We consider the semilinear wave equationThis equation admits an explicit spatially homogeneous blow up solution ψ T given bywhere T > 0 and κp is a p-dependent constant. We prove that the blow up described by ψ T is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag.

We study the blowup behavior for the focusing energysupercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability in H 2 × H 1 of the ODE blowup profile.

Abstract. We consider co-rotational wave maps from (3 + 1) Minkowski space into the threesphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper [11]. In particular, we prove that f0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1 2 . The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

We study the semilinear wave equation ∂ 2 t ψ − ∆ψ = |ψ| p−1 ψ for p > 3 with radial data in three spatial dimensions. There exists an explicit solution which blows up at t = T > 0 given bywhere cp is a suitable constant. We prove that the blow up described by ψ T is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that lead to a solution which converges to ψ T as t → T − in the backward lightcone of the blow up point (t, r) = (T, 0).

We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions d ≥ 5. We prove that for every p > d+3 d−1 there exists an open set of radial initial data in Hsuch that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions. 1(1.5)By setting Ψ T (τ ) := (ψ T 1 (τ, ·), ψ T 2 (τ, ·)) this can be written aswhere L 0 represents the linear part of the right hand side of Eq. (1.5). To formulate the following statement we define for k ∈ N 0 H k rad (B d ) := {u ∈ H k (B d ) : u is radial}.

For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution u * T , which is defined on the whole space and exists in all supercritical dimensions d ≥ 5. For d = 7, we analyze its stability properties without any symmetry assumptions and prove the existence of a co-dimension one Lipschitz manifold consisting of initial data whose solutions blowup in finite time and converge asymptotically to u * T (modulo space-time shifts and Lorentz boosts) in the backward lightcone of the blowup point. The underlying topology is strictly above scaling.

We consider Ornstein-Uhlenbeck operators on L 2 (R d ) perturbed by a radial potential V . Under weak assumptions on V we prove a spectral mapping theorem for the generated semigroup. The proof relies on a perturbative construction of the resolvent, based on angular separation, and the Gearhart-Prüß Theorem.

In this paper, we consider the heat flow for Yang-Mills connections on R 5 × SO(5). In the SO(5)−equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reactiondiffusion equation for which an explicit self-similar blowup solution was found by Weinkove [57]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in L ∞ . 7 9