Related papers: On the wave equation with space dependent coeffici…
We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a…
We investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type. The main novelty in this paper is that both the damping and the coupling coefficients are non smooth.…
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type…
To understand an oft-observed but poorly understood phenomenon in which a solitary wave in a dispersive equation slowly deteriorates due to a persistent emission of radiation (i.e. a ``radiating solitary wave''), we propose a bare-bones…
In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a…
Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\hbox{div}(b(x)\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies…
We consider the initial-value problem for a one-dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we…
We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the…
In this paper we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belonging to $L^1_{loc}([0, +\infty)).$ Under suitable assumptions, we show…
We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial…
In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the…
We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and $L^2$-data. Our first contribution is a direct proof of well-posedness of the…
In this paper we consider a class of $p$-evolution equations of arbitrary order with variable coefficients depending on time and space variables $(t,x)$. We prove necessary conditions on the decay rates of the coefficients for the…
In this paper, we investigate a class of semilinear wave equations in non-cylindrical time-dependent domains, subject to exterior homogeneous Dirichlet conditions. Under mild regularity and monotonicity assumptions on the evolving spatial…
We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time…
In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent mass and speed of pro\-pagation. We introduce a classification of the potential term, which clarifies whether the solution…
A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from…
For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…
We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. We also obtain some ill-posedness/weak ill-posedness results. The proof…
We develop a general theory for the existence, uniqueness, and higher regularity of solutions to wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities. These singularities may be of geometric type (cone…