We study the Cauchy problem for the one-dimensional wave equation \[ \partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0. \] The potential V is assumed to be smooth with asymptotic behavior \[ V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty. \] ...
We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy pro ...
For the critical focusing wave equation □u=u5 on R3+1 in the radial case, we construct a family of blowup solutions which are obtained from the stationary solutions W(r) by means of a dynamical rescaling $\lambda(t)\frac{1}{2}W(\la ...
We consider the hyperboloidal initial value problem for the cubic focusing wave equation (-partial derivative(2)(t) + Delta(x))nu(t, x) + nu(t, x)(3) = 0, x is an element of R-3 Without symmetry assumptions, we prove the existence of a codimension-4 Lipsch ...
We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in -dimensional Yang-Mills theory. A certain self-similar solution of this model is conjectured to act as an attractor for generic large data evolutions. Assumin ...
We study the semilinear wave equation ∂t2ψ−Δψ=∣ψ∣p−1ψ for p>3 with radial data in three spatial dimensions. There exists an explicit solution which blows up at t=T>0 given by $\displaystyl ...
We consider the critical focusing wave equation (−∂2t+Δ)u+u5=0 in R1+3 and prove the existence of energy class solutions which are of the form u(t,x)=tμ2W(tμx)+η(t,x) in the forward lightcone {(t,x)∈R×R3:|x|≤t,t≫1} where W(x)=(1+13|x|2)−12 is the ground st ...
We prove nonlinear stability of the fundamental self-similar solution of the wave equation with a focusing power nonlinearity tt-=p for [image omitted] in the radial case. The proof is based on a semigroup formulation of the wave equation in similarity coo ...
We present a rigorous functional analytic setting to study the radial wave equation in similarity coordinates. As an application we analyze linear stability of the fundamental self-similar solution of the wave equation with a focusing power nonlinearity. T ...
We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues wit ...
