Abstract
We study in a rectangle Q_T = (0, T)×(0, 1) global well-posedness of nonhomo-geneous initial-boundary value problems for general odd-order quasilinear partial differential equations. This class of equations includes well-known Korteweg–de Vries and Kawahara equations which model the dynamics of long small-amplitude waves in various media. Our study is motivated by physics and numerics and our main goal is to formulate a correct nonhomogeneous initial-boundary value problem under consideration in a bounded interval and to prove the existence and uniqueness of global in time weak and regular solutions in a large scale of Sobolev spaces as well as to study decay of solutions while t → ∞.
Highlights
The analogous equality can be written for problem (1.1)–(1.4) in the case of zero boundary data
In [20] we proved that this phenomenon takes place for general dispersive equations of odd-orders for homogeneous boundary data and the right-hand side
We generalize this result proving the exponential stability for small nonhomogeneous boundary data and the right-hand side
Summary
The analogous equality can be written for problem (1.1)–(1.4) in the case of zero boundary data. In the general case one has to make this data zero with the help of a certain auxiliary function. In our paper [20] we constructed a solution of an initial-boundary value problem for the linear homogeneous equation ut + (−1)l+1∂x2l+1u = 0 (1.5)
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