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

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  • 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.
    Mathematics Subject Classification: 35B40 (primary), 35B35, 35K15 (secondary).

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