On linearization of hyperbolic equations with integral load in the main part using an a priori estimate of their solutions

A priori estimates are established for solutions of one-dimensional inhomogeneous hyperbolic equations with an integral load in the main part, which has the form a(s) = sp, for p = 1, 0.5 and -1, with inhomogeneous initial and homogeneous boundary conditions. Here s is the integral over the space variable of the square of the modulus of the derivative of the solution of the equation with respect to x. Examples of linearization of loaded equations by substituting the right-hand sides of the estimates for a(s) are given.