Existence and optimal control for periodic parabolic equations with nonlocal terms

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

“…in agreement with assumption (H 2 ) of [2]. The literature about systems of degenerate parabolic equations is scarce and, to our knowledge, there are no papers dealing with the periodic problem (for systems of nondegenerate parabolic equations see, for example, [9]).…”

“…[1], [2], [3], [4], [12], [13], [14], [17], [18] and [23]. For this reason we look at the functions a, b as control parameters, i.e.…”

“…A similar optimization problem has been considered in [2] for a single reac-tion diffusion equation of the form:…”

“…Φ(u) ≥ C u(t) The authors of [2] pointed out that monotonicity methods fail here due to the presence of the nonlocal term and thus cannot be employed to prove the existence of periodic solutions of (2) (see [2,Theorem 0]). In fact they adopted a topological approach based on a priori bounds and the continuation property of Leray-Schauder topological degree.…”

“…As already noticed, for a single equation a coercivity assumption was considered in both [2] and [14]. In particular, competitive systems (those with K 2 ≤ 0 and K 3 ≤ 0 a.e.…”