Regularity of the solution of the Cauchy problem for a higher-order parabolic equation

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

“…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.…”

“…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].…”

“…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).…”

“…For this purpose we show in Section 2 that the solution to the Cauchy problem for an operator with constant coefficients belongs to the space C 2,α λ,α (R n+1 + ), λ > 0. In Section 4, we establish an a priori estimate for the solution to the problem (1.10) by a modified method of [3,4].…”

The Cauchy problem for parabolic equations with unbounded coefficients

“…The Cauchy problem for parabolic systems with uniformly Hölder continuous and bounded coef ficients with right hand side having a power singularity at t = 0 was considered in [3] (in a bounded layer) and [4] (in a half space); it was proved that this problem is solvable in the weighted Hölder space in which the derivatives of elements may increase in a certain way as the plane t = 0 is approached. In [5] (see also [6]), the solvability of the Cauchy problem was proved for a parabolic equation in the same weighted Hölder class as in [4] under substantially relaxed conditions on the coefficients of the equation, namely, under the assumption that all coefficients are locally Hölder continuous (with an exact indication of the character of the Hölder property) and the lower order coeffi cients increase as t → +0 no faster than a certain power function.…”