Lower regularity solutions of the non-homogeneous boundary-value problem for a higher order Boussinesq equation in a quarter plane. (August 2022) - Li, Shenghao; Chen, Min; Yang, Xin; Zhang, Bing-Yu
Abstract: We continue to study the initial–boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: u t t − u x x + β u x x x x − u x x x x x x + ( u 2 ) x x = 0, x, t ∈ R +, u ( x, 0 ) = φ ( x ), u t ( x, 0 ) = ψ ′ ′ ( x ), u ( 0, t ) = h 1 ( t ), u x x ( 0, t ) = h 2 ( t ), u x x x x ( 0, t ) = h 3 ( t ), where β = ± 1 . We show that the problem is locally analytically well-posed in the space H s ( R + ) for any s > − 3 4 with the initial-value data ( φ, ψ ) ∈ H s ( R + ) × H s − 1 ( R + ) and the boundary-value data ( h 1, h 2, h 3 ) ∈ H s + 1 3 ( R + ) × H s − 1 3 ( R + ) × H s − 3 3 ( R + ) .