No CrossRef data available.
Article contents
Dispersive and Strichartz estimates for 3D wave equation with a Laguerre potential
Part of:
Hyperbolic equations and systems
Harmonic analysis in several variables
Equations of mathematical physics and other areas of application
Published online by Cambridge University Press: 07 March 2024
Abstract
Dispersive and Strichartz estimates are obtained for solutions to the wave equation with a Laguerre potential in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schrödinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schrödinger semigroup, obtain a Gaussian-type upper bound, establish Bernstein-type inequalities, and finally pass to the Müller–Seeger’s subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel–Tao.
MSC classification
- Type
- Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society
References
Ablowitz, M. J. and Fokas, A. S., Introduction and applications of complex variables. 2nd ed., Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions with formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1972.Google Scholar
Andrews, G. E., Askey, R., and Roy, R., Special functions, Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
Beals, M.,
Optimal
${L}^{\infty }$
decay for solutions to the wave equation with a potential
. Comm. Partial Differential Equations 19(1994), nos. 7–8, 1319–1369.CrossRefGoogle Scholar
${L}^{\infty }$
decay for solutions to the wave equation with a potential
. Comm. Partial Differential Equations 19(1994), nos. 7–8, 1319–1369.CrossRefGoogle ScholarBeals, M. and Strauss, W.,
${L}^p$
estimates for the wave equation with a potential
. Comm. Partial Differential Equations 18(1993), nos. 7–8, 1365–1397.CrossRefGoogle Scholar
${L}^p$
estimates for the wave equation with a potential
. Comm. Partial Differential Equations 18(1993), nos. 7–8, 1365–1397.CrossRefGoogle ScholarBeceanu, M.,
New estimates for a time-dependent Schrödinger equation
. Duke Math. J. 159(2011), no. 3, 417–477.CrossRefGoogle Scholar
Beceanu, M.,
Structure of wave operators for a scaling-critical class of potentials
. Amer. J. Math. 136(2014), no. 2, 255–308.CrossRefGoogle Scholar
Beceanu, M. and Goldberg, M.,
Schrödinger dispersive estimates for a scaling-critical class of potentials
. Comm. Math. Phys. 314(2012), no. 2, 471–481.CrossRefGoogle Scholar
Bizoń, P., Chmaj, T., and Tabor, Z.,
On blowup for semilinear wave equations with a focusing nonlinearity
. Nonlinearity 17(2004), no. 6, 2187–2201.CrossRefGoogle Scholar
Bongioanni, B. and Torrea, J. L.,
Sobolev spaces associated to the harmonic oscillator
. Proc. Indian Acad. Sci. Math. Sci. 116(2006), no. 3, 337–360.CrossRefGoogle Scholar
Bui, T. A., Duong, X. T., and Hong, Y.,
Dispersive and Strichartz estimates for the three-dimensional wave equation with a scaling-critical class of potentials
. J. Funct. Anal. 271(2016), no. 8, 2215–2246.CrossRefGoogle Scholar
Burq, N., Planchon, F., Stalker, J. G., and Tahvildar-Zadeh, A. S.,
Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential
. J. Funct. Anal. 203(2003), no. 2, 519–549.CrossRefGoogle Scholar
Burq, N., Planchon, F., Stalker, J. G., and Tahvildar-Zadeh, A. S.,
Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay
. Indiana Univ. Math. J. 53(2004), no. 6, 1665–1680.CrossRefGoogle Scholar
D’Ancona, P. and Fanelli, L.,
${L}^p-$
boundedness of the wave operator for the one dimensional Schrödinger operator
. Comm. Math. Phys. 268(2006), no. 2, 415–438.CrossRefGoogle Scholar
${L}^p-$
boundedness of the wave operator for the one dimensional Schrödinger operator
. Comm. Math. Phys. 268(2006), no. 2, 415–438.CrossRefGoogle ScholarD’Ancona, P. and Pierfelice, V.,
On the wave equation with a large rough potential
. J. Funct. Anal. 227(2005), no. 1, 30–77.CrossRefGoogle Scholar
D’Ancona, P., Pierfelice, V., and Ricci, F.,
On the wave equation associated to the Hermite and the twisted Laplacian
. J. Fourier Anal. Appl. 16(2010), no. 2, 294–310.CrossRefGoogle Scholar
Fanelli, L., Felli, V., Fontelos, M. A., and Primo, A.,
Time decay of scaling critical electromagnetic Schrödinger flows
. Comm. Math. Phys. 324(2013), no. 3, 1033–1067.CrossRefGoogle Scholar
Fanelli, L., Zhang, J., and Zheng, J.,
Dispersive estimates for 2D-wave equations with critical potentials
. Adv. Math. 400(2022), Article no. 108333, 46 pp.CrossRefGoogle Scholar
Georgiev, V. and Visciglia, N.,
Decay estimates for the wave equation with potential
. Comm. Partial Differential Equations 28(2003), nos. 7–8, 1325C1369.CrossRefGoogle Scholar
Ginibre, J. and Velo, G.,
Generalized Strichartz inequalities for the wave equation
. J. Funct. Anal. 133(1995), no. 1, 50–68.CrossRefGoogle Scholar
Goldberg, M.,
Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials
. Amer. J. Math. 128(2006), no. 3, 731–750.CrossRefGoogle Scholar
Goldberg, M.,
Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials
. Geom. Funct. Anal. 16(2006), no. 3, 517–536.Google Scholar
Goldberg, M.,
Dispersive estimates for Schrödinger operators with measure-valued potentials in
${\mathbb{R}}^3$
. Indiana Univ. Math. J. 61(2012), no. 6, 2123–2141.CrossRefGoogle Scholar
${\mathbb{R}}^3$
. Indiana Univ. Math. J. 61(2012), no. 6, 2123–2141.CrossRefGoogle ScholarGoldberg, M. and Schlag, W.,
Dispersive estimates for the Schrödinger operators in dimensions one and three
. Comm. Math. Phys. 251(2004), no. 1, 157–178.CrossRefGoogle Scholar
Goldberg, M., Vega, L., and Visciglia, N.,
Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials
. Int. Math. Res. Not. IMRN 2006(2006), Article no. 13927, 16 pp.Google Scholar
Hardy, G. H.,
Note on a theorem of Hilbert
. Math. Z. 6(1920), nos. 3–4, 314–317.CrossRefGoogle Scholar
Journé, J.-L., Soffer, A., and Sogge, C. D.,
Decay estimates for Schrödinger operators
. Comm. Pure Appl. Math. 44(1991), no. 5, 573–604.CrossRefGoogle Scholar
Keel, M. and Tao, T.,
Endpoint Strichartz estimates
. Amer. J. Math. 120(1998), no. 5, 955–980.CrossRefGoogle Scholar
Killip, R., Miao, C., Visan, M., Zhang, J., and Zheng, J.,
Sobolev spaces adapted to the Schrödinger operator with inverse-square potential
. Math. Z. 288(2018), nos. 3–4, 1273–1298.CrossRefGoogle Scholar
Koch, H. and Tataru, D.,
${L}^p$
eigenfunction bounds for the Hermite operator
. Duke Math. J. 128(2005), no. 2, 369–392.CrossRefGoogle Scholar
${L}^p$
eigenfunction bounds for the Hermite operator
. Duke Math. J. 128(2005), no. 2, 369–392.CrossRefGoogle ScholarKrieger, J. and Schlag, W.,
On the focusing critical semi-linear wave equation
. Amer. J. Math. 129(2007), no. 3, 843–913.CrossRefGoogle Scholar
Miao, C., Su, X., and Zheng, J.,
The
${W}^{s,p}$
-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential
. Trans. Amer. Math. Soc. 376(2023), no. 3, 1739–1797.CrossRefGoogle Scholar
${W}^{s,p}$
-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential
. Trans. Amer. Math. Soc. 376(2023), no. 3, 1739–1797.CrossRefGoogle ScholarMizutani, H.,
Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential
. J. Differ. Equ. 263(2017), no. 7, 3832–3853.CrossRefGoogle Scholar
Müller, C., Spherical harmonics, Lecture Notes in Mathematics, 17, Springer, Berlin, 1966.CrossRefGoogle Scholar
Müller, D. and Seeger, A.,
Sharp
${L}^p$
bounds for the wave equation on groups of Heisenberg type
. Anal. PDE 8(2015), no. 5, 1051–1100.CrossRefGoogle Scholar
${L}^p$
