Smooth solutions to the heat equation which are nowhere analytic in time

Yang, Xin; Zeng, Chulan; Zhang, Qi

doi:10.1090/proc/16323

Smooth solutions to the heat equation which are nowhere analytic in time
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by Xin Yang, Chulan Zeng and Qi S. Zhang;

Proc. Amer. Math. Soc. 152 (2024), 2821-2830

DOI: https://doi.org/10.1090/proc/16323

Published electronically: May 23, 2024

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Abstract:

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $\delta >0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+\delta$, which is nowhere analytic in time.

References

G. G. Bilodeau, The origin and early development of nonanalytic infinitely differentiable functions, Arch. Hist. Exact Sci. 27 (1982), no. 2, 115–135. MR 677684, DOI 10.1007/BF00348345
Emile Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. (3) 12 (1895), 9–55 (French). MR 1508908, DOI 10.24033/asens.406
Georg Cantor, Ueber ein neues und allgemeines Condensationsprincip der Singularitäten von Functionen, Math. Ann. 19 (1882), no. 4, 588–594 (German). MR 1510146, DOI 10.1007/BF01446674
A. Cauchy, Mémoire sur l’emploi du calcul des limites dans l’intégration des équations aux dérivées partielles, Comptes Rendus XV (1842), 44–59
C. Cellerier, Note sur les principes fondamentaux de l’analyse, Bull. Sci. Math. 14 (1890), 142–160
Hongjie Dong and Xinghong Pan, Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space, J. Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53, 20. MR 4145847, DOI 10.1007/s00021-020-00515-5
Hongjie Dong, Chulan Zeng, and Qi S. Zhang, Time analyticity for nonlocal parabolic equations, SIAM J. Math. Anal. 55 (2023), no. 3, 1883–1915. MR 4598823, DOI 10.1137/22M1490740
Hongjie Dong and Qi S. Zhang, Time analyticity for the heat equation and Navier-Stokes equations, J. Funct. Anal. 279 (2020), no. 4, 108563, 15. MR 4095801, DOI 10.1016/j.jfa.2020.108563
Paul du Bois-Reymond, Ueber den Gültigkeitsbereich der Taylor’schen Reihenentwickelung, Math. Ann. 21 (1883), no. 1, 109–117 (German). MR 1510189, DOI 10.1007/BF01442615
Luis Escauriaza, Santiago Montaner, and Can Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM J. Math. Anal. 49 (2017), no. 5, 4064–4092. MR 3713902, DOI 10.1137/15M1039705
Yoshikazu Giga, Time and spatial analyticity of solutions of the Navier-Stokes equations, Comm. Partial Differential Equations 8 (1983), no. 8, 929–948. MR 699168, DOI 10.1080/03605308308820290
J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpement de taylor, J. Math. Pures Appl. 8 (1924), no. 4, 101–186
Herrmann Hankel, Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen, Math. Ann. 20 (1882), no. 1, 63–112 (German). MR 1510154, DOI 10.1007/BF01443329
Charles Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 33 (1969), 386–405. MR 245989, DOI 10.1007/BF00247697
Gen Komatsu, Global analyticity up to the boundary of solutions of the Navier-Stokes equation, Comm. Pure Appl. Math. 33 (1980), no. 4, 545–566. MR 575737, DOI 10.1002/cpa.3160330405
Sophie von Kowalevsky, Zur Theorie der partiellen Differentialgleichung, J. Reine Angew. Math. 80 (1875), 1–32 (German). MR 1579652, DOI 10.1515/crll.1875.80.1
Camille Laurent and Lionel Rosier, Exact controllability of semilinear heat equations in spaces of analytic functions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), no. 4, 1047–1073. MR 4104834, DOI 10.1016/j.anihpc.2020.03.001
Matthias Lerch, Ueber die Nichtdifferentiirbarkeit gewisser Functionen, J. Reine Angew. Math. 103 (1888), 126–138 (German). MR 1580160, DOI 10.1515/crll.1888.103.126
Kyûya Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad. 43 (1967), 827–832. MR 247304
Dietrich Morgenstern, Unendlich oft differenzierbare nicht-analytische Funktionen, Math. Nachr. 12 (1954), 74 (German). MR 64830, DOI 10.1002/mana.19540120106
A. Tychonoff, Théorèmes d’unicité pour l’équation de la chaleur, Mat. Sb. 42 (1935), no. 2, 199–216
PawełWalczak, A proof of some theorem on the $C^{\infty }$-functions of one variable which are not analytic, Demonstratio Math. 4 (1972), 209–213. MR 335724
D. V. Widder, Analytic solutions of the heat equation, Duke Math. J. 29 (1962), 497–503. MR 157127, DOI 10.1215/S0012-7094-62-02950-2
Chulan Zeng, Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations, Commun. Pure Appl. Anal. 21 (2022), no. 3, 749–783. MR 4389599, DOI 10.3934/cpaa.2021197
Q. S. Zhang, A formula for backward and control problems of the heat equation, arXiv:2005.08375, 2020.
Qi S. Zhang, A note on time analyticity for ancient solutions of the heat equation, Proc. Amer. Math. Soc. 148 (2020), no. 4, 1665–1670. MR 4069203, DOI 10.1090/proc/14830