Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness. (English) Zbl 1476.35297

Summary: We study the Hermite operator \(H = - \Delta + | x |^2\) in \(\mathbb{R}^d\) and its fractional powers \(H^\beta\), \(\beta > 0\) in phase space. Namely, we represent functions \(f\) via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform \(V_g f (g\) being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of \(V_g f\), that is in terms of membership to modulation spaces \(M^{p , q}\), \(0 < p, q \leq \infty \). We prove the complete range of fixed-time estimates for the semigroup \(e^{- t H^\beta}\) when acting on \(M^{p , q} \), for every \(0 < p, q \leq \infty \), exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for \(H^\beta\) with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay \(e^{- c t}\) as the solution of the corresponding linear equation, where \(c = d^\beta\) is the bottom of the spectrum of \(H^\beta \). Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to \(M^{\infty , 1})\).


35R11 Fractional partial differential equations
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
42B35 Function spaces arising in harmonic analysis
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI arXiv


[1] Bényi, Á.; Gröchenig, K.; Okoudjou, K. A.; Rogers, L. G., Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246, 2, 366-384 (2007) · Zbl 1120.42010
[2] Bényi, Á.; Oh, T.; Pocovnicu, O., On the probabilistic Cauchy theory for nonlinear dispersive PDEs, (Landscapes of Time-Frequency Analysis. Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. (2019), Birkhäuser/Springer: Birkhäuser/Springer Cham), 1-32 · Zbl 1418.35389
[3] Bényi, Á.; Oh, T., Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math., 228, 5, 2943-2981 (2011) · Zbl 1229.42021
[4] Bényi, Á.; Okoudjou, K. A., Modulation Spaces (2020), Birkhäuser/Springer: Birkhäuser/Springer New York · Zbl 1476.35001
[5] Bényi, Á.; Okoudjou, K. A., Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41, 3, 549-558 (2009) · Zbl 1173.35115
[6] Bhimani, D. G., The nonlinear heat equations with fractional Laplacian & harmonic oscillator in modulation spaces (2019), Preprint
[7] Bhimani, D. G.; Balhara, R.; Thangavelu, S., Hermite multipliers on modulation spaces, (Analysis and Partial Differential Equations: Perspectives from Developing Countries. Analysis and Partial Differential Equations: Perspectives from Developing Countries, Springer Proc. Math. Stat., vol. 275 (2019), Springer: Springer Cham), 42-64 · Zbl 1414.35183
[8] Bhimani, D. G.; Grillakis, M.; Okoudjou, K. A., The Hartree-Fock equations in modulation spaces, Commun. Partial Differ. Equ., 45, 9, 1088-1117 (2020) · Zbl 1448.35420
[9] Chen, J.; Ding, Y.; Deng, Q.; Fan, D., Estimates on fractional power dissipative equations in function spaces, Nonlinear Anal., Theory Methods Appl., 75, 2959-2974 (2012) · Zbl 1242.42013
[10] Cordero, E., On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces, J. Pseudo-Differ. Oper. Appl., 12, 1, Article 13 pp. (2021) · Zbl 1466.35357
[11] Cordero, E.; Nicola, F.; Rodino, L., Schrödinger equations with rough Hamiltonians, Discrete Contin. Dyn. Syst., 35, 10, 4805-4821 (2015) · Zbl 1334.35458
[12] Cordero, E.; Rodino, L., Time-Frequency Analysis of Operators (2020), De Gruyter · Zbl 07204958
[13] Feichtinger, H. G., Banach convolution algebras of Wiener type, (Functions, Series, Operators, Vol. I, II. Functions, Series, Operators, Vol. I, II, Budapest, 1980. Functions, Series, Operators, Vol. I, II. Functions, Series, Operators, Vol. I, II, Budapest, 1980, Colloq. Math. Soc. János Bolyai, vol. 35 (1983), North-Holland: North-Holland Amsterdam), 509-524
[14] Feichtinger, H. G., Generalized amalgams, with applications to Fourier transform, Can. J. Math., 42, 3, 395-409 (1990) · Zbl 0733.46014
[15] Feichtinger, H. G., Modulation spaces on locally compact Abelian groups, (Krishna, M.; Radha, R.; Thangavelu, S., Wavelets and Their Applications (2003), Allied Publishers), 99-140, Technical Report, University Vienna, 1983, and also
[16] Feichtinger, H. G.; Gröchenig, K.; Li, K.; Wang, B., Navier-Stokes equation in super-critical spaces \(E_{p , q}^s\), Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 38, 1, 139-173 (2021) · Zbl 1464.35177
[17] Folland, G. B., Harmonic Analysis in Phase Space (1989), Princeton University Press · Zbl 0682.43001
[18] Garofalo, N., Fractional thoughts (2017), Preprint · Zbl 1423.35397
[19] Galperin, Y. V.; Samarah, S., Time-frequency analysis on modulation spaces \(M_{p , q}^m, 0 < p, q \leq \infty \), Appl. Comput. Harmon. Anal., 16, 1, 1-18 (2004) · Zbl 1040.42025
