Stabilization of a solution to the Cauchy problem for a nondivergence parabolic equation with growing lower order coefficients

We obtain sharp sufficient conditions on the growth of lower order coefficients of a second-order parabolic equation under which a solution to the Cauchy problem stabilizes to zero uniformly in x on every compact set K ∈ ℝ ^N in some classes of growing initial functions.     

Stabilization of the solution of the Cauchy problem for a parabolic equation in Nondivergence form with growing lower coefficients

We study sharp sufficient conditions on the growing lower coefficients of a parabolic equation guaranteeing the stabilization of the solution of the Cauchy problem to zero in some classes of growing initial functions.     

Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

The Cauchy problem for the equation with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen Banach space of functions on an infinite-dimensional separable real Hilbert space. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 663–670, June, 1994.     

Fundamental solution and Cauchy problem for a parabolic system with unbounded coefficients

The purpose of this paper is to obtain a fundamental solution matrix for a second order weakly coupled parabolic system and to solve the associated initial-value problem under a global assumption on the growth at infinity of the coefficients and the initial functions.     

On stabilization of solutions of the Cauchy problem for a parabolic equation with lower-order coefficients

In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation

Cauchy problem for a quasilinear parabolic equation with a large initial gradient and low viscosity

The Cauchy problem for a quasilinear parabolic equation with a small parameter ɛ multiplying the highest derivative is considered. The derivative of the initial function is on the order of O(1/ρ), where ρ is another small parameter. Asymptotic expansions of the solution in powers of ɛ and ρ are constructed in various forms.     

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

We prove the unique solvability of the Cauchy problem in a weighted Hölder space for a linear parabolic equation of order 2m under the condition that the lower coefficients and the right-hand side of the equation can have certain growth when approaching the plane that is the support of the initial data, while the higher coefficients do not necessarily satisfy the Dini condition near this plane.We construct a smoothness scale of solutions of the Cauchy problem in the corresponding weighted Hölder spaces.     

Stabilization of solution of cauchy problem for a non-divergent parabolic equation

Conditions for stabilization of solution of Cauchy problem for a linear second-order parabolic equation in nondivergent form are studied.     

The Cauchy problem for a parabolic equation with coefficients of “white noise” type

Existence and uniqueness conditions are found for a solution possessing a first moment. Sufficient conditions are given for the second moment to be bounded.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 18–28, 1987.     

On the Cauchy problem for a second order semilinear parabolic equation with factored linear part

Communicated by J. A. Goldstein