Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting

Dorothea-Enrica von Criegern, Gabriele Grillo, Dario Monticelli

TL;DR

This work proves nonexistence of global nonnegative weak solutions for a broad class of parabolic inequalities on complete noncompact Riemannian manifolds, incorporating Porous Medium, p-Laplacian, and doubly nonlinear diffusion with a (time-dependent) potential. The authors develop a test-function framework, deriving integral a priori estimates that tie the solution's growth to weighted volume growth and the potential’s behavior at infinity; two main hypotheses HP1 and HP2 encode these geometric-analytic conditions. Under the condition $q>\max(p+m-2,1)$, the results show that any global solution must vanish, extending nonexistence phenomena to manifold settings without curvature constraints but with controlled weighted volume growth. The outcomes yield concrete Euclidean corollaries and cover time-dependent and separated potentials, providing a unifying approach to nonexistence in reaction-diffusion-type parabolic problems on manifolds with variable geometry and diffusion mechanisms.

Abstract

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a potential term. Our results reveal the interplay between the manifold's geometry, the power nonlinearity, and the potential's behavior at infinity. Using a test function argument, we identify explicit parameter ranges where nonexistence holds.

Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting

TL;DR

This work proves nonexistence of global nonnegative weak solutions for a broad class of parabolic inequalities on complete noncompact Riemannian manifolds, incorporating Porous Medium, p-Laplacian, and doubly nonlinear diffusion with a (time-dependent) potential. The authors develop a test-function framework, deriving integral a priori estimates that tie the solution's growth to weighted volume growth and the potential’s behavior at infinity; two main hypotheses HP1 and HP2 encode these geometric-analytic conditions. Under the condition , the results show that any global solution must vanish, extending nonexistence phenomena to manifold settings without curvature constraints but with controlled weighted volume growth. The outcomes yield concrete Euclidean corollaries and cover time-dependent and separated potentials, providing a unifying approach to nonexistence in reaction-diffusion-type parabolic problems on manifolds with variable geometry and diffusion mechanisms.

Abstract

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a potential term. Our results reveal the interplay between the manifold's geometry, the power nonlinearity, and the potential's behavior at infinity. Using a test function argument, we identify explicit parameter ranges where nonexistence holds.
Paper Structure (7 sections, 10 theorems, 142 equations, 1 table)

This paper contains 7 sections, 10 theorems, 142 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Applications
  4. Proof of Theorems \ref{['thm:Thm1']} and \ref{['thm:Thm2']}
  5. Proof of Theorem \ref{['thm:Thm1']}
  6. Proof of Theorem \ref{['thm:Thm2']}
  7. Proof of Corollaries \ref{['thm:cor1']}-- \ref{["thm:lastcor'"]}

Key Result

Theorem 1.1

Let $p>1$, $m\geqslant1$, $q>\max(p+m-2, 1)$ and $V>0$ a.e. in $M\times (0,\infty)$, $V\in L^1_{loc}(M\times (0,\infty))$. Let further $0\leqslant f \leqslant K$ for some $K>0$ and $a\in\mathrm{Lip}_{\mathrm{loc}}(M)$, $a>0$ a.e. in $M$. If $u$ is a global nonnegative solution to Problem main eq and

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • ...and 18 more