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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

Filippo Gazzola 1,

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.
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, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583
References:
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show all references

References:
[1]

71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999.  Google Scholar

[2]

J. Diff. Eq., 174 (2001), 442-463. doi: 10.1006/jdeq.2000.3940.  Google Scholar

[3]

J. Operator Theory, 36 (1996), 179-198.  Google Scholar

[4]

Adv. Diff. Eq., 13 (2008), 959-976.  Google Scholar

[5]

J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3.  Google Scholar

[6]

Nonlin. Anal., 75 (2012), 194-210. doi: 10.1016/j.na.2011.08.022.  Google Scholar

[7]

Nonlin. Anal., 75 (2012), 3510-3530. 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]

C. R. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5.  Google Scholar

[10]

C. R. Acad. Sci. Paris, 335 (2002), 805-810. doi: 10.1016/S1631-073X(02)02567-0.  Google Scholar

[11]

Adv. Diff. Eq., 9 (2004), 1009-1038.  Google Scholar

[12]

Izdat. "Nauka", Moscow, (1964).  Google Scholar

[13]

Disc. Cont. Dynam. Syst., 21 (2008), 1129-1157. doi: 10.3934/dcds.2008.21.1129.  Google Scholar

[14]

Nonlin. Diff. Eq. Appl., 5 (2009), 597-655. doi: 10.1007/s00030-009-0025-x.  Google Scholar

[15]

Nonlinearity, 18 (2005), 717-746. doi: 10.1088/0951-7715/18/2/014.  Google Scholar

[16]

Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131.  Google Scholar

[17]

Progress in Nonlinear Differential Equations and their Applications 56, Boston (MA) etc.: Birkhäuser, (2004). doi: 10.1007/978-1-4612-2050-3.  Google Scholar

[18]

Nonlinearity, 17 (2004), 1075-1099. doi: 10.1088/0951-7715/17/3/017.  Google Scholar

[19]

Calc. Var., 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7.  Google Scholar

[20]

Disc. Cont. Dynam. Syst. S., 1 (2008), 83-87.  Google Scholar

[21]

Nonlinear Analysis, 70 (2009), 2965-2973. doi: 10.1016/j.na.2008.12.039.  Google Scholar

[22]

Proc. Roy. Soc. London A., 441 (1993), 423-432. doi: 10.1098/rspa.1993.0071.  Google Scholar

[23]

Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

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