A nonlinear boundary value problem for some third-order equation (Q2713898)

The author studies the following boundary value problem: Find a~solution to the equation NEWLINE\[NEWLINE u_{tt}-{\partial\over\partial x}F(u_x)-u_{xxt}=f(x,t) \quad (0<x<1,\;0<t<T<+\infty) NEWLINE\]NEWLINE satisfying NEWLINE\[NEWLINE \begin{gathered} u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad 0<x<1, \\ u_x(0,t)-\varphi_0(u_t(0,t))=\psi_0(u(0,t)),\quad 0<t<T, \\ u_x(1,t)-\varphi_1(u_t(0,t))=\psi_1(u(1,t)),\quad 0<t<T. \end{gathered} NEWLINE\]NEWLINE Under some conditions on \(F(\eta)\), \(\varphi_0(\xi)\), \(\varphi_1(\xi)\), \(\psi_0(\xi)\), and \(\psi_1(\xi)\), the problem has a~regular solution for all~\(T\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].