The Carleman formula for the Helmholtz equation on the plane.
The Cauchy problem and decay rates for strong solutions of a Boussinesq system.
The Cauchy problem and Hadamard's example
The Cauchy problem and self-similar solutions for a nonlinear parabolic equation
The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation.
The Cauchy problem for a nonlinear Wheeler-DeWitt equation
The Cauchy problem for a short-wave equation.
The Cauchy problem for a strongly degenerate quasilinear equation
We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation , where , and is a convex function of with linear growth as , satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.
The Cauchy problem for a system of Maxwell equations.
The Cauchy problem for coupled Yang-Mills and scalar fields in the Lorentz gauge
The Cauchy problem for degenerate parabolic equations in Gevrey classes
The Cauchy problem for Hartree-Fock time-dependent equations
The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable
We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.
The Cauchy problem for linear hyperbolic systems in
The Cauchy problem for non linear wave equations in domains with moving boundary
The Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space
The Cauchy problem for one class of parabolic pseudodifferential systems with nonsmooth symbols.
The Cauchy problem for operators with injective symbol in the Lebesgue space in a domain.
The Cauchy problem for second-order elliptic systems on the plane.
The Cauchy problem for semilinear weakly hyperbolic equations in Hilbert spaces