On The Method Of Lower And Upper Solutions For The Heat Equation On A Polygonal Domain

The purpose of this paper is to prove the existence of a solution in thernpresence of lower and upper solutions for the nonlinear periodic-Dirichletrnheat equation on a polygonal domain Ω of the plane in weighted Lp-Sobolevrnspaces.rnConsider the problem;rn∂tu − Δu = f(x, t, u), in Ω × (−π, π),rnu = 0, on ∂Ω × (−π, π),rnu(•,−π) = u(•, π) in Ω,rnloc (Ω)}, with a real parameter μ and r(x)rnthe distance from x to the set of corners of Ω. We prove some existencernresults of this problem in presence of lower and upper solutions well-orderedrnor not. We first give existence results in an abstract setting obtained usingrndegree theory. We secondly apply them for polygonal domains of the planernunder geometrical constraints

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