On the Cauchy problem for the singular parabolic equations with gradient term

The Cauchy problem for a singular parabolic equation with gradient term of the form

ut−div(|Du|^p−2Du)=|Duqσ^|     

On the approximate solution of the singular problem of Cauchy for ordinary differential equations

In the present paper an approximate solution of the singular problem of Cauchy for the ordinary differential equation of mth order is constructed and, by the method of finite differences, sufficient conditions are found for the convergence to the exact solution when the mesh width tends to zero.     

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

The uniqueness and existence of BV-solutions for Cauchy problem of the form are proved.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

On the Cauchy problem for nonlinear parabolic equations with variable density

We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density. Sufficient conditions for uniqueness or nonuniqueness in L ^∞(IR ^N  × (0, T)) (N ≥ 3) are established in dependence of the behavior of the density at infinity. We deal with conditions at infinity of Dirichlet type, and possibly inhomogeneous.     

On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations

We present the results of an investigation and some applications of fundamental solutions of the Cauchy problem for a new class of parabolic equations. In these equations: (i) there exist three groups of spatial variables, one basic and two auxiliary, (ii) different weights of spatial variables from the basic group with respect to the time variable are admitted, (iii) degeneracies in variables from the auxiliary groups are present, (iv) a degeneracy on the initial hyperplane is present. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Ternopil' Academy of Economics, Ternopil'. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 13–19, April–June, 1998.     

Line method approximations to the Cauchy problem for nonlinear parabolic differential equations

Summary The Cauchy problemu _t =f(x, t, u, u _x , u _xx ),u(x, o)=(x),xR, is treated with the longitudinal method of lines. Existence, uniqueness, monotonicity and convergence properties of the line method approximations are investigated under the classical assumption that satisfies an inequality |(x)|<=conste ^{Bx 2} . We obtain generalizations of the works of Kamynin [4], who got similar results in the case of the one dimensional heat equation when is allowed to grow likee ^{Bx 2–}, >0, and of Walter [11], who proved convergence in the case of nonlinear parabolic differential equations under the growth condition |(x)|<=conste ^B |x|     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

Cauchy problem for doubly singular parabolic equation with gradient source

This paper examines the Cauchy problem for doubly singular parabolic equation with a source term depending solely on the gradient. We establish the local and global existence of solutions when initial data is merely a function in (). Moreover, the uniform ‐estimates and gradient estimates of solutions are obtained.     

Uniqueness of a solution to the Cauchy problem for parabolic systems

The Cauchy problem for a second-order Petrovskii parabolic system with bounded and continuous coefficients that are Dini-continuous with respect to space variables is proved to have a unique classical solution in the Tikhonov class.     

On asymptotic proximity of solutions of the Cauchy problem for second-order parabolic equations

A theorem about asymptotic (as t) proximity of weak fundamental solutions of the Cauchy problem is proved for divergent second-order parabolic equations. It is assumed that the coefficients have derivatives generalized in the Sobolev sense. A possible application of this theorem to establishing the uniform proximity of weak solutions of the Cauchy problem is also discussed.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 693–700, May, 1995.     

Existence and uniqueness of solution of the Cauchy problem for singular systems of integro-differential equations

Sufficient conditions are established for the existence of a unique solution of the Cauchy problem for singular systems of integro-differential equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1716–1720, December, 1993.     

The Cauchy problem for some classes of quasi-linear parabolic equations

Theorems are established concerning the solubility in the large of the Cauchy problem for quasi-linear parabolic second-order equations.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 295–300, September, 1969.