The Cauchy problem is studied for an homogeneous quantum kinetic equation describing the Compton effect. Since the collision kernel commonly used in physics is highly singular, numerical simulations are performed for related collision kernels to get a preliminary insight into the behavior of the solutions. Some of the numerical results are then given a theoretical explanation. Global existence ...

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation

A singular Cauchy-Nicoletti problem for a system of three ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder’s principle is used. Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained.

We have recently solved the inverse spectral problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev Petviashvili (dKP) equation, arising as commutation ...

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: ut + (|u|ρ−1u)x + Huxx = 0, where H is the Hilbert transform, x, t ∈ R, when the initial data are small enough. If the power ρ of the nonlinearity is greater than 3, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the...

By using integration method, we give a directly proof that the Cauchy process on a circle satisfies the Chapman-Kolmogorov equation.

We constract an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator.

We discuss an integral form of the Cauchy initial value problem for the nonlinear Schrödinger equation with variable coefficients. Some special and limiting cases are outlined.

We have recently solved the inverse spectral problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev Petviashvili (dKP) equation, arising as commutation ...