Singular perturbations for a class of quasi-linear hyperbolic equations
Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms.
Singular solutions to systems of conservation laws and their algebraic aspects
We discuss the definitions of singular solutions (in the form of integral identities) to systems of conservation laws such as shocks, δ-, δ’-, and -shocks (n = 2,3,...). Using these definitions, the Rankine-Hugoniot conditions for δ- and δ’-shocks are derived. The weak asymptotics method for the solution of the Cauchy problems admitting δ- and δ’-shocks is briefly described. The algebraic aspects of such singular solutions are studied. Namely, explicit formulas for flux-functions of singular solutions...
Singularités des solutions d'équations d'ondes semi-linéaires
Singularities of the scattering kernel for generic obstacles
Singularities of the scattering kernel for non convex obstacles
Singularities of the scattering kernel for trapping obstacles
Singularity formation in systems of non-strictly hyperbolic equations.
Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic...
Singularly perturbed telegraph equations with applications in the random walk theory.
Sistemi iperbolici di leggi di conservazione
This survey paper provides a brief introduction to the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After reviewing some basic concepts, we describe the fundamental theorem of Glimm on the global existence of BV solutions. We then outline the more recent results on uniqueness and stability of entropy weak solutions. Finally, some major open problems and research directions are discussed in the last section.
Slepian's inequality and commuting semigroups
Small analytic solutions to nonlinear weakly hyperbolic systems
Small data scattering for nonlinear Schrödinger wave and Klein-Gordon equations
Small data scattering for nonlinear Schrödinger equations (NLS), nonlinear wave equations (NLW), nonlinear Klein-Gordon equations (NLKG) with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space with order . The assumptions on the nonlinearities are described in terms of power behavior at zero and at infinity such as for NLS and NLKG, and for NLW.
Small solutions to semi-linear wave equations with radial data of critical regularity.
Small time periodic solutions of fully nonlinear telegraph equations in more spatial dimensions
Small time-periodic solutions to a nonlinear equation of a vibrating string
In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.
Smooth approximations of global in time solutions to scalar conservation laws.
Smooth solutions of the heat and wave equations.
Smoothly symmetrizable hyperbolic systems of partial differential equations.