On The Method of Lower and Upper Solutions for the Heat Equation on a Polygonal Domain
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Date
2016-07
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Addis Ababa University
Abstract
The purpose of this paper is to prove the existence of a solution in the
presence of lower and upper solutions for the nonlinear periodic-Dirichlet
heat equation on a polygonal domain Ω of the plane in weighted Lp-Sobolev
spaces.
Consider the problem;
∂tu − Δu = f(x, t, u), in Ω × (−π, π),
u = 0, on ∂Ω × (−π, π),
u(•,−π) = u(•, π) in Ω,
loc (Ω)}, with a real parameter μ and r(x)
the distance from x to the set of corners of Ω. We prove some existence
results of this problem in presence of lower and upper solutions well-ordered
or not. We first give existence results in an abstract setting obtained using
degree theory. We secondly apply them for polygonal domains of the plane
under geometrical constraints
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Keywords
Method of Lower and Upper Solutions