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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].
34A34 Nonlinear ordinary differential equations and systems, general theory
34D05 Asymptotic properties of solutions to ordinary differential equations