The behaviour of the Gaussian beam in an anisotropic medium with an interface between two media.
The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II
The BV solution of the parabolic equation with degeneracy on the boundary
Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.
The Cauchy problem for a nonlinear Wheeler-DeWitt equation
The Cauchy problem for coupled Yang-Mills and scalar fields in the Lorentz gauge
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 semilinear weakly hyperbolic equations in Hilbert spaces
The Cauchy problem for the coupled Klein-Gordon-Schrödinger system
We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...
The Cauchy problem for the Vlasov-Maxwell equations
The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations
In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant...
The Child–Langmuir limit for semiconductors : a numerical validation
The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical comparisons...
The Child–Langmuir limit for semiconductors: a numerical validation
The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical...
The Convergence of Successive Approximations for Boundary Value Problems of Hyperbolic Equations in the Banach Space
The correct use of the Lax–Friedrichs method
We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of...
The correct use of the Lax–Friedrichs method
We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version...
The critical case for a semilinear weakly hyperbolic equation.
The critical nonlinear wave equation in two space dimensions
Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.