Publications de Roland Donninger

A vector field method on the distorted Fourier side and decay for wave equations with potentials

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.

...

2016

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

Joachim Krieger, Roland Donninger, Willie Wai Yeung Wong

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 ...

Duke Univ Press2015

Exotic blow up solutions for the $\Box u^5$-focussing wave equation in $\mathbb{R}^3$

Min Huang, Joachim Krieger, Roland Donninger

For the critical focusing wave equation

\Box u = u^5

on

\mathbb{R}^{3+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 ...

Michigan Mathematical Journal2014

Nondispersive Decay For The Cubic Wave Equation

Roland Donninger

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 ...

Mathematical Science Publ2014

Stable self-similar blowup in energy supercritical Yang-Mills theory

Roland Donninger

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 ...

Springer Verlag2014

Stable blow up dynamics for energy supercritical wave equations

Roland Donninger

We study the semilinear wave equation

\displaystyle \partial _t^2 \psi -\Delta \psi =\vert\psi \vert^{p-1}\psi

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 ...

American Mathematical Society2014

Nonscattering solutions and blowup at infinity for the critical wave equation

Joachim Krieger, Roland Donninger

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 ...

Springer Verlag2013

Stable self-similar blow up for energy subcritical wave equations

Roland Donninger

We consider the semilinear wave equation partial derivative(2)(t)psi - Delta psi = vertical bar psi vertical bar(p-1)psi for 1 < p

International Press2012

Nonlinear Stability of Self-Similar Solutions for Semilinear Wave Equations

Roland Donninger

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 ...

Taylor & Francis2010

The radial wave operator in similarity coordinates

Roland Donninger

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 ...

American Institute of Physics2010

Asymptotics and analytic modes for the wave equation in similarity coordinates

Roland Donninger

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 ...

Springer Verlag2009