Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle

Previous Article
Transition semigroups corresponding to Lipschitz dissipative systems
DCDS Home
This Issue
Next Article
Complex Neumann type boundary problem and decomposition of Lebesgue spaces

January & February 2004, 10(1&2): 193-200. doi: 10.3934/dcds.2004.10.193

Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle

Jesus Ildefonso Díaz ^1, and Jacqueline Fleckinger-Pellé ^2,

1.	Dpto. Matemática Aplicada, Univ. Complutense, 28040 Madrid, Spain
2.	CEREMATH – UMR MIP, Université Toulouse 1, Place A.France, F–31042 Toulouse Cedex, France

Received March 2003 Revised August 2003 Published October 2003

Abstract
Related Papers

We study the positivity, for large time, of the solutions to the heat equation $\mathcal Q_a(f,u^0)$:

$\mathcal Q_a(f,u^0)\qquad$ $\partial_tu-\Delta u=au+f(t,x),$ in $Q=]0,\infty [ \times \Omega, $

$u(t,x)=0\qquad$ $(t,x)\in ]0,\infty [ \times \partial \Omega,$

$u(0,x)=u^0(x), \qquad x\in \Omega,$

where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ and $a\in\mathbb R$. We obtain some sufficient conditions for having a finite time $t_p>0$ (depending on $a$ and on the data $u^0$ and $f$ which are not necessarily of the same sign) such that $ u(t,x)>0 \forall t>t_p, a. e. x\in\Omega$.

Keywords: parabolic problems., heat equation, Maximum and antimaximum principle.

Mathematics Subject Classification: 35B30, 35B50, 35K2.

Citation: 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

[1]	Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024
[2]	Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[3]	Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
[4]	Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[5]	Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[6]	H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[7]	Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[8]	Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[9]	E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209
[10]	Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[11]	Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[12]	Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003
[13]	Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[14]	Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17
[15]	Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034
[16]	Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291
[17]	Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[18]	Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[19]	H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[20]	Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences