Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

Sallah Eddine Boutiah; Abdelaziz Rhandi; Cristian Tacelli

doi:10.3934/dcds.2019033

Article Contents

2019, Volume 39, Issue 2: 803-817. Doi: 10.3934/dcds.2019033

This issue Previous Article Determination of initial data for a reaction-diffusion system with variable coefficients Next Article On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces

Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

1.
Department of Mathematics, University Ferhat Abbas Setif-1, Setif 19000, Algeria
2.
Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo Ⅱ, 132, 84084 Fisciano (SA), Italy

* Corresponding author: Abdelaziz Rhandi
* Corresponding author: Abdelaziz Rhandi

Received: November 30, 2017

Revised: May 31, 2018

Published: January 2019

This work has been supported by the M.I.U.R. research project Prin 2015233N54 "Deterministic and Stochastic Evolution Equations". The second and the third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

In this paper we prove that the heat kernel $k$ associated to the operator $A: = (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ satisfies

$\begin{eqnarray*}k(t,x,y) &\leq&c_1e^{\lambda_0 t+ c_2t^{-\gamma}}\left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(\beta-\alpha)}}{1+|y|^\alpha}\\&&\times\exp\left(-\frac{\sqrt{2}}{\beta-\alpha+2}\left(|x|^{\frac{\beta-\alpha+2}{2}}+ |y|^{\frac{\beta-\alpha+2}{2}}\right)\right)\end{eqnarray*}$

for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $\lambda_0$ is the largest eigenvalue of the operator $A$, and $\gamma = \frac{\beta-\alpha+2}{\beta+\alpha-2}$, in the case where $N>2,\,\alpha>2$ and $\beta>\alpha -2$. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.

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Mathematics Subject Classification: Primary: 35K08, 35K20; Secondary: 35J10, 47D06.

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