Behavior of solutions of Burgers’s equation with nonlocal boundary conditions. II
Author:
Keng Deng
Journal:
Quart. Appl. Math. 52 (1994), 553-567
MSC:
Primary 35Q53; Secondary 35B40, 35K60
DOI:
https://doi.org/10.1090/qam/1292205
MathSciNet review:
MR1292205
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Abstract: We study the large-time behavior of positive solutions of Burgers’s equation ${u_t} = {u_{xx}} + \varepsilon u{u_x}, 0 < x < 1, t > 0\left ( {\varepsilon > 0} \right )$, subject to the nonlocal boundary condition: $- {u_x}\left ( {0, t} \right ) - \frac {1}{2}\varepsilon {u^2}\left ( {0, t} \right ) = a{u^p}\left ( {0, t} \right ){\left ( {\int _0^1 u \left ( {x, t} \right )dx} \right )^q},u\left ( {1, t} \right ) = 0 \\ \left ( {0 < p, q < \infty } \right )$. The steady-state problem is analyzed in detail, and the result about finite-time blow-up is proved.
K. Deng, Behavior of solutions of Burgers’ equation with nonlocal boundary conditions (I), J. Differential Equations (to appear)
K. Deng, M. K. Kwong, and H. A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’s equation, Quart. Appl. Math. 50, 173–200 (1992)
H. A. Levine, Stability and instability for solutions of Burgers’ equation with a semilinear boundary condition, SIAM J. Math. Anal. 19, 312–336 (1988)
K. Deng, Behavior of solutions of Burgers’ equation with nonlocal boundary conditions (I), J. Differential Equations (to appear)
K. Deng, M. K. Kwong, and H. A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’s equation, Quart. Appl. Math. 50, 173–200 (1992)
H. A. Levine, Stability and instability for solutions of Burgers’ equation with a semilinear boundary condition, SIAM J. Math. Anal. 19, 312–336 (1988)
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Article copyright:
© Copyright 1994
American Mathematical Society