A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy problem for a linear operator equation and proving that this problem has a global solution whose limit at infinity solves the original linear equation.

We consider radial solutions to the Cauchy problem for the linear wave equation with a small short–range electromagnetic potential (the“square version”of the massless Dirac equation with a potential) and zero initial data. We prove two a priori estimates that imply, in particular, a dispersive estimate.

The exact solution of the Cauchy problem for a generalized ”linear” vectorial Fokker-Planck equation is found using the disentangling techniques of R. Feynman and algebraic (operational) methods. This approach may be considered as a generalization of the Masuo Suzuki’s method for solving the 1-dimensional linear Fokker-Planck equation.

In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution. keywords: semilinear dissipative wave equation, pointwise estimates, Green function. MSC...

By using the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satifying finite mass, “inertia”, energy and entropy.

We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation i Ψt + 2 2 Ψxx + |Ψ |2Ψ = 0, 1, with analytic initial data of the form Ψ (x,0; ) = A(x)e i S(x) is approximately described by a particular solution to the Painlevé-I equation.

The stability of the functional equation f(x composite function y) = H(f(x), f(y)) (x, y in S) is investigated, where H is a homogeneous function and composite function is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.