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Convergence of semilinear parabolic flows with general initial data

Daniel Restrepo

Abstract

We analyze the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional. We prove that, for general initial data (including data without compact support) the flow converges to a unique ground state. The argument relies on a sharp stability estimate for almost critical points of the energy, providing a flexible framework for establishing convergence of gradient flows associated with constrained minimization problems in R^n. As an application, we strengthen previous convergence results of Cortazar (1999) and Feireisl (1997).

Convergence of semilinear parabolic flows with general initial data

Abstract

We analyze the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional. We prove that, for general initial data (including data without compact support) the flow converges to a unique ground state. The argument relies on a sharp stability estimate for almost critical points of the energy, providing a flexible framework for establishing convergence of gradient flows associated with constrained minimization problems in R^n. As an application, we strengthen previous convergence results of Cortazar (1999) and Feireisl (1997).
Paper Structure (14 sections, 15 theorems, 196 equations)

This paper contains 14 sections, 15 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Extension to volume constrained flows.
  4. Strategy of the proof
  5. Organization of the paper
  6. Background material
  7. Proof of the main results
  8. Setup.
  9. The $M$-bubble energy inequality and the role of assumption \ref{['eq concat0']}
  10. Linear stability and energy deficit estimate
  11. Proof of Theorem \ref{['thm1']}
  12. Quantiative weak interaction estimate
  13. Multibubble stability dichotomy:
  14. Interaction estimates and related lemmata

Key Result

Theorem 1.1

Let $n\geq 3$ and where $a_i\geq 0$, $a_0>0$, $\sum_{i=1}^La_i>0$, and $1<r<p_i < \frac{n}{n-2}$. Let $u_0 \in W^{1,2}(\mathbb{R}^n)$ with $u_0 \geq 0$ almost everywhere in $\mathbb{R}^n$ and let $u$ be a solution to eq generalflow. Then, $u$ is the unique solution to eq generalflow, $u(t,x)>0$ for $t>0$, and either or $\xi$ is the unique radial positive solution of eq gs and $x^* \in \mathbb{R

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 21 more