Energy estimates and global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients singular in time

Well-posedness for Hyperbolic Problems

Well-posedness and Long Time Behavior of a Parabolic-hyperbolic Phase-field System with Singular Potentials

In this article, we study the long time behavior of a phase-field parabolichyperbolic system arising from the phase-field theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation ...

Generic Well-Posedness for a Class of Equilibrium Problems

We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category, we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed.

Global Well-posedness and Scattering for a Class of Nonlinear Schrödinger Equations below the Energy Space

We prove global well-posedness and scattering for the nonlinear Schrödinger equation with power-type nonlinearity { iut +∆u = |u|u, 4 n < p < 4 n−2 , u(0, x) = u0(x) ∈ H(R), n ≥ 3, below the energy space, i.e., for s < 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the H x-norm of the solution, and hence global well-posedness for initi...

Inverse hyperbolic problems with time - dependent coefficients

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism ...