Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms

Abstract

In this paper, the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of Leray-Schauder's degree.

Local and Global Existence and Blow-Up of Solutions to a Polytropic Filtration System With Nonlinear Memory and Nonlinear Boundary Conditions

“…When (p 1 , p 2 ) = (2, 2) and (m 1 , m 2 ) = (1, 1) the connection with the flow in porous media is by now classical. When (m 1 , m 2 ) ≥ (1, 1) and (p 1 , p 2 ) > (2, 2), the system models the non-stationary, polytropic flow of a fluid in a porous medium whose tangential stress has a power dependence on the velocity of the displacement under polytropic conditions (non-Newtonian elastic filtration); it has been intensively studied (see [16,17,21] and references therein). The nonlocal growth terms present a more realistic model of a population [6,10,14,18].…”

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…Similar topological methods are also employed to a great extent for the existence of non-negative periodic solutions of degenerate and doubly degenerate parabolic equations, see [3], [9], [20], [30], [31], [38], [42], [44], [45], [48], [55], [56], [58], [59], [60], [61], [62], [63], [64], [67], [68]. Nonlocal models to study aggregation in biological systems with degenerate diffusion are proposed in several papers, see the recent [12], [43] and the references therein.…”

“…[3], [5], [6], [7], [21], [22], [25], [26], [29], [31], [35], [42], [50], [56], [57], [60], [67], [68]. We also recall the related problems faced in [23] and [24] also for higher order operators, and in [19] for p = 2 and N = 1.…”

Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources

“…When p ¼ 2 and m > 1 the connection with the flow in porous media is by now classical. When m P 1 and p > 2, the equation models the non-stationary, polytropic flow of a fluid in a porous medium whose tangential stress has a power dependence on the velocity of the displacement under polytropic conditions (non-Newtonian elastic filtration); it has been intensively studied (see [24,25,33,38,44] and references therein). We refer to [7] for further information on these phenomena.…”

“…During the recent years, many authors have focused their eyes on the problems of semilinear and quasilinear equations with nonlocal terms, see, for example, [2,9,10,12,46]. While, due to the relevant connections to gas or fluid flows media and population dynamics, periodic problems for quasilinear and doubly degenerate parabolic equations with nonlocal terms have been the subject of extensive study, see [1,5,25,48,38] and references therein.…”