American Institute of Mathematical Sciences

Advanced
March  2008, 1(1): 83-87. doi: 10.3934/dcdss.2008.1.83

Eventual local positivity for a biharmonic heat equation in RN

Filippo Gazzola 1, and  Hans-Christoph Grunau 2,

1. 

Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano
2. 

Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany

Received  September 2006 Revised  September 2007 Published  December 2007

We study the positivity preserving property for the Cauchy problem for the linear fourth order heat equation. Although the complete positivity preserving property fails, we show that it holds eventually on compact sets.
Keywords: eventual local positivity., biharmonic heat equation, instantaneous change of sign.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35K3.
Citation: Filippo Gazzola, Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in RN. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 83-87. doi: 10.3934/dcdss.2008.1.83
[1]

Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129
[2]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193
[3]

Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
[4]

Hans-Christoph Grunau, Guido Sweers. A clamped plate with a uniform weight may change sign. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 761-766. doi: 10.3934/dcdss.2014.7.761
[5]

Dario Pighin, Enrique Zuazua. Controllability under positivity constraints of semilinear heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 935-964. doi: 10.3934/mcrf.2018041
[6]

Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465
[7]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[8]

Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711
[9]

Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
[10]

Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847
[11]

Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020008
[12]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305
[13]

Hailong Ye, Jingxue Yin. Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1743-1755. doi: 10.3934/dcdsb.2017083
[14]

Marc Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinetic & Related Models, 2015, 8 (2) : 281-308. doi: 10.3934/krm.2015.8.281
[15]

Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405
[16]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411
[17]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991
[18]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
[19]

Zongming Guo, Xiaohong Guan, Yonggang Zhao. Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2613-2636. doi: 10.3934/dcds.2019109
[20]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences