By means of the divergence
theorem and certain well-known inequalities, the author presents explicit a priori
pointwise bounds for the solution of the linear and nonlinear second initial-boundary
value problem of the parabolic type. The desired result is obtained by using the
parabolic form of Green’s second identity with an appropriately defined
parametrix serving as the first function of the identity and the difference of
the solution and an arbitrary function which approximates the given data
as the second. By means of various well-known inequalities, the unknown
integrals in the resulting expression are bounded in terms of volume and surface
integrals of the square of known functions. In the linear case the form of the
bound is such that it may be improved by employing the Rayleigh-Ritz
technique.
