Abstract
Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.
Highlights
Evolution equations with several time-like variables are encountered in various models of science and technology, in mathematical models of multiparameter Brownian motion [1], theory of boundary layers [2], mathematical models of diffusion of pollutants in water flows [3], transport theory [4], mathematical models of age structured biological population dynamics [5, 6], mathematical finance [7], mechanics, physics, and cosmology [8, 9]
We show characteristic properties of some spaces of vectorvalued functions, which are used to investigate the nonclassical problem for ultraparabolic equation
Let V and H be separable Hilbert spaces, such that V is dense in H and continuously embedded in it
Summary
Evolution equations with several time-like variables are encountered in various models of science and technology, in mathematical models of multiparameter Brownian motion [1], theory of boundary layers [2], mathematical models of diffusion of pollutants in water flows [3], transport theory [4], mathematical models of age structured biological population dynamics [5, 6], mathematical finance [7], mechanics, physics, and cosmology [8, 9]. Applying methods of the theory of semigroups ultraparabolic integrodifferential equation with classical initial conditions with respect to time variables in the spaces of Holder continuous vector-functions with values in a Banach space was investigated by Lorenzi [28] and ultraparabolic equations arising in age structured population dynamics with integral condition with respect to one of time variables were studied by many researchers; see [6, 29, 30] and their references. In the case of classical initial conditions, that is, for α = 0, the well-posedness results are obtained either in spaces of continuous functions or in spaces of distributions with respect to time variables, for which the trace operator on boundary of domain of time variables cannot be defined. In the present paper we obtain wellposedness theorem for nonlocal problem (1), (2), which in the case of α = 0 gives new existence, uniqueness, and continuous dependence result for classical problem for the abstract ultraparabolic equation.
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