On a coefficient concerning an ill-posed Cauchy problem and the singularity detection with the wavelet transform

We study the Cauchy problem for 2nd order weakly hyperbolic equations. F. Colombini, E. Jannelli and S. Spagnolo showed a coefficient degenerating at an infinite number of points, with which the Cauchy problem is ill-posed Gevrey classes. Moreover, we olso report numerical results of the singularity detection with wavelet trasform for coefficient functions

The Log-effect for 2 by 2 hyperbolic systems

AbstractIn the present paper we are interested to extend the Log-effect from wave equations with time-dependent coefficients to 2 by 2 strictly hyperbolic systems ∂tU−A(t)∂xU=0. Besides the effects of oscillating entries of the matrix A=A(t) and interactions between the entries of A we have to take into consideration the system character itself. We will prove by tools from phase space analysis results about H∞ well- or ill-posedness. The precise loss of regularity is of interest. Finally, we discuss the cone of dependence property

A Note on Wave Equation in Einstein & de Sitter Spacetime

We consider the wave propagating in the Einstein & de Sitter spacetime. The covariant d'Alembert's operator in the Einstein & de Sitter spacetime belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the $L^p - L^q$ estimates for solutions

Weakly hyperbolic systems with Hölder continuous coefficients

AbstractWe study the Cauchy Problem for a hyperbolic system with multiple characteristics and non-smooth coefficients depending on time. We prove in particular that, if the leading coefficients are α-Hölder continuous, and the system has size m⩽3, then the Problem is well posed in each Gevrey class of exponent s<1+α/m

Klein-Gordon Type Equations with a Singular Time-dependent Potential

In this note we study Klein-Gordon type Cauchy problems with a time-dependent singular potential. We ask for the influence of the sign and the singularity order of the potential on the regularity of solutions with respect to time