[20] Gramchev, T.; Pilipović, S.; Rodino, L., Classes of degenerate elliptic operators in Gelfand-Shilov spaces, (New Developments in Pseudo-Differential Operators. New Developments in Pseudo-Differential Operators, Oper. Theory Adv. Appl., vol. 189 (2009), Birkhäuser: Birkhäuser Basel), 15-31 · Zbl 1170.47031
[21] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser: Birkhäuser Boston, MA · Zbl 0966.42020
[22] Helffer, B., Théorie Spectrale pour des Opérateurs Globalement Elliptiques, Astérisque, vol. 112 (1984), Société Mathématique de France: Société Mathématique de France Paris · Zbl 0541.35002
[23] Huang, Q.; Fan, D.; Chen, J., Critical exponent for evolution equations in modulation spaces, J. Math. Anal. Appl., 443, 1, 230-242 (2016) · Zbl 1339.35033
[24] Iwabuchi, T., Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differ. Equ., 248, 8, 1972-2002 (2010) · Zbl 1185.35166
[25] Janssen, A. J.E. M., Hermite function description of Feichtinger’s space \(S_0\), J. Fourier Anal. Appl., 11, 5, 577-588 (2005) · Zbl 1102.42011
[26] Kobayashi, M.; Sugimoto, M., The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260, 11, 3189-3208 (2011) · Zbl 1232.46033
[27] Koch, H.; Tataru, D., \( L^p\) eigenfunction bounds for the Hermite operator, Duke Math. J., 128, 2, 369-392 (2005) · Zbl 1075.35020
[28] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, 5, Article 056108 pp. (2002)
[29] Manna, R., The Cauchy problem for non-linear higher order Hartree type equation in modulation spaces, J. Fourier Anal. Appl., 25, 1319-1349 (2019) · Zbl 1420.35368
[30] Mizoguchi, N.; Souplet, P., Optimal condition for blow-up of the critical \(L^q\) norm for the semilinear heat equation, Adv. Math., 335, Article 106763 pp. (2019) · Zbl 1420.35139
[31] Nicola, F., Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267, 3, 727-743 (2014) · Zbl 1296.35225
[32] Nicola, F.; Rodino, L., Global Pseudo-Differential Calculus on Euclidean Spaces (2010), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1257.47002
[33] Pierfelice, V., Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials, Asymptot. Anal., 47, 1-2, 1-18 (2006) · Zbl 1100.35020
[34] Rauhut, H., Coorbit space theory for quasi-Banach spaces, Stud. Math., 180, 3, 237-253 (2007) · Zbl 1122.42018
[35] Rauhut, H., Wiener amalgam spaces with respect to quasi-Banach spaces, Colloq. Math., 109, 2, 345-362 (2007) · Zbl 1125.46022
[36] Ruzhansky, M.; Sugimoto, M.; Wang, B., Modulation spaces and nonlinear evolution equations, (Evolution Equations of Hyperbolic and Schrödinger Type. Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., vol. 301 (2012), Birkhäuser/Springer Basel AG: Birkhäuser/Springer Basel AG Basel), 267-283 · Zbl 1256.42038
[37] Shubin, M. A., Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0616.47040
[38] Tao, T., Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics (2006), Amer. Math. Soc. · Zbl 1106.35001
[39] Tataru, D., Phase space transforms and microlocal analysis, (Phase Space Analysis of Partial Differential Equations. Vol. II (2004), Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup.: Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup. Pisa), 505-524 · Zbl 1111.35143
[40] Toft, J., Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes, Anal. Appl., 15, 3, 353-389 (2017) · Zbl 1459.47019
[41] Toft, J., Continuity properties for modulation spaces, with applications to pseudo-differential calculus I., J. Funct. Anal., 207, 2, 399-429 (2004) · Zbl 1083.35148
[42] Thangavelu, S., Lectures on Hermite and Laguerre Expansions (1993), Princeton University Press: Princeton University Press Princeton, NJ, USA · Zbl 0791.41030
[43] Thangavelu, S., A note on fractional powers of the Hermite operator (2018), Preprint
[44] Vázquez, J. L., The mathematical theories of diffusion: nonlinear and fractional diffusion, (Bonforte, M.; Grillo, G., Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Lecture Notes in Mathematics, vol. 218 (2017), Springer: Springer Cham), 205-278 · Zbl 1492.35151
[45] Wang, B., Exponential Besov spaces and their applications to certain evolution equations with dissipation, Commun. Pure Appl. Anal., 3, 4, 883-919 (2004) · Zbl 1072.46023
[46] Wang, B.; Huang, C., Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differ. Equ., 239, 1, 213-250 (2007) · Zbl 1219.35289
[47] Wang, B.; Huo, Z.; Hao, C.; Guo, Z., Harmonic Analysis Method for Nonlinear Evolution Equations. I (2011), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1254.35002
[48] Wang, B.; Zhao, L.; Guo, B., Isometric decomposition operators, function spaces \(E_{p , q}^\lambda\) and applications to nonlinear evolution equations, J. Funct. Anal., 233, 1, 1-39 (2006) · Zbl 1099.46023
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