabolic problems with nonlocal inequalities, Z. Angew. Math. Mech., 70, 202-206 (1990) · Zbl 0709.35018
[4] Byszewski, L., Strong maximum principles for parabolic nonlinear problems with non-local inequalities together with integrals, J. Appl. Math. Stochastic. Ann., 3, 65-79 (1990) · Zbl 0726.35023
[5] Chabrowski, J., On non-local problems for parabolic equations, Nagoya Math. J., 93, 109-131 (1984) · Zbl 0506.35048
[6] Chabrowski, J., On the non-local problem with a functional for parabolic equation, Funkcial. Ekvac., 27, 101-123 (1984) · Zbl 0568.35046
[7] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston/London/Melbourne · Zbl 0658.35003
[8] Lakshmikantham, V.; Leela, S., Reaction-diffusion systems and vector Lyapunow functions, Differential Integral Equations, 1, 41-47 (1988) · Zbl 0727.35069
[9] Redheffer, R.; Walter, W., Das Maximumprinzip in unbeschränkten Gebieten für parabolische Ungleichungen mit Funktionalen, Math. Ann., 226, 155-170 (1977) · Zbl 0328.35005
[10] Szarski, J., Strong maximum principle for nonlinear parabolic differential-functional inequalities in arbitrary domains, Ann. Polon. Math., 29, 207-217 (1974)
[11] Walter, W., On the strong maximum principle for parabolic differential equations, (Proc. Edinburgh Math. Soc., 29 (1986)), 93-96, (2) · Zbl 0603.35039
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.