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February  2019, 39(2): 803-817. doi: 10.3934/dcds.2019033

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

Sallah Eddine Boutiah 1, , Abdelaziz Rhandi 2,, and  Cristian Tacelli 2,

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

Received  December 2017 Revised  June 2018 Published  November 2018

Fund Project: 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).

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.
Keywords: Schrödinger type operator, semigroup, heat kernel estimates.
Mathematics Subject Classification: Primary: 35K08, 35K20; Secondary: 35J10, 47D06.
Citation: Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033
References:
[1]

D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, Rev. Mat. Iberoam., 28 (2012), 879-906.  doi: 10.4171/RMI/695.  Google Scholar

[2]

S. E. Boutiah, F. Gregorio, A. Rhandi and C. Tacelli, Elliptic operators with unbounded diffusion, drift and potential terms, J. Differential Equations, 264 (2018), 2184-2204.  doi: 10.1016/j.jde.2017.10.020.  Google Scholar

[3]

A. Canale, A. Rhandi and C. Tacelli, Schrödinger-type operators with unbounded diffusion and potential terms, Annali Scuola Normale Superiore di Pisa Cl. Sci.(5), 16 (2016), 581-601.   Google Scholar

[4]

A. Canale, A. Rhandi and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, Z. Anal. Anwend., 36 (2017), 377-392.  doi: 10.4171/ZAA/1593.  Google Scholar

[5]

A. Canale and C. Tacelli, Optimal kernel estimates for a Schrödinger type operator, Riv. Mat. Univ. Parma, 7 (2016), 341-450.   Google Scholar

[6]

E. B. Davies, Heat kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[7]

T. Durante, R. Manzo and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponent, Ricerche Mat., 65 (2016), 289-305.  doi: 10.1007/s11587-016-0284-x.  Google Scholar

[8]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^{p}$ and $C_{b}$-spaces, Discrete and Continuous Dynamical Systems A, 18 (2007), 747-772.  doi: 10.3934/dcds.2007.18.747.  Google Scholar

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[10]

M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations, Advances in Mathematics, 287 (2016), 600-639.  doi: 10.1016/j.aim.2015.09.029.  Google Scholar

[11]

L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, Chapman & Hall/CRC, 2007.  Google Scholar

[12]

L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.  doi: 10.1007/s00028-014-0249-z.  Google Scholar

[13]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205.  doi: 10.1007/s002330010129.  Google Scholar

[14]

G. Metafune and C. Spina, Elliptic operators with unbounded coefficients in $L^{p}$ spaces, Annali Scuola Normale Superiore di Pisa Cl. Sci.(5), 11 (2012), 303-340.   Google Scholar

[15]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients, Discrete and Continuous Dynamical Systems A, 32 (2012), 2285-2299.  doi: 10.3934/dcds.2012.32.2285.  Google Scholar

[16]

G. Metafune, C. Spina and C. Tacelli, Elliptic operators with unbounded diffusion and drift coefficients in $L^{p}$ spaces, Adv. Diff. Equat., 19 (2014), 473-526.   Google Scholar

[17]

G. Metafune, C. Spina and C. Tacelli, On a class of elliptic operators with unbounded diffusion coefficients, Evol. Equ. Control Theory, 3 (2014), 671-680.  doi: 10.3934/eect.2014.3.671.  Google Scholar

[18]

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.  Google Scholar

[19]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monogr. Ser., 31, Princeton Univ. Press, 2005.  Google Scholar

show all references

References:
[1]

D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, Rev. Mat. Iberoam., 28 (2012), 879-906.  doi: 10.4171/RMI/695.  Google Scholar

[2]

S. E. Boutiah, F. Gregorio, A. Rhandi and C. Tacelli, Elliptic operators with unbounded diffusion, drift and potential terms, J. Differential Equations, 264 (2018), 2184-2204.  doi: 10.1016/j.jde.2017.10.020.  Google Scholar

[3]

A. Canale, A. Rhandi and C. Tacelli, Schrödinger-type operators with unbounded diffusion and potential terms, Annali Scuola Normale Superiore di Pisa Cl. Sci.(5), 16 (2016), 581-601.   Google Scholar

[4]

A. Canale, A. Rhandi and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, Z. Anal. Anwend., 36 (2017), 377-392.  doi: 10.4171/ZAA/1593.  Google Scholar

[5]

A. Canale and C. Tacelli, Optimal kernel estimates for a Schrödinger type operator, Riv. Mat. Univ. Parma, 7 (2016), 341-450.   Google Scholar

[6]

E. B. Davies, Heat kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[7]

T. Durante, R. Manzo and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponent, Ricerche Mat., 65 (2016), 289-305.  doi: 10.1007/s11587-016-0284-x.  Google Scholar

[8]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^{p}$ and $C_{b}$-spaces, Discrete and Continuous Dynamical Systems A, 18 (2007), 747-772.  doi: 10.3934/dcds.2007.18.747.  Google Scholar

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[10]

M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations, Advances in Mathematics, 287 (2016), 600-639.  doi: 10.1016/j.aim.2015.09.029.  Google Scholar

[11]

L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, Chapman & Hall/CRC, 2007.  Google Scholar

[12]

L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.  doi: 10.1007/s00028-014-0249-z.  Google Scholar

[13]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205.  doi: 10.1007/s002330010129.  Google Scholar

[14]

G. Metafune and C. Spina, Elliptic operators with unbounded coefficients in $L^{p}$ spaces, Annali Scuola Normale Superiore di Pisa Cl. Sci.(5), 11 (2012), 303-340.   Google Scholar

[15]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients, Discrete and Continuous Dynamical Systems A, 32 (2012), 2285-2299.  doi: 10.3934/dcds.2012.32.2285.  Google Scholar

[16]

G. Metafune, C. Spina and C. Tacelli, Elliptic operators with unbounded diffusion and drift coefficients in $L^{p}$ spaces, Adv. Diff. Equat., 19 (2014), 473-526.   Google Scholar

[17]

G. Metafune, C. Spina and C. Tacelli, On a class of elliptic operators with unbounded diffusion coefficients, Evol. Equ. Control Theory, 3 (2014), 671-680.  doi: 10.3934/eect.2014.3.671.  Google Scholar

[18]

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.  Google Scholar

[19]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monogr. Ser., 31, Princeton Univ. Press, 2005.  Google Scholar

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