Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms

Genni Fragnelli, Paolo Nistri, Duccio Papini
2011 Discrete and Continuous Dynamical Systems. Series A  
The aim of the paper is to provide conditions ensuring the existence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satisfying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here L 2 (Q T ), on the solutions of certain homotopies.
more » ... his is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biological species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates. and we look for continuous weak solutions. Here m,p [u] := u t − div(|∇u m | p−2 ∇u m ) ( n,q [v] is similarly defined), Ω is an open bounded domain of R N with smooth 2000 Mathematics Subject Classification. Primary: 35K65, 35B10; Secondary: 47H11. Key words and phrases. Doubly degenerate parabolic equations, non-negative periodic solutions, topological degree. Research supported by the grant P.R.I.N. 2008 "Controllo Nonlineare: metodi geometrici e applicazioni". 1 2 GENNI FRAGNELLI, PAOLO NISTRI AND DUCCIO PAPINI boundary ∂Ω, Q T := Ω × (0, T ), T > 0, τ i ∈ (0, +∞), m, n > 1, s m = |s| m−1 s, p, q > 2, and K i , a, b ∈ L ∞ (Q T ), i = 1, 2, 3, 4, are extended to Ω×R by T -periodicity. Let A[u] := div(|∇u m | p−2 ∇u m ) and observe that Received xxxx 20xx; revised xxxx 20xx.
doi:10.3934/dcds.2011.31.35 fatcat:eoa7fuzxgzcotauonnokullije