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Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

Xinye Li, Christof Melcher

Abstract

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-Δ)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(ξ) = \pm \lvert ξ\rvert^2 ξ$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.

Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

Abstract

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation , where satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.
Paper Structure (12 sections, 19 theorems, 151 equations)

This paper contains 12 sections, 19 theorems, 151 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Well-posedness in local BMO spaces
  4. Basic estimates for the biharmonic heat equation
  5. Existence and uniqueness
  6. Higher regularity
  7. Stability of global solutions
  8. Main idea
  9. The linear operator $\mathcal{L}_{a,b}$
  10. Asymptotics of $u$
  11. The coercive case
  12. The non-coercive case

Key Result

Theorem 1

Suppose $F$ satisfies eq:lip. There exists $\rho>0$ with the following property:

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • ...and 23 more