Abstract
The present paper is a survey concerned with certain aspects of solvability and well-posedness of initial and initial-boundary value problems for various quasilinear evolution equations of the third order. This class includes, for example, Korteweg - de Vries (KdV) and Zakharov - Kuznetsov (ZK) equations.
Highlights
The present paper is a survey concerned with certain aspects of solvability and well-posedness of initial and initial-boundary value problems for various quasilinear evolution equations of the third order
The potentials J, Q, R were used in [39] to construct a suitable solution of the linear problem in QT (7), (2), (21) and similar result on local well-posedness of the problem (1), (2), (21) was established under natural assumptions on the boundary data
For u0 ∈ L2(R2−), u2 ∈ Hts,/y3,s((0, T ) × R), s > 3/2, u3 ∈ L2((0, T ) × R), proved in [40]. This result is based on the first conservation law (45), the local smoothing effect of the (46) type and certain properties of the boundary potential J2
Summary
The present paper is a survey concerned with certain aspects of solvability and well-posedness of initial and initial-boundary value problems for various quasilinear evolution equations of the third order. Were constructed with the use of the rest roots of the equation (35) in [39] for the problem in Π−T (Q(·, 0 − 0; μ) = Rx(·, 0 − 0; μ) = μ, Qx(·, 0 − 0; μ) = R(·, 0 − 0; μ) = 0), and local well-posedness for the problem (1), (2), (20) was proved under natural assumptions on the boundary data.
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