Construction of two-bubble solutions for energy-critical wave equations

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k > 2. The concentrating speed of the second bubble is exponential for the first two models and a power function in the last case.Comment: 44 pages; the new version fixes an error in the proof of Theorem

Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of the family of solitons $W_\lambda$, asymptotically decomposes in the energy space as a sum of a bubble $W_\lambda$ and an asymptotic profile $(u_0^*, u_1^*)$, where $\lim_{t\to 0}\lambda(t)/t = 0$ and $(u^*_0, u^*_1) \in \dot H^1\times L^2$. We construct a blow-up solution of this type such that $(u^*_0, u^*_1)$ is any pair of sufficiently regular functions with $u_0^*(0) > 0$. For these solutions the concentration rate is $\lambda(t) \sim t^4$. We also provide examples of solutions with concentration rate $\lambda(t) \sim t^{\nu + 1}$ for $\nu > 8$, related to the behaviour of the asymptotic profile near the origin.Comment: 39 pages; the new version takes into account the remarks of the referee

Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case

We consider the focusing energy-critical wave equation in space dimension $N\in \{3, 4, 5\}$ for radial data. We study type II blow-up solutions which concentrate one bubble of energy. It is known that such solutions decompose in the energy space as a sum of the bubble and an asymptotic profile. We prove bounds on the blow-up speed in the case when the asymptotic profile is sufficiently regular. These bounds are optimal in dimension $N = 5$. We also prove that if the asymptotic profile is sufficiently regular, then it cannot be strictly negative at the origin.Comment: 22 page

Small time heat kernel asymptotics at the cut locus on surfaces of revolution

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.Comment: Accepted on Annales IHP - Analyse Non Lineair

Bubble decomposition for the harmonic map heat flow in the equivariant case

We consider the harmonic map heat flow for maps from the plane taking values in the sphere, under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times -- the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by the authors' recent work on the soliton resolution problem for equivariant wave maps.Comment: arXiv admin note: text overlap with arXiv:2106.1073

Strichrtz estimates for Klein-Gordon equations with moving potentials

We study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are established. The results and estimates obtained in this paper will be used to study the interaction of solitons and the stability of multisolitons of nonlinear Klein-Gordon equations.Comment: 53 page