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Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

Pablo Àlvarez-Caudevilla, Matthieu Bonnivard, Antoine Lemenant

Abstract

In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case. We provide a strong convergence result for the solution by use of energetic methods and $Γ$-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

Abstract

In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case. We provide a strong convergence result for the solution by use of energetic methods and -convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.
Paper Structure (12 sections, 14 theorems, 116 equations)

This paper contains 12 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. The stationary problem
  3. Existence and regularity of solutions for $(P_\lambda)$
  4. Uniqueness of solution for $(P_\infty)$
  5. Convergence of $u_\lambda$
  6. First energy bound
  7. Second energy bound
  8. Weak convergence of solutions
  9. Strong convergence in $L^2(0,T;H^1(\Omega))$
  10. Simon's exponential estimate
  11. The stationary case
  12. The parabolic case

Key Result

Theorem 1

For all $\lambda >0$, let $u_\lambda$ be the solution of $(P_\lambda)$ with $f_\lambda \in L^2(\Omega\times (0,T))$ and $g_\lambda \in H_0^1(\Omega)$. Assume that $a:\Omega \times [0,T]\to \mathbb{R}^+$ is a Lipschitz function which satisfies Assume also that the initial condition $g_\lambda$ satisfies converges weakly to $g$ in $L^2(\Omega)$, and that $f_\lambda$ converges weakly to $f$ in $L^2

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Lemma 1
  • ...and 21 more