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Global existence for certain fourth order evolution equations

Rafael Granero-Belinchón, Martina Magliocca

TL;DR

This work establishes global-in-time weak solutions for two fourth-order nonlinear parabolic PDEs arising in epitaxial growth and thin-film dynamics, using Wiener spaces $A^s$ as the functional framework. The authors build spectral-approximation schemes, derive sharp a priori estimates in $A^s$ spaces, and apply compactness (via Simon-type results) to pass to the limit in nonlinear terms, obtaining global existence and decay under explicit smallness conditions. A key contribution is proving global existence with initial data in the scale-critical space $A^0$, leveraging the equation’s inherent scaling $u_\lambda(x,t)=u(\lambda x,\lambda^4 t)$. This framework both extends prior energy-based results and provides a robust method for handling high-order parabolic systems with nonlinear diffusion and Hessian-type terms.

Abstract

In this paper we establish three global in time results for two fourth order nonlinear parabolic equations. The first of such equations involves the Hessian and appears in epitaxial growth. For such equation we give conditions ensuring the global existence of solution. For certain regime of the parameters, our size condition involves the norm in a critical space with respect to the scaling of the equation and improves previous existing results in the literature for this equation. The second of the equations under study is a thin film equation with a porous medium nonlinearity. For this equation we establish conditions leading to the global existence of solution.

Global existence for certain fourth order evolution equations

TL;DR

This work establishes global-in-time weak solutions for two fourth-order nonlinear parabolic PDEs arising in epitaxial growth and thin-film dynamics, using Wiener spaces as the functional framework. The authors build spectral-approximation schemes, derive sharp a priori estimates in spaces, and apply compactness (via Simon-type results) to pass to the limit in nonlinear terms, obtaining global existence and decay under explicit smallness conditions. A key contribution is proving global existence with initial data in the scale-critical space , leveraging the equation’s inherent scaling . This framework both extends prior energy-based results and provides a robust method for handling high-order parabolic systems with nonlinear diffusion and Hessian-type terms.

Abstract

In this paper we establish three global in time results for two fourth order nonlinear parabolic equations. The first of such equations involves the Hessian and appears in epitaxial growth. For such equation we give conditions ensuring the global existence of solution. For certain regime of the parameters, our size condition involves the norm in a critical space with respect to the scaling of the equation and improves previous existing results in the literature for this equation. The second of the equations under study is a thin film equation with a porous medium nonlinearity. For this equation we establish conditions leading to the global existence of solution.
Paper Structure (4 sections, 3 theorems, 93 equations)

This paper contains 4 sections, 3 theorems, 93 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorems \ref{['teo:ex-hessianA2']} and \ref{['teo:ex-hessianA0']}
  3. Proof of Theorem \ref{['teo:ex-porous']}
  4. Conclusion

Key Result

Theorem 1.3

Let $K_3>0$ in eq:pb-hessian and consider $u_0\in A^2$ be a zero mean initial data such that and Then there exists at least one global weak solution of eq:pb-hessian in the sense of Definition def:ex-hessian Furthermore, the solution satisfies and

Theorems & Definitions (5)

  • Definition 1.1: Weak solution to \ref{['eq:pb-hessian']}
  • Definition 1.2: Weak solution to \ref{['eq:pb-porous-thin-v']}
  • Theorem 1.3: Existence result to \ref{['eq:pb-hessian']} for $K_3>0$
  • Theorem 1.4: Existence result to \ref{['eq:pb-hessian']} for $K_3=0$
  • Theorem 1.5: Existence result to \ref{['eq:pb-porous-thin-v']}