Related papers: On the wave equation with space dependent coeffici…
We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very…
We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very…
This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations…
This paper addresses a wave equation on a exterior domain in R^{d}(d odd) with nonlinear time dependent dissipation. Under a microlocal geometric condition we prove that the decay rates of the local energy functional are obtained by solving…
We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$…
In this work, we consider a system of multidimensional wave equations coupled by velocities with one localized fractional boundary damping. First, using a general criteria of Arendt- Batty, by assuming that the boundary control region…
We propose a model for frequency-dependent damping in the linear wave equation. After proving well-posedness of the problem, we study qualitative properties of the energy. In the one-dimensional case, we provide an explicit analysis for…
We consider the operator $H:= \partial_t -\Delta+V$ in 2D or 3D waveguide. With an adapted global Carleman estimate with singular weight functions we give a stability result for the time dependent part of the potential for this particular…
In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of…
The covariance of the d'Alembert equation for acoustic phenomena, described by mechanical waves in one or three spatial dimensions, under Galilean transformations, is demonstrated without the need to abandon the hypothesis that time is…
We consider a system of nonlinear equations which can be reduced to a degenerate parabolic equation. In the case $x\in\bR^2$ we obtained necessary conditions for the existence of a weakly singular solution of heat wave type…
This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficient. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of…
In this paper we study the long time behavior for a semilinear wave equation with space-dependent and nonlinear damping term. After rewriting the equation as a first order system, we define a class of approximate solutions that employ…
We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation at both low and high frequency in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in…
This paper is concerned with weighted energy estimates for solutions to wave equation $\partial_t^2u-\Delta u + a(x)\partial_tu=0$ with space-dependent damping term $a(x)=|x|^{-\alpha}$ $(\alpha\in [0,1))$ in an exterior domain $\Omega$…
In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and…
We present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature most results concerning existence and uniqueness are obtained under…
We obtain local (i.e., linearized) convergence conditions for iterative methods that seek solitary waves with prescribed values of quadratic conserved quantities of multi-component Hamiltonian nonlinear wave equations. These conditions…
We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation…
We consider the stable dependence of solutions to wave equations on metrics in C^{1,1} class. The main result states that solutions depend uniformly continuously on the metric, when the Cauchy data is given in a range of Sobolev spaces. The…