Nonlinear boundary dissipation for a coupled system of Klein-Gordon equations.
Nonlinear Cauchy problems with small analytic data and the lifespan of their solutions
We study the lifespan of solutions to fully nonlinear second-order Cauchy problems with small real- or complex-analytic data. The nonlinear term is an analytic function in u, ū and their derivatives. We give an outline of the proof based on the method of majorants and the fixed point technique.
Nonlinear compressible vortex sheets in two space dimensions
We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized...
Nonlinear effectively hyperbolic equations
Nonlinear feedback stabilization of a rotating body-beam without damping
Nonlinear feedback stabilization of a rotating body-beam without damping
This paper deals with nonlinear feedback stabilization problem of a flexible beam clamped at a rigid body and free at the other end. We assume that there is no damping and the feedback law proposed here consists of a nonlinear control torque applied to the rigid body and either a boundary control moment or a nonlinear boundary control force or both of them applied to the free end of the beam. This nonlinear feedback, which insures the exponential decay of the beam vibrations, extends the linear...
Nonlinear Hyperbolic Smoothing at a Focal Point
The nonlinear dissipative wave equation in dimension has strong solutions with the following structure. In the solutions have a focusing wave of singularity on the incoming light cone . In that is after the focusing time, they are smoother than they were in . The examples are radial and piecewise smooth in
Nonlinear Kirchhoff-Carrier wave equation in a unit membrane with mixed homogeneous boundary conditions.
Nonlinear oscillations of completely resonant wave equations
Nonlinear Pulse Propagation
This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.
Nonlinear Small Data Scattering for the Wave and Klein-Gordon Equation.
Nonlinear stability and convergence of finite-differnce methods for the "good" Boussinesq equation.
Nonlinear stability of centered rarefaction waves of the Jin-Xin relaxation model for conservation laws.
Nonlinear transmission problem with a dissipative boundary condition of memory type.
Nonlinear vibrations of completely resonant wave equations
We present recent existence results of small amplitude periodic and quasi-periodic solutions of completely resonant nonlinear wave equations. Both infinite-dimensional bifurcation phenomena and small divisors difficulties occur. The proofs rely on bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques and variational methods.
Nonlinear Wave Equation with Vanishing Potential
We study the Cauchy problem for utt − ∆u + V (x)u^5 = 0 in 3–dimensional case. The function V (x) is positive and regular, in particular we are interested in the case V (x) = 0 in some points. We look for the global classical solution of this equation under a suitable hypothesis on the initial energy.
Non-Lipschitz coefficients for strictly hyperbolic equations
In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.
Nonlocal and initial problems for quasilinear nonstrictly hyperbolic equations with general solutions represented by superpositions of arbitrary functions.
Nonlocal problem for the hyperbolic system of differential equation of the first order.
Nonlocal problems for first order functional partial differential equations
Local existence of generalized solutions to nonlocal problems for nonlinear functional partial differential equations of first order is investigated. The proof is based on the bicharacteristics and successive approximations methods.