
Previous Article
Asymptotic analysis of a simple model of fluidstructure interaction
 NHM Home
 This Issue

Next Article
Stability and Riesz basis property of a starshaped network of EulerBernoulli beams with joint damping
Large time behavior of nonlocal aggregation models with nonlinear diffusion
1.  Westfälische WilhelmsUniversität Münster, Institutfür Numerische und Angewandte Mathematik, Einsteinstr. 62, D 48149 Münster, Germany 
2.  Division of Mathematics for Engineering, Piazzale E. Pontieri, 2 Monteluco di Roio, 67040 L'Aquila, Italy 
Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients.
All these results are obtained via a reformulation of the equations considered using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension.
[1] 
Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems  A, 2010, 27 (1) : 301323. doi: 10.3934/dcds.2010.27.301 
[2] 
Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems  A, 2015, 35 (4) : 13911407. doi: 10.3934/dcds.2015.35.1391 
[3] 
Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems  B, 2017, 22 (2) : 407419. doi: 10.3934/dcdsb.2017019 
[4] 
Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete & Continuous Dynamical Systems  A, 2010, 27 (4) : 15531570. doi: 10.3934/dcds.2010.27.1553 
[5] 
Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 463473. doi: 10.3934/cpaa.2005.4.463 
[6] 
Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533556. doi: 10.3934/cpaa.2017027 
[7] 
Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems  A, 1995, 1 (2) : 237252. doi: 10.3934/dcds.1995.1.237 
[8] 
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems  A, 2017, 37 (11) : 59435977. doi: 10.3934/dcds.2017258 
[9] 
Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 17071714. doi: 10.3934/cpaa.2011.10.1707 
[10] 
P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151159. doi: 10.3934/cpaa.2004.3.151 
[11] 
Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems  A, 1997, 3 (3) : 383400. doi: 10.3934/dcds.1997.3.383 
[12] 
Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 16171637. doi: 10.3934/cpaa.2010.9.1617 
[13] 
Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems  S, 2016, 9 (2) : 383392. doi: 10.3934/dcdss.2016002 
[14] 
Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2020, 10 (1) : 113140. doi: 10.3934/mcrf.2019032 
[15] 
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems  A, 2019, 0 (0) : 115. doi: 10.3934/dcds.2019229 
[16] 
Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxisfluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems  A, 2010, 28 (4) : 14371453. doi: 10.3934/dcds.2010.28.1437 
[17] 
Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems  A, 2017, 37 (6) : 34353465. doi: 10.3934/dcds.2017146 
[18] 
Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic FokkerPlanck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 14271441. doi: 10.3934/krm.2018056 
[19] 
Kazuo Yamazaki, Xueying Wang. Global wellposedness and asymptotic behavior of solutions to a reactionconvectiondiffusion cholera epidemic model. Discrete & Continuous Dynamical Systems  B, 2016, 21 (4) : 12971316. doi: 10.3934/dcdsb.2016.21.1297 
[20] 
Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Nonexistence of positive stationary solutions for a class of semilinear PDEs with random coefficients. Networks & Heterogeneous Media, 2010, 5 (4) : 745763. doi: 10.3934/nhm.2010.5.745 
2018 Impact Factor: 0.871
Tools
Metrics
Other articles
by authors
[Back to Top]