-
Previous Article
Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities
- DCDS Home
- This Issue
-
Next Article
Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations
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 |
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. |
show all references
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. |
[1] |
Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068 |
[2] |
John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349 |
[3] |
Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure and Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519 |
[4] |
Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393 |
[5] |
Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 |
[6] |
Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617 |
[7] |
Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234 |
[8] |
Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189 |
[9] |
Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667 |
[10] |
Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4367-4381. doi: 10.3934/dcdss.2021112 |
[11] |
Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032 |
[12] |
Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935 |
[13] |
José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1 |
[14] |
Feliz Minhós, João Fialho. Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities. Conference Publications, 2013, 2013 (special) : 555-564. doi: 10.3934/proc.2013.2013.555 |
[15] |
Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033 |
[16] |
To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694 |
[17] |
Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure and Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245 |
[18] |
John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291 |
[19] |
Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225 |
[20] |
John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]