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Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile

Danielle Hilhorst, Sabrina Roscani, Piotr Rybka

Abstract

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.

Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile

Abstract

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.
Paper Structure (7 sections, 9 theorems, 70 equations, 1 figure)

This paper contains 7 sections, 9 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On the content of the paper
  3. Existence and uniqueness of the solution
  4. Self-similar, lower and upper solutions
  5. Self-similar solution
  6. Lower and upper solutions
  7. Convergence

Key Result

Theorem 1

Suppose that $(u_0, b_0)$ satisfies the conditions Let $(u,s)$ be the corresponding solution of Problem (NSP). Then, 1) $\lim\limits_{t\to\infty}s(t)/\sqrt{t+1} = \omega$; 2) ${\lim\limits_{t\to\infty}\sup_{x/\sqrt{t+1}\in[0,\omega]} \left| u(x,t) - U\left(\frac{x}{\sqrt{t+1}}\right)\right|} =0.$

Figures (1)

  • Figure 1: Lower and upper solutions

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Definition 1
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • ...and 6 more