Solvability of the Cauchy Problem for a Parabolic Equation with Unbounded Coefficients

Abstract

UDC 517.95We consider the Cauchy problem in the half-space for a parabolic equation with unbounded initial data and lower order coefficients. We establish the solvability of the problem in the weighted space of Hölder functions such that the functions, together with their derivatives, admit exponential growth as t → ∞ and, near the plane t = 0, the derivatives grow not faster than a power function. Bibliography: 11 titles.The classical solvability of the Cauchy problem for a second order parabolic equation with bounded and uniformly Hölder coefficients is well known (cf., for example, [1, 2]). In [3] (cf. also [4]), it is proved that this problem is solvable in the weighted space of Hölder functions such that their higher order derivatives may increase in a certain way near the plane t = 0 under the assumption that the lower order coefficients grow not faster than power functions as t → +0 and all the coefficients are locally Hölder continuous (moreover, the character of the Hölder continuity is indicated).In this paper, we establish the solvability of the Cauchy problem for a second order parabolic equation in a weighted space of Hölder functions whose derivatives are not necessarily bounded near the plane t = 0. While approaching the plane t = 0, the lower order coefficients and their Hölder coefficients are allowed to grow faster than it was admitted in [3,4]. As in [3], the higher order coefficients do not necessarily satisfy the Dini condition near t = 0, which prohibits construction of the classical fundamental solution even for equations with rather good properties (cf. [5]). Thereby the methods of [1,2] are not applicable in the case under consideration.The result of this paper was announced in [6]. The Main ResultWe denote by R n+1 (n 1) the Euclidean space of points P = (x, t), where x = (x 1 , . . . , x n ); R n+1 + = {P ∈ R n+1 : t > 0}. In R n+1 + , we consider the parabolic operator with real coefficients