Slow growth of solutions of super-fast diffusion equations with unbounded initial data

TL;DR

This paper investigates the behavior of positive solutions to super-fast diffusion equations with unbounded initial data, revealing how their slow growth over time depends on initial spatial growth, and introduces new self-similar solutions.

Contribution

It provides an explicit relationship between initial spatial growth and temporal solution growth, and introduces a new class of self-similar solutions for super-fast diffusion equations.

Findings

Solutions exhibit slow temporal growth depending on initial spatial growth

A new class of self-similar solutions is identified

Explicit dependence of growth rate on initial data is established

Abstract

We study positive solutions of the super-fast diffusion equation in the whole space with initial data which are unbounded as $|x|\to\infty$. We find an explicit dependence of the slow temporal growth rate of solutions on the initial spatial growth rate. A new class of self-similar solutions plays a significant role in our analysis.