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On steady solutions of the Kuramoto-Sivashinsky equation. (English) Zbl 1006.34010
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 223, 45-51 (2002).
The author studies the Kuramoto-Sivashinsky equation $u_t+ u_{xxxx}+ u_{xx}+ \textstyle{{1\over 2}}u^2_x= 0,\quad u= u(x,t).\tag{1}$ This equation can be reduced, by using an appropriate transformation, into one of the following third-order ordinary differential equations
(2) $$\lambda y'''(x)+ y'(x)= 1- y(x)^2$$, $$\lambda:= c^2/2$$, $$c\approx\sqrt{1.2}$$,
(3) $$\lambda y'''(x)+ y'(x)= -y(x)^2$$,
and, respectively,
(4) $$\varepsilon y'''(x)+ y'(x)= \cos y(x)$$, $$\varepsilon> 0$$.
The attention is focused on two problems:
(i) a proof of the nonexistence of monotonic global solutions;
(ii) the existence of solutions which blow up on bounded intervals.
The first main result states that if $$\lambda> 8/27$$ in (2) (resp. $$\varepsilon\geq 32/27$$ in (4)), then there is no solution $$y$$ satisfying $$y'> 0$$ and $$y(x)\to \pm 1$$ (resp. $$y(x)\to \pm\pi/2$$) as $$x\to\pm\infty$$, respectively.
To give the second result, the author supplements (2) or (3) with the initial condition
(5) $$y(0)= 0$$, $$y'(0)= -\beta< 0$$, $$y''(0)= 0$$.
For all large values of $$\beta> 0$$, there exists finite $$x_\beta> 0$$ such that there are solutions $$y$$ to (2), (5) verifying $$y'(x)< 0$$ on $$-x_\beta< x< x_\beta$$ and $$y(x)\to \mp\infty$$ as $$x\to \pm x_\beta$$, respectively. Moreover, there hold $\limsup_{x\uparrow x_\beta} (x_\beta- x)^3(- y(x))\leq 30\sqrt{10} \lambda,\;\liminf_{x\uparrow x_\beta} (x_\beta- x)^3(-y(x))\geq 24\lambda.$ Connected to (3), (5), there also exist similar blow-up solutions with possibly different $$x_\beta$$.
The proof of these results is based on the reduction of the third-order equation into a second-order one, which is assured by the monotonicity of the solution.
For the entire collection see [Zbl 0972.00046].
##### MSC:
 34A34 Nonlinear ordinary differential equations and systems, general theory 34D05 Asymptotic properties of solutions to ordinary differential equations