Abstract
We prove a result on existence and uniqueness of weak solutions for a diffusion problem associated with nonlinear diffusions of nonlocal type studied by Chipot and Lovat (1999) by an application of the fixed point result of Schauder. Moreover, making use of Faedo‐Galerkin approximation, coupled with some technical ideas, we establish a result on existence of periodic solution.
Highlights
In this work, we are going to study some questions concerning to the existence, uniqueness, and periodic solution for the parabolic problem ut − a l(u) Δu + f (u) = h in Q = Ω × (0, T), u(x, t) = 0 on Σ = Γ × (0, T), u(x, 0) = u0(x) in Ω, (1.1)where Ω is a smooth bounded open subset of RN with regular boundary Γ
We are going to present a simple extension of the results contained in [4], in which f = f (u) depends on the state u, where we study, among other things, the case in which u is periodic
Concerning problem (1.1), we will suppose that a : R → R is continuous and that for some constants m, M, 0 < m ≤ a(ξ) ≤ M, ∀ξ ∈ R, l : L2(Ω) −→ R is a continuous nonlinear form, (2.1) (2.2)
Summary
DE MENEZES Received 7 May 2005; Revised 20 September 2005; Accepted 28 November 2005 Dedicated to Professor L. A. Medeiros on the occasion of his 80th birthday. We prove a result on existence and uniqueness of weak solutions for a diffusion problem associated with nonlinear diffusions of nonlocal type studied by Chipot and Lovat (1999) by an application of the fixed point result of Schauder. Making use of Faedo-Galerkin approximation, coupled with some technical ideas, we establish a result on existence of periodic solution
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