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

Daniel López-García,Rosa Pardo

doi:10.3846/mma.2020.9792

Abstract

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.

Highlights

The main goal of this paper is to solve a singular linear heat equation with a coefficient of the time derivative depending on t, and Dirichlet boundary conditions
Where Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω, −∞ < β ≤ 1 and T > 0 is some fixed time
In Theorems 7 and 8 we provide sufficient conditions on the source term f for the existence of a unique weak solution of (1.15) and we analyze its corresponding regularity, which again is not the usual one

Summary

Introduction

The main goal of this paper is to solve a singular linear heat equation with a coefficient of the time derivative depending on t, and Dirichlet boundary conditions. The case β = 1 is a limit case appearing in symmetric boundary layers like (1.17), when the tangential velocity w(0) = 0, it is is such that w/(tz) remains positive for t > 0, and 0 < z < l, so that the underlying operator is essentially t(∂/∂t) − (1/z)(∂2/∂z2), or more generally, t(∂/∂t) + L, where L is a partial differential operator with respect to the normal coordinate In this situation, no initial conditions with respect to t can be prescribed, due to the singularity, and the temperature on the symmetry axis have to be obtained formally by letting t = 0 in the PDE contained in (1.17).

Main results

The standard parabolic problem