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Nonexistence results for parabolic equations involving the \({\varvec{p}}\)-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds

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Abstract

The main goal of this paper is twofold. The first one is to investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation on a noncompact Riemannian manifold M,

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u}{\partial t}=\varDelta _{p,g} u+V(x)u^{p-1}+ \lambda u^q &{} \text {in}\quad \varOmega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\ge 0 &{} \text {in} \quad \varOmega ,\\ u(x,t)=0 &{} \text {on } \partial \varOmega \times (0,T), \end{array}\right. } \end{aligned}$$

where \(1<p<2\), \(V\in L_ {\text {loc}}^1(\varOmega ) \), \(q>0 \), \(\lambda \in {\mathbb {R}}\), \(\varOmega \) is bounded and has a smooth boundary in M and \(\varDelta _{p,g}\) is the p-Laplacian on M. The second one is to obtain Hardy- and Leray-type inequalities with remainder terms on a Riemannian manifold M that provide us concrete potentials to use in the partial differential equation we are interested in. In particular, we obtain explicit (mostly sharp) constants for these inequalities on the hyperbolic space \({\mathbb {H}}^n\).

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Authors and Affiliations

  1. Department of Mathematics, University of Memphis, Memphis, TN, 38152, USA

    Gisèle Ruiz Goldstein & Jerome A. Goldstein

  2. Department of Mathematics, Faculty of Humanities and Social Sciences, Istanbul Commerce University, Beyoglu, Istanbul, Turkey

    Ismail Kömbe

  3. Engineering Faculty, KTO Karatay University, Konya, Turkey

    Sümeyye Bakim

Authors
  1. Gisèle Ruiz Goldstein
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  2. Jerome A. Goldstein
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  3. Ismail Kömbe
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  4. Sümeyye Bakim
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Correspondence to Ismail Kömbe.

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Dedicated to Matthias Hieber on the occasion of his 60th birthday.

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Goldstein, G.R., Goldstein, J.A., Kömbe, I. et al. Nonexistence results for parabolic equations involving the \({\varvec{p}}\)-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds. J. Evol. Equ. 21, 3675–3701 (2021). https://doi.org/10.1007/s00028-021-00691-5

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