Asymptotic expansions for degenerate parabolic equations

We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.     

Nonlocal solvability of the Cauchy problem for second-order nonlinear parabolic equations

We prove the existence of smooth solutions of the Cauchy problem for some second-order nonlinear parabolic equations subject to natural smoothness conditions on the right side of the equation and on the initial function.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 581–586, April, 1974.     

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.     

Asymptotic structure for solutions of the Cauchy problem for Burgers type equations

Large time asymptotic structure for solutions of the Cauchy problem for a generalized Burgers equation is determined. In particular, Gelfand’s question about location of viscous shock waves for such equations is answered.     

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|     

Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations

We show that the solutions of the initial value problems for a large class of Burgers type equations approach with time to the sum of appropriately shifted wave-trains and of diffusion waves.

Résumé

Nous montrons que les solutions du problème de Cauchy pour une grande classe d'équations de type de Burgers sont approchées en temps grand vers des sommes d'ondes de diffusion et d'ondes progressives adéquatement translatées.     

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.     

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.     

Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations

The uniqueness of solutions for Cauchy problem of the form $$\frac{{\partial u}}{{\partial t}} = \Delta A(u) + \sum\limits_{i = 1}^N {\frac{{\partial b^i (u)}}{{\partial x_i }} + c(u)} $$ is studied. It is proved that ifu ∈BVx and A(u) is strictly increasing, the solution is unique.     

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.     

Asymptotic expansions for parabolic systems

We consider a parabolic system in a half space. A theorem, similar to one proved by Meyers and Pazy for elliptic equations outside the unit ball is proved, namely, if the coefficients, the right side, and the initial conditions of the parabolic system have asymptotic expansions at infinity with respect to the space variable, then so does the solution of the corresponding Cauchy problem. Some generalizations and examples are given.     

Uniqueness classes for the solutions of the cauchy problem for nonlinear degenerate parabolic equations

Translated from Matematicheskie Zametki, Vol. 48, No. 6, pp. 118–125, December, 1990.     

Asymptotic behavior of periodic solutions of parabolic equations with weakly nonlinear perturbation

Translated from Matematicheskie Zametki, Vol. 57, No. 3, pp. 369–376, March, 1995.