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Application of Hardy inequalities for some singular parabolic equations

N. Kutev, T. Rangelov

TL;DR

This work extends Hardy-inequality thresholds to a class of singular parabolic equations with four Hardy potentials that are singular on the boundary and/or at the origin. By employing truncated approximations and energy estimates anchored to optimal Hardy constants, it establishes global existence for $\mu$ below the constants and finite-time blow-up for $\mu$ above them. The analysis connects spectral thresholds (via weighted eigenvalues) with nonlinear diffusion dynamics through a separated-variable sub-solution and a comparison principle, highlighting the critical role of Hardy constants in long-time behavior. The results deepen our understanding of how boundary and origin singularities shape global solvability and blow-up in nonlinear diffusion models with singular potentials.

Abstract

Boundary value problems for non-linear parabolic equations with singular potentials are considered. Existence and non-existence results as an application of different Hardy inequalities are proved. Blow-up conditions are investigated too.

Application of Hardy inequalities for some singular parabolic equations

TL;DR

This work extends Hardy-inequality thresholds to a class of singular parabolic equations with four Hardy potentials that are singular on the boundary and/or at the origin. By employing truncated approximations and energy estimates anchored to optimal Hardy constants, it establishes global existence for below the constants and finite-time blow-up for above them. The analysis connects spectral thresholds (via weighted eigenvalues) with nonlinear diffusion dynamics through a separated-variable sub-solution and a comparison principle, highlighting the critical role of Hardy constants in long-time behavior. The results deepen our understanding of how boundary and origin singularities shape global solvability and blow-up in nonlinear diffusion models with singular potentials.

Abstract

Boundary value problems for non-linear parabolic equations with singular potentials are considered. Existence and non-existence results as an application of different Hardy inequalities are proved. Blow-up conditions are investigated too.

Paper Structure

This paper contains 3 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Existence of global solution
  3. Finite time blow-up

Key Result

Theorem 2.1

Suppose $\Omega$ is a bounded $C^2$ smooth domain in $\mathbb{R}^n$, $n\geq2$, with nonnegative mean curvature $H(x)\geq0$ and $p>1, p\neq n$. Then if $\mu<\left(\frac{p-1}{p}\right)^p$, $u_0(x)\in L^2(\Omega)$ problem eq1 with $W(x)$ in eq001i) has a global solution

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof : Proof of Theorems \ref{['th1']} - \ref{['th4']}
  • Corollary 2.1
  • proof
  • Remark 2.1
  • Lemma 3.1
  • proof
  • ...and 6 more