American Institute of Mathematical Sciences

Advanced
October  2000, 6(4): 947-973. doi: 10.3934/dcds.2000.6.947

Stabilization of positive solutions for analytic gradient-like systems

Peter Takáč 1,

1. 

Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock, Germany

Received  September 1999 Revised  August 2000 Published  August 2000

The long-time dynamical properties of an arbitrary positive solution $u(t)$, $t\ge 0$, to autonomous gradient-like systems are investigated. These evolutionary systems are generated by semilinear parabolic Dirichlet problems where coefficients and nonlinearities are allowed to be unbounded near the boundary $\partial \Omega$­ of the underlying bounded domain ­$\Omega\subset \mathbb R^N$. Analyticity of the potential is used to show that every positive solution of the system asymptotically approaches a (single) steady-state solution. A key tool in the proof is a Lojasiewicz-Simon-type inequality. Weighted Lebesgue and Sobolev spaces are employed. Important applications include the nonlinear heat and porous medium equations that contain nonlinearities which are not necessarily analytic on the boundary of the domain.
Keywords: Parabolic Dirichlet problem, stabilization of positive solutions, autonomous gradient-like system, Lojasiewicz-Simon-type inequality., weighted Sobolev space, holomorphic semigroup.
Mathematics Subject Classification: 35K55, 47H20, 35B40, 35B5.
Citation: Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947
[1]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[2]

Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092
[3]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[4]

Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020311
[5]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[6]

Tommaso Leonori, Martina Magliocca. Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131
[7]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[8]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
[9]

Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
[10]

Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023
[11]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[12]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063
[13]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[14]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[15]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
[16]

E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731
[17]

John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493
[18]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076
[19]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011
[20]

Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences