Mixed initial-boundary value problem for the capillary wave equation
Summary: We study the mixed initial-boundary value problem for the capillary wave equation: \(i u_t +| u|^2 u =| \partial_x|^{3/2} u\), \(t>0\), \(x>0\); \(u(x, 0)=u_0(x)\), \(x>0\); \(u(0,t)+\beta u_x(0,t)=h(t)\), \(t > 0\), where \(| \partial_x|^{3 / 2} u = (1 / \sqrt{2 \pi}) \int_0^\infty\left(\operatorname{sign}(x-y)/\sqrt{| x - y|}\right) u_{y y}(y)\;dy\). 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.
scientific article; zbMATH DE number 6625305
scientific article