A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations

We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.     

Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type

In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as t → +0 for one class of degenerate parabolic equations of the Kolmogorov type with 2^?b overrightarrow {2b} -parabolic part whose coefficients are continuous functions that depend only on t.     

Properties of the fundamental solutions and uniqueness theorems for the solutions of the Cauchy problem for one class of ultraparabolic equations

For one class of degenerate parabolic equations of the Kolmogorov type, we establish the property of normality, the convolution formula, the property of positivity, and a lower bound for the fundamental solution. We also prove uniqueness theorems for the solutions of the Cauchy problem for the classes of functions with bounded growth and for the class of nonnegative functions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1482–1496, November, 1998.     

Fundamental solution matrix of the Cauchy problem for a class of systems of Kolmogorov type equations

We consider a system of linear degenerate parabolic equations that generalize diffusion equations with inertia; the coefficients of the system depend on time. We construct the fundamental solution matrix of the Cauchy problem, analyze its properties, and estimate its derivatives.     

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.     

Systems of equations of Kolmogorov type

We consider one class of degenerate parabolic systems of equations of the type of diffusion equation with Kolmogorov inertia. For systems whose coefficients may depend only on the time variable, we construct a fundamental matrix of solutions of the Cauchy problem and obtain estimates for this matrix and all its derivatives. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1650–1663, December, 2008.     

The boundary-value problem for degenerate ultraparabolic equations of the Sobolev type

In this paper we study the first boundary-value problem for a class of degenerate equations of the Sobolev type and prove existence and uniqueness theorems for regular solutions to the considered problem.     

On the regularity of positive solutions of a class of Choquard type equations

This paper is concerned with positive solutions of a class of Choquard type equations. Such equations are equivalent to integral systems involving the Bessel potential and the Riesz potential. By using two regularity lifting lemmas introduced by Chen and Li [2], we study the regularity for integrable solutions u. We first use the Hardy–Littlewood–Sobolev inequality to obtain an integrability result. Then, it is improved to ${u \in L^s(R^n)}$ for all ${s \in [1, \infty]}$ by an iteration. Next, we use the properties of the contraction map and the shrinking map to prove that u is Lipschitz continuous. Finally, we establish the smoothness of u by a bootstrap argument. Our technique can also be used to handle other integral systems involving the Riesz potential or the Bessel potential, such as the Hartree type equations.     

Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.     

On the maximum principle for ultraparabolic equations

We prove the maximum principle and various modifications of it for one class of degeneration of parabolic equations.     

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations

In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family of left invariant operators on a free nilpotent Lie group. The fundamental solution of the operator is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is .

     

Oscillation of the solutions of a certain class of first-order functional-differential equations of neutral type

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1370–1375, October, 1989.     

On the solutions of a class of difference equations

In this note we investigate the solutions of a class of difference equations and prove that Conjectures 4.8.2, 4.8.3, 5.4.6 and 6.10.3 proposed by M. Kulenovic and G. Ladas in [M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002] are true.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form