# American Institute of Mathematical Sciences

March  2022, 21(3): 749-783. doi: 10.3934/cpaa.2021197

## Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations

 University of California, Riverside, 900 University Ave, Riverside, CA 92521, USA

Received  October 2021 Revised  November 2021 Published  March 2022 Early access  November 2021

In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order $p$. The potentials include all the nonnegative ones. For the first two equations, we prove if $u$ satisfies some growth conditions in $(x,t)\in \mathrm{M}\times [0,1]$, then $u$ is analytic in time $(0,1]$. Here $\mathrm{M}$ is $R^d$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that $u(x,t)$ is analytic in time at $t = 0$. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.

For the nonlinear heat equation with power nonlinearity of order $p$, we prove that a solution is analytic in time $t\in (0,1]$ if it is bounded in $\mathrm{M}\times[0,1]$ and $p$ is a positive integer. In addition, we investigate the case when $p$ is a rational number with a stronger assumption $0<C_3 \leq |u(x,t)| \leq C_4$. It is also shown that a solution may not be analytic in time if it is allowed to be $0$. As a lemma, we obtain an estimate of $\partial_t^k \Gamma(x,t;y)$ where $\Gamma(x,t;y)$ is the heat kernel on a manifold, with an explicit estimation of the coefficients.

An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable $x$, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.

Citation: Chulan Zeng. Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations. Communications on Pure and Applied Analysis, 2022, 21 (3) : 749-783. doi: 10.3934/cpaa.2021197
##### References:
 [1] G. Barbatis and E. Davies, Sharp Bounds on Heat Kernels of Higher Order Uniformly Elliptic Operators, J. Operat. Theor., 36 (1997), 18 pp. [2] P. Buser, A note on the isoperimetric constant, Annales scientifiques de l'École Normale Supérieure, 15 (1982), 213-230. [3] J. Cheeger and T. H. Tobias, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144 (1996), 189-237.  doi: 10.2307/2118589. [4] H. Dong and D. Kim, On the Lp-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3. [5] H. Dong and X. Pan, Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space, J. Math. Fluid Mech., 22 (2020), 20 pp. doi: 10.1007/s00021-020-00515-5. [6] H. Dong and Q. S. Zhang, Time analyticity for the heat equation and Navier-Stokes equations, J. Funct. Anal.., 279 (2020), 15 pp. doi: 10.1016/j.jfa.2020.108563. [7] L. Escauriaza, S. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM J. Math. Anal., 49 (2017), 4064-4092.  doi: 10.1137/15M1039705. [8] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Basel, Boston, Berlin: Birkh$\ddot{a}$user, 1993. [9] Y. Giga, Time and spatial analyticity of solutions of the Navier-Stokes equations, Commun. Partial Differ. Equ., 8 (1983), 929-948.  doi: 10.1080/03605308308820290. [10] Z. Grujić and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differ. Equ., 154 (1999), 42-54.  doi: 10.1006/jdeq.1998.3562. [11] H. Emmanuel, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, in Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [12] V. G. Papanicolaou, E. Kallitsi and G. Smyrlis, Analytic solutions of the heat equation, preprint, arXiv: 1906.02233 [13] Q. Hou and L. Saloff-Coste, Time regularity for local weak solutions of the heat equation on local Dirichlet spaces, preprint, arXiv: 1912.12998 [14] G. Komatsu, Global analyticity up to the boundary of solutions of the Navier-Stokes equation, Commun. Pure Appl. Math., 33 (1980), 545-566.  doi: 10.1002/cpa.3160330405. [15] L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, in London Mathematical Society Lecture Note Series Cambridge: Cambridge University Press, 2001. [16] P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139105798. [17] Z. Lin and P. Pan, Some remarks on regularity criteria of axially symmetric Navier-Stokes equations, Commun. Pure Appl. Anal., 18 (2019), 1333-1350.  doi: 10.3934/cpaa.2019064. [18] F. Lin and Q. S. Zhang, On ancient solutions of the heat equations, Commun. Pure Appl. Math., 72 (2019), 2006-2028.  doi: 10.1002/cpa.21820. [19] Z. Li and Q. S. Zhang, Regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials, J. Differ. Equ., 1 (2017), 57-87.  doi: 10.1016/j.jde.2017.02.029. [20] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203. [21] G. Łysik and S. Michalik, Formal solutions of semilinear heat equations, J. Math. Anal. Appl., 341 (2008), 372-385.  doi: 10.1016/j.jmaa.2007.10.005. [22] K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43 (1967), 827-832. [23] K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.  doi: 10.1016/0362-546X(91)90100-F. [24] B. Rodgers and T. Tao, The De Bruijn-Newman constant is non-negative, Forum Math., 8 (2020), 62 pp. doi: 10.1017/fmp.2020.6. [25] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Cambridge, MA, 1994. [26] D. V. Widder, Analytic solutions of the heat equation, in Contributions to Nonlinear Functional Analysis (eds. E.H. Zarantonello and Author 2), Duke Math. J., 29 (1962), 497–503. [27] B. Wong and Q. S. Zhang, Refined gradient bounds, Poisson equations and some applications to open k$\ddot{a}$hler manifolds, Asian J. Math., 7 (2003), 1-28.  doi: 10.4310/AJM.2003.v7.n3.a4. [28] Q. S. Zhang, A note on time analyticity for ancient solutions of the heat equation, Proc. Amer. Math. Soc., 148 (2019), 1665-1670.  doi: 10.1090/proc/14830. [29] Q. S. Zhang, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture, CRC Press, Boca Raton, 2011.

