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Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation. (English) Zbl 0845.35111

The interest of the paper is focused on the smoothing properties of the KdV equation for the infinite interval case. And to this end one introduces the operator \(A_\alpha\) defined by \((A_\alpha w)(x)= - w'''(x)\) with domain \[ D(A_\alpha)\equiv \{w\in H^3(0, 1)\mid w(1)= w(0), w'(1)= \alpha w'(0), w'' (1)= w''(0)\}. \] The spectral properties of \(A_\alpha\) are studied in detail. Then the various smoothing properties are given. One shows that the initial boundary value problem for the KdV equation is well posed in \(H^{3n+ 1}(0, 1)\) for any \(n\geq 0\). And then the small amplitude solutions of the KdV equation are shown to decay exponentially to the mean value of the initial state in \(L^2(0, 1)\) as \(t\uparrow \infty\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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