Unstable Periodic Solutions of a Parabolic Problem in the Presence of Non-well-ordered Lower and Upper Solutions

Abstract

We prove the solvability of the Dirichlet-periodic problem for a semilinear parabolic equation, assuming the existence of a lower solution : and an upper solution ;, which do not satisfy the ordering condition : ;. Our results yield the existence of unstable solutions, localized by means of : and ;, requiring weak regularity of the coefficients of the equation. Academic Press

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

“…We introduce a concept of order stability and order instability, adapted to the present setting from [24,Chapter I]. Our analysis follows patterns developed in [17,18,16].…”

“…Here we use the notion of order stability: for a discussion of the relationships between this concept and other classical ones considered in the literature we refer to [24,18,21]. It is worthy to point out that our stability, or instability, conclusions will follow, as in [17,18,16], without assuming any additional regularity hypotheses on the function f besides the L ∞ -Carathéodory conditions.…”

Lower and upper solutions for elliptic problems in nonsmooth domains

“…More recently, this kind of result was generalized to consider unbounded perturbations of λ 1 u (see C. De Coster and M. Henrard [8] and the references therein, as well as [9] for other types of extensions in the framework of parabolic problems).…”

“…In the non-well-ordered case we consider only situations where "f is a bounded perturbation of the first eigenvalue" as in [4]. Extension of the asymptotic behaviour can be done arguing as in [8,9]. Using an argument of [7], we prove also an extension of Amann's three solutions theorem asking less ordering relations between the lower and upper solutions.…”

Periodic solutions of a class of degenerate parabolic system with delays

“…Periodic semi-linear reaction diffusion equations are of particular interests since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and have been extensively studied by many researchers (of [1,2,4,7,12,17,[24][25][26]28,29]). At the same time, much attentions have also been paid to periodic semi-linear parabolic equations with delays where time lag is taken into consideration in nonlinear reaction functions (of [11,21,27,31,[33][34][35]).…”