On the linearization of parabolic equations with integral load in the main part by a priori estimation of their solutions
A method of linearization of one-dimensional inhomogeneous parabolic equations with an integral load in the main part is described. The integral load here is some function a(s), where s is the integral over the spatial variable from the square of the modulus of the derivative of the solution of the equation in x. Inhomogeneous initial and homogeneous boundary conditions are considered. In the case of a linear function a(s), as well as in two cases when a(s) is not linear, a priori estimates of the derivative of the solution of the problem in the L2 space are established. The right-hand sides of these estimates are used to linearize the corresponding equations. Examples of linearization of loaded equations by this method are given.