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Large-time behavior of solutions to certain quasilinear parabolic equations in several space dimensions


Authors: Patricia Bauman and Daniel Phillips
Journal: Proc. Amer. Math. Soc. 96 (1986), 237-240
MSC: Primary 35B40; Secondary 35K55
DOI: https://doi.org/10.1090/S0002-9939-1986-0818451-2
MathSciNet review: 818451
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Abstract: We consider the Cauchy problem, $ {u_t} + {\text{div}}f(u) = \Delta u$ for $ x \in {{\mathbf{R}}^n},t > 0$ with $ u(x,0) = {u_0}(x)$. For $ n = 1$, suppose $ f'' > 0$ and $ \smallint \left\vert {{u_0} - \phi } \right\vert dx < \infty $ where $ \phi $ is piecewise constant and $ \phi (x) \to {u^ + }({u^ - })$ as $ x \to + \infty ( - \infty )$. A result of Il'in and Oleinik states that if $ \phi (x - kt)$ is an entropy solution of $ {u_t} + {\text{div}}f(u) = 0$, then $ u(x,t)$ approaches a traveling wave solution, $ \tilde u(x - kt)$, as $ t \to \infty $, with $ \tilde u(x) \to {u^ + }({u^ - })$ as $ x \to + \infty ( - \infty )$. We give two examples which show that this result does not hold for $ n \geqslant 2$.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0818451-2
Article copyright: © Copyright 1986 American Mathematical Society

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