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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 and Continuous Dynamical Systems, 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.

[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.

[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. 

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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. 

[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.

[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. 

[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.

[18]

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

[19]

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

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.

[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.

[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. 

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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. 

[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.

[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. 

[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.

[18]

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

[19]

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

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