A Priori Estimates for Derivative Solutions of One-Dimensional Inhomogeneous Heat Conduction Equations with an Integral Load in the Main Part

Abstract

This article considers the second initial-boundary value problem with homogeneous boundary conditions for a one-dimensional modified heat equation. The modification consists in replacing the temperature-conductivity coefficient with an integral load. In our case, it has the form of a power function of the integral of the square of the modulus of the derivative of the solution of the equation with respect to the spatial variable. Equations with such a load are associated with some practically important parabolic equations with a power nonlinearity in the main part. This makes it possible to use previously found solutions of loaded problems to start the successive approximation to solutions of the 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 integral load: the square of the norm of the derivative of the solution with respect to x in the space L2 in natural, inverse to natural, and integer negative powers. The corresponding a priori inequalities are established. Their right sides are used to pass to linearized equations. Examples of linearization of heat conduction equations with an integral load in the main part are given.