Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

Abstract

This article deals with the heat equation @tu @2x u = f in D; D = {(t; x) 2 R2 : a < t < b; (t) < x < +1} with the function satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f 2 L2(D) there exists a unique solution u such that u; @tu; @jx u 2 L2 (D) ; j = 1; 2: The proof is based on the domain decomposition method. This work complements the results obtained in [10].

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

“…Whereas secondorder parabolic equations in bounded non-cylindrical domains are well studied (see for instance [2,5,7,[15][16][17][18] and the references therein), the literature concerning unbounded non-cylindrical domains does not seem to be very rich. The regularity of the heat equation solution in a nonsmooth and unbounded domain (in the x direction) is obtained in [3,6,8] and [4].…”

“…The transformation (t, x) −→ (t ′ , x ′ ) = (t, φ (t) x + φ 1 (t)) leads to the following lemma: Lemma 5.2. If w ∈ H 1,2 (Ω 2 ), then w| Γ T 1 ∈ H 1 (Γ T1 ) , w| x=φ1(t) ∈ H 3 4 (Γ 1,2 ) and w| x=φ2(t) ∈ H 3 4 (Γ 2,2 ), where Γ i,2 are the parts of the boundary of Ω 2 where x = φ i (t) , i = 1, 2.…”