Related papers: On Morawetz estimates for the elastic wave equatio…
Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a…
We study linear dispersive equations in dimension one and two for a class of radial nonhomogeneous phases. L 1 $\rightarrow$ L $\infty$ type estimates, Strichartz estimates, local Kato smoothing and Morawetz type estimates are provided. We…
In this paper we establish that the time-harmonic elasticity problem in a half-strip with non-homogeneous Dirichlet conditions on its boundary section and traction-free conditions on its upper and lower boundaries, has a unique weak…
We study the wave equation on the real line with a potential that falls off like $|x|^{-\alpha}$ for $|x| \to \infty$ where $2 < \alpha \leq 4$. We prove that the solution decays pointwise like $t^{-\alpha}$ as $t \to \infty$ provided that…
We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse spacetime norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by…
We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the…
We consider the variational wave equation in one-dimensional space with stochastic forcing by an additive noise. Blow-up of local smooth solutions is established, and global existence is proved in the class of weak martingale solutions.
In this work we give a few new Strichartz estimates of radial solutions to the wave equation. These Strichartz estimates still use $L^p L^q$ type norms in each channel-like region $\{(x,t): |t|+2^k < |x| < |t|+2^{k+1}\}$, with weaker…
We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^\alpha$. We solve them in a special case while a more detailed study is contained…
The inhomogeneous wave equations for the scalar, vector, and Hertz potentials are derived starting from retarded charge, current, and polarization densities and then solved in the reciprocal (or k-) space to obtain general solutions, which…
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H\"{o}lder's inequalities…
We study the homogeneous wave equation with radially symmetric data in four or higher space dimensions. Using some new integral representations for the Riemann operator, we establish weighted decay estimates for the solution.
We establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there…
We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and…
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $\alpha \in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the…
We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild-de Sitter spacetimes with parameters $(M,\Lambda)$. These are integrated decay statements…
A new representation for solutions of Maxwell's equations is derived. Instead of being expanded in plane waves, the solutions are given as linear superpositions of spherical wavelets dynamically adapted to the Maxwell field and…
We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes…
We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a…
The Euler-$\alpha$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $\alpha$. We show that there exists $\beta>1$ such that weak solutions to the two and three dimensional…