Mixed Initial-Boundary Value Problem for the Capillary Wave Equation

Abstract EN

We study the mixed initial-boundary value problem for the capillary wave equation: i u t + u 2 u = ∂ x 3 / 2 u , t > 0 , x > 0 ; u ( x , 0 ) = u 0 ( x ) , x > 0 ; u ( 0 , t ) + β u x ( 0 , t ) = h ( t ) , t > 0 , where ∂ x 3 / 2 u = ( 1 / 2 π ) ∫ 0 ∞ sign x - y / x - y u y y ( y ) d y .

We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions.

Also we are interested in the study of the asymptotic behavior of solutions.