American Institute of Mathematical Sciences

Advanced
March  2004, 3(1): 151-159. doi: 10.3934/cpaa.2004.3.151

Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations

P. R. Zingano 1,

1. 

Departamento de Matematica Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil

Received  December 2002 Revised  September 2003 Published  January 2004

We examine the large time behavior of the $L^{1}$ norm of solutions $ u(\cdot,t) $ to nonlinear parabolic equations $u_{t} + f(u)_{x} = (\kappa(u) u_{x})_{x}$ in 1-D with (arbitrary) initial states $ u(\cdot,0) $ in $ L^{1}(\mathbb{R}) $, where $ \kappa(u) $ is positive. If $ u(\cdot,t) $, ũ$(\cdot,t) $ are any solutions having the same mass, say $m$, then one has $\| u(\cdot,t) -$ ũ$(\cdot,t) \|_{L^{1}(\mathbb{R})} \rightarrow 0$ as $t \rightarrow \infty $, and the limiting value for the $L^{1}$ norm of either solution is the absolute value of $m$. Other results of interest are also discussed.
Keywords: Asymptotic behavior, $L^1$ norm, advection-difusion equations..
Mathematics Subject Classification: 35B40 (primary), 35B35, 35K15 (secondary.
Citation: P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[1]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
[2]

Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243
[3]

Pia Heins, Michael Moeller, Martin Burger. Locally sparse reconstruction using the $l^{1,\infty}$-norm. Inverse Problems & Imaging, 2015, 9 (4) : 1093-1137. doi: 10.3934/ipi.2015.9.1093
[4]

Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046
[5]

Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations & Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034
[6]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[7]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733
[8]

Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061
[9]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
[10]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[11]

Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251
[12]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[13]

Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463
[14]

Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[15]

Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
[16]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[17]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041
[18]

Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669
[19]

C. García Vázquez, Francisco Ortegón Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure & Applied Analysis, 2005, 4 (3) : 589-612. doi: 10.3934/cpaa.2005.4.589
[20]

Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure & Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences