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

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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

Abstract
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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.

Keywords: parabolic equations, moments of solutions., Fourth order.

Mathematics Subject Classification: 35K3.

Citation: Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems - A, 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, 71 (1999). Google Scholar
[2]	G. Barbatis, Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients,, J. Diff. Eq., 174 (2001), 442. doi: 10.1006/jdeq.2000.3940. Google Scholar
[3]	G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniformly elliptic operators,, J. Operator Theory, 36 (1996), 179. Google Scholar
[4]	E. Berchio, On the sign of solutions to some linear parabolic biharmonic equations,, Adv. Diff. Eq., 13 (2008), 959. Google Scholar
[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. doi: 10.1016/S0022-247X(03)00062-3. Google Scholar
[6]	J. W. Cholewa and A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in $\mathbbR^N$,, Nonlin. Anal., 75 (2012), 194. doi: 10.1016/j.na.2011.08.022. Google Scholar
[7]	J. W. Cholewa and A. Rodriguez-Bernal, Dissipative mechanism of a semilinear higher order parabolic equations in $\mathbbR^N$,, Nonlin. Anal., 75 (2012), 3510. doi: 10.1016/j.na.2012.01.011. Google Scholar
[8]	J. W. Cholewa and A. Rodriguez-Bernal, On the Cahn-Hilliard equation in $H^1(\mathbbR^N)$,, to appear in J. Diff. Eq.., (). Google Scholar
[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. doi: 10.1016/S0764-4442(00)00124-5. Google Scholar
[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. doi: 10.1016/S1631-073X(02)02567-0. Google Scholar
[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. Google Scholar
[12]	S. D. Eidelman, Parabolicheskie sistemy,, Izdat., (1964). Google Scholar
[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. doi: 10.3934/dcds.2008.21.1129. Google Scholar
[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. doi: 10.1007/s00030-009-0025-x. Google Scholar
[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. doi: 10.1088/0951-7715/18/2/014. Google Scholar
[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. doi: 10.1512/iumj.2002.51.2131. Google Scholar
[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, 56 (2004). doi: 10.1007/978-1-4612-2050-3. Google Scholar
[18]	V. A. Galaktionov and J. F. Williams, On very singular similarity solutions of a higher-order semilinear parabolic equation,, Nonlinearity, 17 (2004), 1075. doi: 10.1088/0951-7715/17/3/017. Google Scholar
[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. doi: 10.1007/s00526-007-0096-7. Google Scholar
[20]	F. Gazzola and H.-Ch. Grunau, Eventual local positivity for a biharmonic heat equation in $\mathbbR^n$,, Disc. Cont. Dynam. Syst. S., 1 (2008), 83. Google Scholar
[21]	F. Gazzola and H.-Ch. Grunau, Some new properties of biharmonic heat kernels,, Nonlinear Analysis, 70 (2009), 2965. doi: 10.1016/j.na.2008.12.039. Google Scholar
[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. doi: 10.1098/rspa.1993.0071. Google Scholar
[23]	E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar

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References:

[1]	G. E. Andrews, R. Askey and R. Roy, "Special Functions,", 71 of Encyclopedia of Mathematics and its Applications, 71 (1999). Google Scholar
[2]	G. Barbatis, Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients,, J. Diff. Eq., 174 (2001), 442. doi: 10.1006/jdeq.2000.3940. Google Scholar
[3]	G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniformly elliptic operators,, J. Operator Theory, 36 (1996), 179. Google Scholar
[4]	E. Berchio, On the sign of solutions to some linear parabolic biharmonic equations,, Adv. Diff. Eq., 13 (2008), 959. Google Scholar
[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. doi: 10.1016/S0022-247X(03)00062-3. Google Scholar
[6]	J. W. Cholewa and A. Rodriguez-Bernal, Linear and semilinear higher order parabolic equations in $\mathbbR^N$,, Nonlin. Anal., 75 (2012), 194. doi: 10.1016/j.na.2011.08.022. Google Scholar
[7]	J. W. Cholewa and A. Rodriguez-Bernal, Dissipative mechanism of a semilinear higher order parabolic equations in $\mathbbR^N$,, Nonlin. Anal., 75 (2012), 3510. doi: 10.1016/j.na.2012.01.011. Google Scholar
[8]	J. W. Cholewa and A. Rodriguez-Bernal, On the Cahn-Hilliard equation in $H^1(\mathbbR^N)$,, to appear in J. Diff. Eq.., (). Google Scholar
[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. doi: 10.1016/S0764-4442(00)00124-5. Google Scholar
[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. doi: 10.1016/S1631-073X(02)02567-0. Google Scholar
[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. Google Scholar
[12]	S. D. Eidelman, Parabolicheskie sistemy,, Izdat., (1964). Google Scholar
[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. doi: 10.3934/dcds.2008.21.1129. Google Scholar
[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. doi: 10.1007/s00030-009-0025-x. Google Scholar
[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. doi: 10.1088/0951-7715/18/2/014. Google Scholar
[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. doi: 10.1512/iumj.2002.51.2131. Google Scholar
[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, 56 (2004). doi: 10.1007/978-1-4612-2050-3. Google Scholar
[18]	V. A. Galaktionov and J. F. Williams, On very singular similarity solutions of a higher-order semilinear parabolic equation,, Nonlinearity, 17 (2004), 1075. doi: 10.1088/0951-7715/17/3/017. Google Scholar
[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. doi: 10.1007/s00526-007-0096-7. Google Scholar
[20]	F. Gazzola and H.-Ch. Grunau, Eventual local positivity for a biharmonic heat equation in $\mathbbR^n$,, Disc. Cont. Dynam. Syst. S., 1 (2008), 83. Google Scholar
[21]	F. Gazzola and H.-Ch. Grunau, Some new properties of biharmonic heat kernels,, Nonlinear Analysis, 70 (2009), 2965. doi: 10.1016/j.na.2008.12.039. Google Scholar
[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. doi: 10.1098/rspa.1993.0071. Google Scholar
[23]	E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar

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