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A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval. (English) Zbl 1446.35049

Summary: This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval \((0, L)\) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the space \(H^s (0, L)\) for any \(s > -2\) with the initial data in \(H^s(0, L)\) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the space \(H^s(0, L)\) for any \(s < -2\) in the sense that the corresponding solution map fails to be in \(C^2\).

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35C15 Integral representations of solutions to PDEs
35D30 Weak solutions to PDEs
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References:

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