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

Abstract

We study the long time behavior of solutions of Burgers’s equation with nonlocal nonlinearities: u t = u x x + ε u u x + 1 2 ( a ∥ u ( ⋅ , t ) ∥ p − 1 + b ) u , 0 > x > 1 , a , ε ∈ R , b > 0 , p > 1 {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 ( 0 , t ) = u ( 1 , t ) = 0 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.