# American Institute of Mathematical Sciences

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

## Stabilization of positive solutions for analytic gradient-like systems

 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.
Citation: Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092 [3] Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311 [4] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [5] Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 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 and Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131 [7] Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 [8] Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073 [9] Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791 [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 and Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023 [11] Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063 [12] José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138 [13] Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete and Continuous Dynamical Systems, 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 and Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061 [15] E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731 [16] Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353 [17] Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076 [18] Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011 [19] Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257 [20] John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493

2020 Impact Factor: 1.392