Strong maximum principle for non-linear parabolic differential-functional inequalities in arbitrary domains

Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations

“…The maximum principle holds under much weaker conditions [26,32]; the decisive conditions are quasimonotonicity, (unilateral) Lipschitz conditions for I and the fact that the functional I does not depend on the "future". As an immediate consequence of the maximum principle we find that every nontrivial solution u of (2.1-2) under the hypotheses 2.1 will satisfy (2.6') 0< u < 1 on R x R+.…”

“…This part uses the theory of functional differential equations [16] and maximum principles for parabolic functional differential equations [26,32] and can therefore be done for more general functionals; for the sake of simplicity I restrict myself to equations (1.11) with one fixed time lag.…”

Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's Condition

“…The results concerning numerical methods, differential functional and difference functional inequalities or the uniqueness theory, appearing in the papers of P. Besala and G. Paszek [1,2], C. V. Pao [22][23][24], R. Redheffer and W. Walter [25,30], J. Szarski [27][28][29] and numerous others, do not apply to nonlinear equations and quasi-linear equations with such a general functional dependence as in our paper.…”

Representation theorems for parabolic systems

“…Let dR be a part of the boundary of R defined by The maximum cannot be attained on dR -E X {0}. Applying the maximum principle (see Szarski (1975) …”