[Paper Review] Anisotropic p-Laplacian Evolution of Fast Diffusion type

Filomena Feo, Juan Luís Vázquez+1 more|arXiv (Cornell University)|May 9, 2021

Nonlinear Partial Differential Equations37 references4 citations

TL;DR

This paper establishes the existence and uniqueness of self-similar fundamental solutions for anisotropic p-Laplacian equations with fast diffusion in all directions, using symmetrization and barrier methods. It proves asymptotic convergence of finite-mass solutions to the self-similar profile under conditions $1 < p_i < 2$ and $\sum 1/p_i < (N+1)/2$, extending Barenblatt-type theory to non-homogeneous anisotropic diffusion.

ABSTRACT

We study an anisotropic, possibly non-homogeneous version of the evolution $p$-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive $L^1$ to $L^\infty$ estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

Motivation & Objective

• To develop a complete existence and regularity theory for the anisotropic p-Laplacian evolution equation with fast diffusion in all directions.
• To establish the existence of a self-similar fundamental solution (Barenblatt-type) under the condition $\sum_{i=1}^N \frac{1}{p_i} < \frac{N+1}{2}$, which ensures finite mass and proper scaling.
• To prove asymptotic convergence of all nonnegative finite-mass solutions to the self-similar profile, characterizing long-time behavior.
• To derive sharp $L^1$ to $L^\infty$ estimates, decay rates, and positivity properties using symmetrization and comparison techniques.
• To extend the classical Barenblatt theory to the anisotropic, non-homogeneous case with different $p_i$ exponents in each spatial direction.

Proposed method

• Formal derivation of the self-similar ansatz $u(x,t) = t^{-\alpha} F(x t^{-\sigma})$, reducing the PDE to a stationary doubly nonlinear anisotropic elliptic equation.
• Application of symmetrization techniques to derive $L^1$ to $L^\infty$ estimates and prove boundedness of solutions.
• Construction of upper and lower barriers to control solution decay and positivity, especially in the fast diffusion regime.
• Use of the inverse-average exponent $\bar{p} = \left(\frac{1}{N}\sum \frac{1}{p_i}\right)^{-1}$ to characterize the critical threshold for existence.
• Proof of uniqueness of the fundamental solution in the homogeneous case ($p_i = p$) via additional regularity and comparison results.
• Analysis of the profile equation $\sum_{i=1}^N \left[ \left(|(F^{m_i})_{y_i}|^{p_i-2}(F^{m_i})_{y_i}\right)_{y_i} + \alpha \sigma_i (y_i F)_{y_i} \right] = 0$ under mass conservation.

Experimental results

Research questions

• RQ1 Under what conditions on the exponents $p_i$ does a self-similar fundamental solution exist for the anisotropic p-Laplacian equation with fast diffusion?
• RQ2 How does the asymptotic behavior of finite-mass solutions relate to the self-similar profile in the anisotropic fast diffusion regime?
• RQ3 What role does the condition $\sum_{i=1}^N \frac{1}{p_i} < \frac{N+1}{2}$ play in ensuring existence and uniqueness of the fundamental solution?
• RQ4 Can symmetrization techniques yield sharp $L^\infty$ estimates in the non-homogeneous anisotropic case?
• RQ5 What are the decay rates and positivity properties of solutions under the given anisotropic fast diffusion structure?

Key findings

• A self-similar fundamental solution exists for the anisotropic p-Laplacian equation when $1 < p_i < 2$ for all $i$ and $\sum_{i=1}^N \frac{1}{p_i} < \frac{N+1}{2}$, with explicit scaling exponents derived from the mass conservation and homogeneity structure.
• Uniqueness of the fundamental solution is proven in the homogeneous case ($p_i = p$) using additional regularity and comparison arguments.
• All nonnegative finite-mass solutions converge asymptotically to the self-similar profile, establishing long-time behavior in the $L^1$-norm.
• Symmetrization techniques yield explicit $L^1$ to $L^\infty$ estimates, with the $L^\infty$ bound depending on the initial mass and the exponent $\bar{p}$.
• The solution profile satisfies a doubly nonlinear anisotropic elliptic equation involving directional derivatives with different $p_i$-Laplacian operators, and the profile decays at a rate consistent with the self-similar scaling.
• The condition $\sum_{i=1}^N \frac{1}{p_i} < \frac{N+1}{2}$ is necessary for the existence of fundamental solutions; without it, the theory breaks down due to lack of integrability or finite speed of propagation.