[Paper Review] On Lipschitz solutions for some forward-backward parabolic equations

Seonghak Kim, Baisheng Yan|arXiv (Cornell University)|Jun 18, 2015

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes the existence of infinitely many Lipschitz solutions to forward-backward parabolic equations with non-monotone diffusion fluxes, using a Baire category method applied to partial differential inclusions. The key contribution is proving such solutions exist under a density condition, which is realized in Perona-Malik and Höllig-type models, including micro-oscillatory behavior.

ABSTRACT

We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona-Malik model in image processing and the other that of Höllig's model related to the Clausius-Duhem inequality in the second law of thermodynamics.

Motivation & Objective

• To establish the existence of Lipschitz solutions for forward-backward parabolic equations with non-monotone diffusion fluxes.
• To address the ill-posedness of initial-boundary value problems in such equations due to backward parabolicity.
• To extend existence results beyond classical parabolic theory by leveraging convex integration and density conditions.
• To demonstrate the coexistence of radial and non-radial Lipschitz solutions in symmetric domains.
• To validate the approach on physically relevant models: Perona-Malik for image processing and Höllig-type for thermodynamics.

Proposed method

• Reformulate the forward-backward parabolic equation as a partial differential inclusion using the flux structure $ A(p) = f(|p|^2)p $.
• Apply the Baire category method to prove existence of solutions in a residual $ G_ au $-set, ensuring dense existence in a Baire space.
• Use a density condition on the flux profile $ \sigma(s) = s f(s^2) $ to guarantee existence of infinitely many Lipschitz solutions.
• Construct approximate solutions via a regularized flux $ \tilde{A}(p) = \tilde{f}(|p|^2)p $ with $ \tilde{f} \in C^{1+\alpha} $, ensuring classical solvability.
• Perform spatial surgery on a solution $ u^* $ in a space-time box away from the central axis to break radial symmetry and generate non-radial solutions.
• Use contradiction in the $ L^\infty $-limit to show that if only finitely many non-radial solutions existed, the non-radial function $ u^*_{nr} $ would be approximated by radial ones, violating symmetry breaking.

Experimental results

Research questions

• RQ1 Under what conditions does the initial-boundary value problem for a forward-backward parabolic equation admit Lipschitz solutions?
• RQ2 Can the Baire category method be applied to prove existence of infinitely many solutions in the context of non-monotone PDEs?
• RQ3 Does the density condition on the flux profile $ \sigma(s) $ ensure existence of Lipschitz solutions in Perona-Malik and Höllig-type models?
• RQ4 Can both radial and non-radial Lipschitz solutions coexist for the same initial-boundary value problem in a ball?
• RQ5 What is the role of micro-oscillatory behavior in the solutions of such equations?

Key findings

• Infinitely many Lipschitz solutions exist for the initial-boundary value problem (1.2) when the initial gradient satisfies $ |Du_0(x_0)| \in (s_1^*, s_2^*) $, as guaranteed by Theorem 3.5.
• For initial data with $ \|Du_0\|_{L^\infty(\Omega)} \leq s_1^* $, infinitely many Lipschitz solutions are constructed by patching a classical solution $ u^* $ with a new solution from Theorem 3.5 after a time $ \bar{t} $ where $ |Du^*| \in (s_1^*, s_2^*) $.
• When $ \min_{\bar{\Omega}} |Du_0| \geq s_2^* $, a similar patching procedure yields infinitely many Lipschitz solutions if $ |Du^*| $ enters the interval $ (s_1^*, s_2^*) $.
• In the case of a radial initial datum $ u_0 $ on a ball $ \Omega = B_R(0) $, there exist infinitely many non-radial Lipschitz solutions, as shown by breaking radial symmetry via surgery on a solution $ u^* $.
• The existence of infinitely many non-radial solutions contradicts the possibility of only finitely many such solutions, proving their abundance via topological argument in the $ L^\infty $-limit.
• The results are realized in two key physical models: the Perona-Malik model (image denoising) and Höllig’s model (thermodynamic phase transitions), both satisfying the required density condition.