An Introduction to Barenblatt Solutions for Anisotropic $p$-Laplace Equations

TL;DR

This paper introduces Barenblatt-type fundamental solutions for anisotropic p-Laplace equations with different exponents and demonstrates their significance in understanding the regularity of solutions.

Contribution

It presents the first construction of Barenblatt solutions for anisotropic p-Laplace equations with varying exponents and explores their role in regularity analysis.

Findings

Construction of Barenblatt solutions for anisotropic p-Laplace equations.

Establishment of the importance of these solutions in regularity theory.

Insights into the behavior of solutions with different anisotropic exponents.

Abstract

We introduce Fundamental solutions of Barenblatt type for the equation $u_t=\sum_{i=1}^N \bigg( |u_{x_i}|^{p_i-2}u_{x_i} \bigg)_{x_i}$, $p_i >2 \quad \forall i=1,..,N$, on $\Sigma_T=\mathbb{R}^N \times[0,T]$, and we prove their importance for the regularity properties of the solutions.