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Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

Ahmad Z. Fino, Michael Ruzhansky, Berikbol T. Torebek

TL;DR

The paper deals with the existence of solutions of some generalized Stefan-type equation in the framework of Orlicz spaces in the context of Or Alicz spaces.

Abstract

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.

Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

TL;DR

The paper deals with the existence of solutions of some generalized Stefan-type equation in the framework of Orlicz spaces in the context of Or Alicz spaces.

Abstract

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.
Paper Structure (9 sections, 11 theorems, 210 equations)

This paper contains 9 sections, 11 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. Results on $\mathbb{R}^n$
  4. Sub-elliptic extensions
  5. Motivation
  6. Porous medium equation
  7. Degenerate parabolic equation
  8. Case of $0\leq q<1$
  9. The case of $q\geq1$

Key Result

Theorem 2.2

Let $m>1,$$\sigma>1$ and let $v_0(\eta)\geq0,\,v_0(\eta)\not\equiv0, \eta\in\mathbb{H}^n$.

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 15 more