Weak Periodic Solution for Semilinear Parabolic Problem with Singular Nonlinearities and $$L^{1}$$ L 1 Data
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Weak Periodic Solution for Semilinear Parabolic Problem with Singular Nonlinearities and L1 Data Abderrahim Charkaoui
and Nour Eddine Alaa
Abstract. We consider a periodic parabolic problem involving singular nonlinearity and homogeneous Dirichlet boundary condition modeled by f ∂u − Δu = γ in QT , ∂t u where T > 0 is a period, Ω is an open regular bounded subset of RN , QT =]0, T [×Ω,γ ∈ R and f is a nonnegative integrable function periodic in time with period T . Under a suitable assumptions on f , we establish the existence of a weak T-periodic solution for all ranges of value of γ. Mathematics Subject Classification. 35B10, 35D30, 35K59, 35K55. Keywords. Periodic solutions, Singular nonlinearity, Weak solutions.
1. Introduction Singular equations present an important role in the development of the mathematical analysis of partial differential equations. As well known these field of boundary value problems arise frequently in the modeling of many real phenomena including, among many others, fluid mechanics, pseudoplastic flow, chemical reactions(the resistivity of the material), nerve impulses (Fitzhugh– Nagumo problems), population dynamics (Lotka–Volterra systems), combustion, morphogenesis, genetics, etc. At the same time, a lot of works are presented (see, e.g. [5,6,8,9,12,16,21,22]) to answer the often asked questions on the existence, uniqueness, regularity and the asymptotic behavior of the solution for the considered problem. Concerning the periodic parabolic problem, there have been manifold papers focusing separately on the existence of classical and weak solutions under either Dirichlet or Neumann boundary conditions (see [1–4,10,11,14,18,20]), these works describe different techniques of functional analysis and numerous methods have been used, we cite Schauder’s fixed point theory, topological degree, the technique of sub and supersolution and the penalty method. As far as we know, only a few papers treat the 0123456789().: V,-vol
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A. Charkaoui, N. E. Alaa
MJOM
parabolic problem with singular nonlinearity related to the periodic case, so all these factors motivated us to take an interest in the analysis of singular periodic problems. The purpose of this work is to study the existence of a non-negative weak periodic solution of the singular parabolic problem modeled by the following equation ⎧ f ∂u ⎪ ⎨ − Δu = γ in QT ∂t u (1) u(0, .) = u(T, .) in Ω ⎪ ⎩ u(t, x) = 0 on ΣT , where Ω is an open regular bounded subset of RN , N ≥ 2, with smooth boundary ∂Ω, T > 0 is the period, QT =]0, T [×Ω, ΣT =]0, T [×∂Ω, γ > 0 and f is a nonnegative integrable function periodic in time with period T . If γ = 0, then problem (1) becomes linear periodic, we refer the readers to Lions (see p. 238 in [20]), where the author assumes that f belongs to L2 (0, T ; H −1 (Ω)) and established the existence and the uniqueness of a periodic solution by using the maximal monotone operators theory combined with the elliptic regularization. This result has been extended to the case where f is only in L1 (QT ) in [11
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