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• W2022693238 abstract "It is shown that the class of functions satisfying Ju(x, t)l < Mealxl forms a uniqueness class for the Cauchy problem for pseudoparabolic equations. The surprising fact is that, unlike the case of parabolic equations, the constant a is not arbitrary but depends on the coefficients of the equation. Introduction. This paper is concerned with determining the uniqueness class for the Cauchy problem for the pseudoparabolic equation Lu Mu, c(x)u, + Mu = 0. (1.1) Here M is an elliptic partial differential operator of second order and c(x) is a positive coefficient. All coefficients are assumed to be bounded. Since solutions of (1.1) are closely related to solutions of the associated parabolic equation Pu = C(x)u, Mu = 0 (1.2) (cf. [1], [3]), one would expect a similarity in the uniqueness class for the Cauchy problem, that is a solution that takes prescribed values on the axis t = 0. For the parabolic equation if Iu(x, t)I < Ce'X for arbitrary fixed a, then there is a unique solution for the Cauchy problem. For equation (1.1) we shall see that the uniqueness class consists of functions of first order growth, that is, Iu(x, t)I < ceaIx1. The surprising fact is that a in this case is no longer arbitrary but depends on the operator L. More specifically it depends on the lower bound for the coefficient c(x) and the modulus of ellipticity of the operator M. We shall illustrate this with an example, deferring the statement of the main theorem until the next section. EXAMPLE. There is a nontrivial solution to the Cauchy problem, UxxtUt + UXX =0, (1.3) U(X, 0) = 0, (1.4) that satisfies the estimate Iu(x, t)I < e V IxI* We let" @default.
• W2022693238 created "2016-06-24" @default.
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• W2022693238 date "1979-02-01" @default.
• W2022693238 modified "2023-09-22" @default.
• W2022693238 title "The uniqueness class for the Cauchy problem for pseudoparabolic equations" @default.
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• W2022693238 doi "https://doi.org/10.1090/s0002-9939-1979-0537083-3" @default.
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