The cauchy problem of a degenerate parabolic equation

This note is concerned with the existence of a weak solution for a degenerate Cauchy problem of parabolic type in then-dimensional spaceR ⁿ. The degenerate property is in the sense that the matrix (a _ij(t,x)) involved in the differential operator is not necessarily positive definite. The essential idea is the construction of a suitable function spaceH and to prove the existence of a weak solution inH.     

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 statistical solution of the parabolic equation

We study the convergence of the statistical solutions of the parabolic equation. Under some mixing condition (in the sense of Rosenblatt) for initial measure and natural assumptions on the coefficients of the equation we prove weak convergence to the Gaussian distribution. Similar results for the hyperbolic equations were obtained in [1–4].     

Existence of a solution of the cauchy problem for an integrodifferential equation of viscoelasticity

Mechanics of Composite Materials - The Cauchy problem for a nonlinear integrodifferential equation is reduced to a nonlinear integral equation for which a theorem of existence and uniqueness is...     

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.     

Critical extinction exponent for a doubly degenerate non-divergent parabolic equation with a gradient source

We concern with the extinction phenomenon of a non-divergent parabolic equation with a nonlinear gradient source. The critical extinction exponent of the solution is obtained.     

The cauchy problem for an elliptic parabolic operator

Summary Necessary and sufficient conditions are established for the existence of a solution of a Cauchy problem which is not well posed in the sense of Hadamard. This research was supported in part by National Science Foundation Grant No. GP 5882 and in part by Air Force Contract AF OSR 396-63     

Uniform meyer solution to the three dimensional cauchy problem for laplace equation

We consider the three dimensional Cauchy problem for the Laplace equation uxx(x,y,z)+ uyy(x,y,z)+ uzz(x,y,z) = 0, x ∈ R,y ∈ R,0 z ≤ 1, u(x,y,0) = g(x,y), x ∈ R,y ∈ R, uz(x,y,0) = 0, x ∈ R,y ∈ R, where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 z 1. The problem is ill-posed, the solution (if it exists) doesn't depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.     

Inverse problem for a parabolic equation

Mathematical Notes -     

Asymptotic behavior of the solution of the cauchy problem for perturbed Klein-Fock-Gordon equation

This paper deals with the Cauchy problem for the perturbed Klein-Fock-Gordon equation. The principal term of an asymptotic expansion of the solution is constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 115–121, 1987.