Mixed Problem for a General 1D Wave Equation with Characteristic Second Derivatives in a Nonstationary Boundary Mode

We use a modified method of characteristics to derive an explicit formula for the unique stable classical solution of a linear mixed problem for a general one-dimensional wave equation in the first quarter plane with time-varying characteristic second derivatives in the boundary condition. We find a criterion for the Hadamard well-posedness of this problem in the form of conditions on the right-hand side of the equation and the initial and boundary data ensuring the unique and stable global solvability of the problem in the set of classical solutions. The well-posedness criterion includes necessary and sufficient smoothness conditions on the initial data of the problem and conditions for the consistency of the boundary condition with the initial conditions and the equation itself. The data smoothness conditions ensure that the solution is twice continuously differentiable, while the consistency conditions are needed for the solution to be smooth across the critical characteristic of the equation.