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August  2013, 33(8): 3583-3597. doi: 10.3934/dcds.2013.33.3583

## On the moments of solutions to linear parabolic equations involving the biharmonic operator

 1 Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32, Milano, 20133

Received  June 2012 Revised  August 2012 Published  January 2013

We consider the solutions to Cauchy problems for the parabolic equation $u_\tau +\Delta^2u=0$ in $\mathbb{R}_+\times\mathbb{R}^n$, with fast decay initial data. We study the behavior of their moments. This enables us to give a more precise description of the sign-changing behavior of solutions corresponding to positive initial data.
Citation: Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583
##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, "Special Functions," 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999. [2] G. Barbatis, Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients, J. Diff. Eq., 174 (2001), 442-463. doi: 10.1006/jdeq.2000.3940. [3] G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniformly elliptic operators, J. Operator Theory, 36 (1996), 179-198. [4] E. Berchio, On the sign of solutions to some linear parabolic biharmonic equations, Adv. Diff. Eq., 13 (2008), 959-976. [5] G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3. [6] J. W. Cholewa and A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in $\mathbb{R}^N2$, Nonlin. Anal., 75 (2012), 194-210. doi: 10.1016/j.na.2011.08.022. [7] J. W. Cholewa and A. Rodriguez-Bernal, Dissipative mechanism of a semilinear higher order parabolic equations in $\mathbb{R}^N2$, Nonlin. Anal., 75 (2012), 3510-3530. doi: 10.1016/j.na.2012.01.011. [8] J. W. Cholewa and A. Rodriguez-Bernal, On the Cahn-Hilliard equation in $H^1(\mathbb{R}^N)$, to appear in J. Diff. Eq.. [9] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5. [10] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range, C. R. Acad. Sci. Paris, 335 (2002), 805-810. doi: 10.1016/S1631-073X(02)02567-0. [11] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Diff. Eq., 9 (2004), 1009-1038. [12] S. D. Eidelman, Parabolicheskie sistemy, Izdat. "Nauka", Moscow, (1964). [13] A. Ferrero, F. Gazzola and H.-Ch. Grunau, Decay and eventual local positivity for biharmonic parabolic equations, Disc. Cont. Dynam. Syst., 21 (2008), 1129-1157. doi: 10.3934/dcds.2008.21.1129. [14] V. A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach, Nonlin. Diff. Eq. Appl., 5 (2009), 597-655. doi: 10.1007/s00030-009-0025-x. [15] V. A. Galaktionov and P. J. Harwin, Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation, Nonlinearity, 18 (2005), 717-746. doi: 10.1088/0951-7715/18/2/014. [16] V. A. Galaktionov and S. I. Pohožaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131. [17] V. A. Galaktionov and J. L. Vázquez, A stability technique for evolution partial differential equations. A dynamical systems approach, Progress in Nonlinear Differential Equations and their Applications 56, Boston (MA) etc.: Birkhäuser, (2004). doi: 10.1007/978-1-4612-2050-3. [18] V. A. Galaktionov and J. F. Williams, On very singular similarity solutions of a higher-order semilinear parabolic equation, Nonlinearity, 17 (2004), 1075-1099. doi: 10.1088/0951-7715/17/3/017. [19] F. Gazzola and H.-Ch. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var., 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7. [20] F. Gazzola and H.-Ch. Grunau, Eventual local positivity for a biharmonic heat equation in $\mathbb{R}^{N}$, Disc. Cont. Dynam. Syst. S., 1 (2008), 83-87. [21] F. Gazzola and H.-Ch. Grunau, Some new properties of biharmonic heat kernels, Nonlinear Analysis, 70 (2009), 2965-2973. doi: 10.1016/j.na.2008.12.039. [22] X. Li and R. Wong, Asymptotic behaviour of the fundamental solution to ${\partial u}/{\partial t}=-(-\Delta)^m u$, Proc. Roy. Soc. London A., 441 (1993), 423-432. doi: 10.1098/rspa.1993.0071. [23] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.

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##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, "Special Functions," 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999. [2] G. Barbatis, Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients, J. Diff. Eq., 174 (2001), 442-463. doi: 10.1006/jdeq.2000.3940. [3] G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniformly elliptic operators, J. Operator Theory, 36 (1996), 179-198. [4] E. Berchio, On the sign of solutions to some linear parabolic biharmonic equations, Adv. Diff. Eq., 13 (2008), 959-976. [5] G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3. [6] J. W. Cholewa and A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in $\mathbb{R}^N2$, Nonlin. Anal., 75 (2012), 194-210. doi: 10.1016/j.na.2011.08.022. [7] J. W. Cholewa and A. Rodriguez-Bernal, Dissipative mechanism of a semilinear higher order parabolic equations in $\mathbb{R}^N2$, Nonlin. Anal., 75 (2012), 3510-3530. doi: 10.1016/j.na.2012.01.011. [8] J. W. Cholewa and A. Rodriguez-Bernal, On the Cahn-Hilliard equation in $H^1(\mathbb{R}^N)$, to appear in J. Diff. Eq.. [9] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5. [10] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range, C. R. Acad. Sci. Paris, 335 (2002), 805-810. doi: 10.1016/S1631-073X(02)02567-0. [11] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohožaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Diff. Eq., 9 (2004), 1009-1038. [12] S. D. Eidelman, Parabolicheskie sistemy, Izdat. "Nauka", Moscow, (1964). [13] A. Ferrero, F. Gazzola and H.-Ch. Grunau, Decay and eventual local positivity for biharmonic parabolic equations, Disc. Cont. Dynam. Syst., 21 (2008), 1129-1157. doi: 10.3934/dcds.2008.21.1129. [14] V. A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach, Nonlin. Diff. Eq. Appl., 5 (2009), 597-655. doi: 10.1007/s00030-009-0025-x. [15] V. A. Galaktionov and P. J. Harwin, Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation, Nonlinearity, 18 (2005), 717-746. doi: 10.1088/0951-7715/18/2/014. [16] V. A. Galaktionov and S. I. Pohožaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131. [17] V. A. Galaktionov and J. L. Vázquez, A stability technique for evolution partial differential equations. A dynamical systems approach, Progress in Nonlinear Differential Equations and their Applications 56, Boston (MA) etc.: Birkhäuser, (2004). doi: 10.1007/978-1-4612-2050-3. [18] V. A. Galaktionov and J. F. Williams, On very singular similarity solutions of a higher-order semilinear parabolic equation, Nonlinearity, 17 (2004), 1075-1099. doi: 10.1088/0951-7715/17/3/017. [19] F. Gazzola and H.-Ch. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var., 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7. [20] F. Gazzola and H.-Ch. Grunau, Eventual local positivity for a biharmonic heat equation in $\mathbb{R}^{N}$, Disc. Cont. Dynam. Syst. S., 1 (2008), 83-87. [21] F. Gazzola and H.-Ch. Grunau, Some new properties of biharmonic heat kernels, Nonlinear Analysis, 70 (2009), 2965-2973. doi: 10.1016/j.na.2008.12.039. [22] X. Li and R. Wong, Asymptotic behaviour of the fundamental solution to ${\partial u}/{\partial t}=-(-\Delta)^m u$, Proc. Roy. Soc. London A., 441 (1993), 423-432. doi: 10.1098/rspa.1993.0071. [23] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.
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