# American Institute of Mathematical Sciences

October  2010, 14(3): 1199-1210. doi: 10.3934/dcdsb.2010.14.1199

## A class of doubly degenerate parabolic equations with periodic sources

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China, China, China, China

Received  May 2009 Revised  March 2010 Published  July 2010

In this paper, we investigate a class of doubly degenerate parabolic equations with periodic sources subject to homogeneous Dirichlet boundary conditions. By means of the theory of Leray-Schauder degree, we establish the existence of non-trivial nonnegative periodic solutions. The key step is how to establish the uniform bound estimate of approximate solutions, for this purpose we will make use of Moser iteration and some results of the eigenvalue problem for the $p$-Laplacian equation.
Citation: Jiebao Sun, Boying Wu, Jing Li, Dazhi Zhang. A class of doubly degenerate parabolic equations with periodic sources. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1199-1210. doi: 10.3934/dcdsb.2010.14.1199
 [1] Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015 [2] Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831 [3] Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35 [4] Genglin Li, Michael Winkler. Nonnegative solutions to a doubly degenerate nutrient taxis system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 687-704. doi: 10.3934/cpaa.2021194 [5] M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885 [6] Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923 [7] Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481 [8] Sergio Polidoro, Annalaura Rebucci, Bianca Stroffolini. Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1385-1416. doi: 10.3934/cpaa.2022023 [9] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [10] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [11] Paul Deuring, Stanislav Kračmar, Šárka Nečasová. Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 967-979. doi: 10.3934/dcdss.2014.7.967 [12] José Godoy, Nolbert Morales, Manuel Zamora. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4137-4156. doi: 10.3934/dcds.2019167 [13] Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005 [14] Yong Liu, Jing Tian, Xuelin Yong. On the even solutions of the Toda system: A degree argument approach. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1895-1916. doi: 10.3934/cpaa.2021075 [15] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [16] Regina Martínez. On the existence of doubly symmetric "Schubart-like" periodic orbits. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 943-975. doi: 10.3934/dcdsb.2012.17.943 [17] Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481 [18] Jinjing Jiao, Guanghua Shi. Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5157-5180. doi: 10.3934/cpaa.2020231 [19] Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138 [20] Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167

2021 Impact Factor: 1.497