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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 |
$k$ |
$A: = (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ |
$\begin{eqnarray*}k(t,x,y) &\leq&c_1e^{\lambda_0 t+ c_2t^{-\gamma}}\left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(\beta-\alpha)}}{1+|y|^\alpha}\\&&\times\exp\left(-\frac{\sqrt{2}}{\beta-\alpha+2}\left(|x|^{\frac{\beta-\alpha+2}{2}}+ |y|^{\frac{\beta-\alpha+2}{2}}\right)\right)\end{eqnarray*}$ |
$t>0,\,|x|,\,|y|\ge 1$ |
$b\in\mathbb{R}$ |
$c_1,\,c_2$ |
$\lambda_0$ |
$A$ |
$\gamma = \frac{\beta-\alpha+2}{\beta+\alpha-2}$ |
$N>2,\,\alpha>2$ |
$\beta>\alpha -2$ |
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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