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##### References:
 [1] G. Barbatis and E. Davies, Sharp Bounds on Heat Kernels of Higher Order Uniformly Elliptic Operators, J. Operat. Theor., 36 (1997), 18 pp. [2] P. Buser, A note on the isoperimetric constant, Annales scientifiques de l'École Normale Supérieure, 15 (1982), 213-230. [3] J. Cheeger and T. H. Tobias, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144 (1996), 189-237.  doi: 10.2307/2118589. [4] H. Dong and D. Kim, On the Lp-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3. [5] H. Dong and X. Pan, Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space, J. Math. Fluid Mech., 22 (2020), 20 pp. doi: 10.1007/s00021-020-00515-5. [6] H. Dong and Q. S. Zhang, Time analyticity for the heat equation and Navier-Stokes equations, J. Funct. Anal.., 279 (2020), 15 pp. doi: 10.1016/j.jfa.2020.108563. [7] L. Escauriaza, S. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM J. Math. Anal., 49 (2017), 4064-4092.  doi: 10.1137/15M1039705. [8] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Basel, Boston, Berlin: Birkh$\ddot{a}$user, 1993. [9] Y. Giga, Time and spatial analyticity of solutions of the Navier-Stokes equations, Commun. Partial Differ. Equ., 8 (1983), 929-948.  doi: 10.1080/03605308308820290. [10] Z. Grujić and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differ. Equ., 154 (1999), 42-54.  doi: 10.1006/jdeq.1998.3562. [11] H. Emmanuel, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, in Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [12] V. G. Papanicolaou, E. Kallitsi and G. Smyrlis, Analytic solutions of the heat equation, preprint, arXiv: 1906.02233 [13] Q. Hou and L. Saloff-Coste, Time regularity for local weak solutions of the heat equation on local Dirichlet spaces, preprint, arXiv: 1912.12998 [14] G. Komatsu, Global analyticity up to the boundary of solutions of the Navier-Stokes equation, Commun. Pure Appl. Math., 33 (1980), 545-566.  doi: 10.1002/cpa.3160330405. [15] L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, in London Mathematical Society Lecture Note Series Cambridge: Cambridge University Press, 2001. [16] P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139105798. [17] Z. Lin and P. Pan, Some remarks on regularity criteria of axially symmetric Navier-Stokes equations, Commun. Pure Appl. Anal., 18 (2019), 1333-1350.  doi: 10.3934/cpaa.2019064. [18] F. Lin and Q. S. Zhang, On ancient solutions of the heat equations, Commun. Pure Appl. Math., 72 (2019), 2006-2028.  doi: 10.1002/cpa.21820. [19] Z. Li and Q. S. Zhang, Regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials, J. Differ. Equ., 1 (2017), 57-87.  doi: 10.1016/j.jde.2017.02.029. [20] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203. [21] G. Łysik and S. Michalik, Formal solutions of semilinear heat equations, J. Math. Anal. Appl., 341 (2008), 372-385.  doi: 10.1016/j.jmaa.2007.10.005. [22] K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43 (1967), 827-832. [23] K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.  doi: 10.1016/0362-546X(91)90100-F. [24] B. Rodgers and T. Tao, The De Bruijn-Newman constant is non-negative, Forum Math., 8 (2020), 62 pp. doi: 10.1017/fmp.2020.6. [25] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Cambridge, MA, 1994. [26] D. V. Widder, Analytic solutions of the heat equation, in Contributions to Nonlinear Functional Analysis (eds. E.H. Zarantonello and Author 2), Duke Math. J., 29 (1962), 497–503. [27] B. Wong and Q. S. Zhang, Refined gradient bounds, Poisson equations and some applications to open k$\ddot{a}$hler manifolds, Asian J. Math., 7 (2003), 1-28.  doi: 10.4310/AJM.2003.v7.n3.a4. [28] Q. S. Zhang, A note on time analyticity for ancient solutions of the heat equation, Proc. Amer. Math. Soc., 148 (2019), 1665-1670.  doi: 10.1090/proc/14830. [29] Q. S. Zhang, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture, CRC Press, Boca Raton, 2011.
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