bounds for the wave equation on groups of Heisenberg type
. Anal. PDE 8(2015), no. 5, 1051–1100.CrossRefGoogle ScholarÓlafsson, G. and Zheng, S.,
Function spaces associated with Schrödinger operators: The Pöschl–Teller potential
. J. Fourier Anal. Appl. 12(2006), no. 6, 653–674.CrossRefGoogle Scholar
Pierfelice, V.,
Decay estimate for the wave equation with a small potential
. NoDEA Nonlinear Differential Equations Appl. 13(2007), nos. 5–6, 511–530.CrossRefGoogle Scholar
Planchon, F., Stalker, J. G., and Tahvildar-Zadeh, A. S.,
${L}^p$
estimates for the wave equation with the inverse-square potential
. Discrete Contin. Dyn. Syst. 9(2003), no. 2, 427–442.CrossRefGoogle Scholar
${L}^p$
estimates for the wave equation with the inverse-square potential
. Discrete Contin. Dyn. Syst. 9(2003), no. 2, 427–442.CrossRefGoogle ScholarPlanchon, F., Stalker, J. G., and Tahvildar-Zadeh, A. S.,
Dispersive estimates for the wave equation with the inverse-square potential
. Discrete Contin. Dyn. Syst. 9(2003), no. 6, 1387–1400.CrossRefGoogle Scholar
Rodnianski, I. and Schlag, W.,
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials
. Invent. Math. 155(2004), no. 3, 451–513.CrossRefGoogle Scholar
Segal, I.,
Space-time decay for solutions of wave equations
. Adv. Math. 22(1976), no. 3, 305–311.CrossRefGoogle Scholar
Simon, B.,
Schrödinger semigroups
. Bull. Amer. Math. Soc. 7(1982), no. 3, 447–526.CrossRefGoogle Scholar
Strichartz, R. S.,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
. Duke Math. J. 44(1977), no. 3, 705–714.CrossRefGoogle Scholar
Szpak, N.,
Relaxation to intermediate attractors in nonlinear wave equations
. Theoret. Math. Phys. 127(2001), no. 3, 817–826.CrossRefGoogle Scholar
Tomas, P. A.,
A restriction theorem for the Fourier transform
. Bull. Amer. Math. Soc. 81(1975), 477–478.CrossRefGoogle Scholar
Watson, G. N., A treatise on the theory of Bessel functions. 2nd ed., Cambridge University Press, Cambridge, 1944.Google Scholar
Weder, R.,
The
${W}^{k,p}$
-continuity of the Schrödinger wave operators on the line
. Comm. Math. Phys. 208(1999), no. 2, 507–520.CrossRefGoogle Scholar
${W}^{k,p}$
-continuity of the Schrödinger wave operators on the line
. Comm. Math. Phys. 208(1999), no. 2, 507–520.CrossRefGoogle ScholarYajima, K.,
Existence of solutions for Schrödinger evolution equations
. Comm. Math. Phys. 110(1987), no. 3, 415–426.CrossRefGoogle Scholar
Yajima, K.,
The
${W}^{k,p}$
-continuity of wave operators for Schrödinger operators
. Proc. Japan Acad. Ser. A Math. Sci. 69(1993), no. 4, 94–98.CrossRefGoogle Scholar
${W}^{k,p}$
-continuity of wave operators for Schrödinger operators
. Proc. Japan Acad. Ser. A Math. Sci. 69(1993), no. 4, 94–98.CrossRefGoogle ScholarYajima, K.,
The
${W}^{k,p}$
-continuity of wave operators for Schrödinger operators
. J. Math. Soc. Japan 47(1995), no. 3, 551–581.CrossRefGoogle Scholar
${W}^{k,p}$
-continuity of wave operators for Schrödinger operators
. J. Math. Soc. Japan 47(1995), no. 3, 551–581.CrossRefGoogle ScholarYajima, K.,
${L}^p$
-boundedness of wave operators for two-dimensional Schrödinger operators
. Comm. Math. Phys. 208(1999), no. 1, 125–152.CrossRefGoogle Scholar
${L}^p$
-boundedness of wave operators for two-dimensional Schrödinger operators
. Comm. Math. Phys. 208(1999), no. 1, 125–152.CrossRefGoogle ScholarYajima, K.,
The
${L}^p$
boundedness of wave operators for Schrödinger operators with threshold singularities. I. The odd dimensional case
. J. Math. Sci. Univ. Tokyo 13(2006), no. 1, 43–93.Google Scholar
${L}^p$
boundedness of wave operators for Schrödinger operators with threshold singularities. I. The odd dimensional case
. J. Math. Sci. Univ. Tokyo 13(2006), no. 1, 43–93.Google Scholar