This paper studies the global well-posedness of solutions to the Cauchy problem of incompressible magneto-hydrodynamics system for large initial data in homogeneous Besov space Ḃ 2 p −1 p,r (R) for 2 < p < ∞ and 1 ≤ r < ∞. In the case of spatial dimension n ≥ 3 we establish the global well-posedness of solution for small data and the local one for large data in Besov space Ḃ n p −1 p,r (R), 1 ≤...

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter. 1991 Mathematics...

The Fokker–Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker– Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leadin...

We consider the Cauchy problem for a n×n strictly hyperbolic system of balance laws

The property of normality and the convolution formula are established for classical fundamental solution Cauchy problem a homogeneous ultraparabolic Kolmogorov-type equation with two groups space variables degeneration. Theorems on integral representations solutions correct solvability presented classes functions exponentially increasing as |x| → ∞ .

We consider the Cauchy problem for a n×n strictly hyperbolic system of balance laws

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on ( , t) near the origin in C and are bounded holomorphic on some horizontal strip in C w.r.t the space variable. In our previous contribution [14], we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of...