The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’ equation

Deng, Keng; Kwong, Man Kam; Levine, Howard A.

doi:10.1090/qam/1146631

Authors: Keng Deng, Man Kam Kwong and Howard A. Levine
Journal: Quart. Appl. Math. 50 (1992), 173-200
MSC: Primary 35Q53; Secondary 35B40, 76D05
DOI: https://doi.org/10.1090/qam/1146631
MathSciNet review: MR1146631
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Abstract: We study the long time behavior of solutions of Burgers’s equation with nonlocal nonlinearities: ${u_t} = {u_{xx}} + \varepsilon u{u_x} + \frac {1}{2}\left ( {a\parallel u\left ( { \cdot , t} \right ){\parallel ^{p - 1}} + b} \right )u, 0 < x < 1, \\ a, \varepsilon \in \mathbb {R}, b > 0, p > 1$, subject to $u\left ( {0, t} \right ) = u\left ( {1, t} \right ) = 0$. A stability-instability analysis is given in some detail, and some finite time blow up results are given.

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