A priori estimates for derivative solutions of one-dimensional inhomogeneous wave equations with an integral load in the main part

In this paper, the second initial-boundary value problem with homogeneous boundary conditions for a one-dimensional modified wave equation is considered. The modification consists in replacing the coefficient at the second spatial derivative with an integral load. In our case, it is a power function of the integral of the squared modulus of the derivative of the equation solution with respect to the spatial variable. Equations with such a load are associated with some practically important hyperbolic equations with a power nonlinearity in the main part. This makes it possible to use previously found solutions of loaded problems to start the process of successive approximation to solutions of nonlinear problems reduced to them. In this case, with respect to the original nonlinear equation, the loaded equation contains a weakened nonlinearity. Linearization of the loaded equation makes it possible to find its approximate solution. The article considers three cases of the integral load. It is the squared norm of the derivative of the solution with respect to $x$ in the space $L_2$ in natural, inverse to natural, and two integer negative powers. The corresponding a priori inequalities are established. Their right-hand side is used to pass to linearized equations. Examples of linearization by this method of wave equations with an integral load in the main part are given.