Vol. 19, No. 2, 1966

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Pointwise bounds for the second initial-boundary value problem of parabolic type

Fred James Bellar, Jr.

Vol. 19 (1966), No. 2, 205–219

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.

Mathematical Subject Classification
Primary: 35.65
Received: 18 June 1965
Revised: 18 October 1965
Published: 1 November 1966
Fred James Bellar, Jr.