多调和热流的一些局部估计及唯一性结果

Abstract

Abstract:

We consider smooth solutions to the poly-harmonic heat equation $\partial_tu+\Delta^{2m}u=0$ on $\mathbb{R} ^n\times (0,T]$. We focus on solutions satisfying a global growth condition: $|\Delta u|(x,t)^2\leq k_0t^{-1/m}$ for some constant $k_0>0$. We establish local estimates (in space and time) for such solutions and demonstrate how these estimates imply uniqueness of smooth solutions within this class. Finally, we provide an example to illustrate the naturalness of this control condition.

Key words: poly-harmonic heat flow, local estimates, uniqueness

CLC Number:

O175.2

Min Gao. Some Local Estimates and a Uniqueness Result for the Entire Poly-Harmonic Heat Flow[J].Acta mathematica scientia,Series A, 2026, 46(6): 2117-2139.

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References 15

[1]	Barbatis G, Gazzola F. Higher order linear parabolic equations. Recent trends in nonlinear partial differential equations. I. Evolution Problems Contemp Math, 2013, 594: 77-97
[2]	Cahn J W, Hilliard J E. Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys, 1958, 28(2): 258-267 doi: 10.1063/1.1744102
[3]	Evans L C. Partial Differential Equations. Providence: American Mathematical Society, 1998
[4]	Evgrafov M A, Postnikov M M. Asymptotic behavior of Green's functions for parabolic and elliptic equations with constant coefficients. Math USSR Sbornik, 1970, 11: 1-24 doi: 10.1070/SM1970v011n01ABEH002060
[5]	Ferreira L C F, Ferreira V A Jr. On the eventual local positivity for polyharmonic heat equations. Proc Amer Math Soc, 2019, 147(10): 4329-4341 doi: 10.1090/proc/2019-147-10
[6]	Ferrero A, Gazzola F, Grunau H C. Decay and eventual local positivity for biharmonic parabolic equations. Discrete Contin Dyn Syst, 2008, 21(4): 1129-1157
[7]	Galaktionov V A, Pohozaev S I. Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators. Indiana Univ Math J, 2002, 51(6): 1321-1338 doi: 10.1512/iumj.2002.51.2131
[8]	Gazzola F, Grunau H C. Eventual local positivity for a biharmonic heat equation in $\mathbb R^n$. Discrete Contin Dyn Syst Ser S, 2008, 1(1): 83-87 doi: 10.3934/dcdss.2008.1.83
[9]	Gazzola F, Grunau H C, Sweers G. Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Berlin: Springer-Verlag, 2010
[10]	Marzuola J L, Onofrei D, Osgood C B. The polyharmonic heat-flow and the Cahn-Hilliard equation on the circle. Comm Partial Differential Equations, 2013, 38(1): 120-143
[11]	Simon M. Local results for flows whose speed or height is bounded by $c/t$. Int Math Res Not IMRN, 2008, 2008: rnn097
[12]	Simon M, Wheeler G. Some local estimates and a uniqueness result for the entire biharmonic heat equation. Adv Calc Var, 2016, 9(1): 77-99 doi: 10.1515/acv-2014-0027
[13]	Stein E, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton University Press, 1971
[14]	Tychonoff A. Théorèmes d'unicité pour l'équation de la chaleur. Mat Sb, 1935, 42(2): 199-216
[15]	Widder D V. Positive temperatures on an infinite rod. Trans Amer Math Soc, 1944, 55: 85-95 doi: 10.1090/tran/1944-055-00

Some Local Estimates and a Uniqueness Result for the Entire Poly-Harmonic Heat Flow

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