On Quasilinear Non-Uniformly Parabolic Equations in General Form

AbstractThe present paper is concerned with the first boundary value problem for a certain class of quasilinear non-uniformly parabolic equations. New a priori estimates of the solution and of its gradient are obtained. These are independent of the smoothness of the coefficients. Existence and uniqueness theorems are proved

A Remark on the Global Solvability of the Cauchy Problem for Quasilinear Parabolic Equations

AbstractThe present paper is concerned with the global solvability of the Cauchy problem for the quasilinear parabolic equations with two independent variables: ut=a(t,x,u,ux)uxx+f(t,x,u,ux). We investigate the case of the arbitrary order of growth of the function f(t,x,u,p) with respect to p when |p|→+∞. Conditions which guarantee the global classical solvability of the problem are given

Estimate of the solution of the Dirichlet problem for parabolic equations and applications

AbstractIn the present paper we consider the Dirichlet problem for quasilinear nonuniformly parabolic equations. A new sufficient condition which guarantees the a priori estimate of the maximum of the modulus of the solution is formulated. A several applications of this estimate are given

Abstract kinetic equations with positive collision operators

We consider "forward-backward" parabolic equations in the abstract form $Jd \psi / d x + L \psi = 0$, $0< x < \tau \leq \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \geq 0$, and $\ker L = 0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $\mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu)$, $0<x<\tau$, $\mu \in \R$, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation $T d \psi/dx = - A \psi (x) + f(x)$, where $T=T^*$ is injective and $A$ satisfies a certain positivity assumption, is considered also.Comment: 20 pages, LaTeX2e, version 2, references have been added, changes in the introductio

On some generalizations of the properties of the multidimensional generalized Erdélyi-Kober operators and their applications

In this paper we investigate the composition of a multidimensional generalized Erdélyi-Kober operator with differential operators of high order. In particular, with powers of the differential Bessel operator. Applications of proved properties to solving the Cauchy problem for a multidimensional polycaloric equation with a Bessel operator are show

Transmutation operators as a solvability concept of abstract singular equations

One of the methods of studying differential equations is the transmutation operators method. Detailed study of the theory of transmutation operators with applications may be found in the literature. Application of transmutation operators establishes many important results for different classes of differential equations including singular equations with Bessel operato