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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 and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
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