The Existence of Periodic Solutions to Reaction-Diffusion Systems with Periodic Data

Abstract

Abstract. The existence of time-periodic solutions is proven for a large class of reactiondiffusion systems in which Dirichlet boundary data, diffusivities, and reaction rates are periodic with common period.

Mathematical Analysis and Pattern Formation for a Partial Immune System Modeling the Spread of an Epidemic Disease

“…The proof of Theorem 4 is very close to the results presented by Morgan et al [14][15][16]. For clarity, we describe briefly the several steps to show existence of strong solutions and boundness of solutions.…”

Controllability Methods for the Computation of Time-Periodic Solutions; Application to Scattering

“…Functional J 1 leads to satisfactory computational results as shown in [3][4][5][6]9] (and also in Section 7.6 of this article); however, it has been shown in [8] that this functional may fail to be strongly "elliptic" (coercive) in some situations with trapping rays. As a cure, Bardos and Rauch have proposed in [8] the functional J 2 , described just below, which has better coercivity properties than J 1 .…”

Periodicity in diffusive age-structured SEIR models

“…Then, using results from [14], we can show that all of our results obtained above are still valid. This more general approach will be the subject of future work.…”

“…System (2.16a)-(2.16e) may be viewed as a system of reaction-diffusion type involving delays. The recent paper [14] establishes criteria sufficient to guarantee the existence of periodic solutions to reaction diffusion systems. It would have been possible to adapt the techniques from [14] and to use duality arguments coupled with an iteration producing L p space-time cylinder bounds for an increasing sequence of p's to obtain the desired a priori L^ bound for £(x,T*,a).…”

“…In [13], periodic solutions are shown for a spatially-dependent Gurtin-MacCamy model with periodic coefficients and a periodic external supply term, and the results are applied to a non-fatal SIR epidemic model. Our work juxtaposes techniques recently developed for diffusive, age-dependent epidemic models [2,[4][5][6] with methods for periodicity in general reaction diffusion systems [14].…”