Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Arezki Kheloufi
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Maximal Regularity ◽  
Dirichlet Boundary Conditions ◽  
Regular Domain

Abstract.New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain

scholarly journals Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

2020 ◽  
pp. 257-274 ◽  
Author(s):  
Louanas Bouzidi ◽  
Arezki Kheloufi
Keyword(s):  
Parabolic Equation ◽  
Sobolev Space ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Dirichlet Boundary ◽  
Global Regularity ◽  
Interpolation Inequality ◽  
Dirichlet Boundary Conditions ◽  
Energy Estimates ◽  
Equation Solution

This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions

scholarly journals A MATHEMATICAL MODEL FOR THE USE OF ENERGY RESOURCES: A SINGULAR PARABOLIC EQUATION

2020 ◽  
Vol 25 (1) ◽  
pp. 88-109
Author(s):  
Daniel López-García ◽  
Rosa Pardo
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Initial Data ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Fixed Time ◽  
Dirichlet Boundary Conditions ◽  
Boundary Layer Flows ◽  
Singular Parabolic Equation

We consider a singular parabolic equation tβut − ∆u = f, for (x,t)∈ Ω × (0,T), arising in symmetric boundary layer flows. Here Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω,β ≤ 1,f = f(t,x) is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data u0 ϵ L2(Ω), there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem tut − ∆u = f. The initial data cannot be arbitrarily chosen. In fact, if f is continuous and f(t) → f0, as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem −∆u = f0, for x ∈ Ω, with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.

scholarly journals On weak solutions of semilinear hyperbolic-parabolic equations

1996 ◽  
Vol 19 (4) ◽  
pp. 751-758 ◽  
Author(s):  
Jorge Ferreira
Keyword(s):  
Parabolic Equation ◽  
Continuous Function ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
A Priori ◽  
A Priori Estimate ◽  
Dirichlet Boundary Conditions ◽  
Priori Estimate

In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation(K1(x,t)u′)′+K2(x,t)u′+A(t)u+F(u)=fwith null Dirichlet boundary conditions and zero initial data, whereF(s)is a continuous function such thatsF(s)≥0,∀s∈Rand{A(t);t≥0}is a family of operators ofL(H01(Ω);H−1(Ω)). For the existence we apply the Faedo-Galerkin method with an unusual a priori estimate and a result of W. A. Strauss. Uniqueness is proved only for some particular classes of functionsF.

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