Special Session 64: Blow-ups and dynamics of nonlinear parabolic equations

Infinite time blow-up for the energy critical heat equation on bounded domains in low dimension

Giacomo Ageno University of Cambridge England Co-Author(s):    Manuel del Pino

Abstract: A positive solution to the Dirichlet problem for the energy-critical heat equation typically decays exponentially fast or blows up in finite time. Threshold behaviors between these two scenarios exist, having been studied since the 1980s. In this talk, I will introduce the first examples of global, unbounded solutions without radial symmetry, precisely describing asymptotic and stability. The low dimension $\{3,4\}$ plays a crucial role, making the heart of the problem nonlocal. In dimension $3$ the analysis reveals a connection with the Brezis-Nirenberg number. This is joint work with Manuel del Pino.

Liouville theorems in the upper half-space and the fully nonlinear Loewner-Nirenberg problem

Jonah Duncan University College London England Co-Author(s):    Luc Nguyen

Abstract: I will discuss recent work with Luc Nguyen concerning a class of fully nonlinear Yamabe-type equations of negative curvature type on the upper half-space. We show that whether the hyperbolic metric is the unique solution is completely determined by a single parameter associated with the equation; in the case of non-uniqueness, we classify all other solutions. I will also discuss implications of non-uniqueness on the validity of certain estimates for the fully nonlinear Loewner-Nirenberg problem on compact manifolds with boundary.

Smooth nonradial stationary 2d Euler flows with compact support

Antonio J. Fernandez Universidad Autonoma de Madrid Spain Co-Author(s):    Alberto Enciso (Instituto de Ciencias Matem\`aticas, Madrid (Spain)); David Ruiz (Universidad de Granada, Granada (Spain))

Abstract: In this talk we will show how to construct nonradial classical solutions to the $2d$ incompressible Euler equations. More precisely, for any positive integer $k$, we will see how to construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial.

Vortex dynamics for the Gross-Pitaevskii equation

Rowan Juneman University of Bath England Co-Author(s):    Manuel del Pino, Monica Musso

Abstract: The Gross-Pitaevskii equation in the plane arises as a physical model for an idealized, two-dimensional superfluid. We construct solutions to this equation with multiple vortices of degree \(\pm1\), corresponding to concentration points of the associated fluid vorticity. The vortex dynamics is described on any finite time interval, and at leading order is governed by the classical Helmholtz-Kirchhoff system. Compared to previous rigorous results of Bethuel-Jerrard-Smets and Jerrard-Spirn, we use a different method based on linearization around an approximate solution. This approach provides a very precise description of the solutions near the vortex set and information on lower order corrections to the vortex dynamics. Moreover, our analysis of the linearized problem is potentially of independent interest in the study of long-time dynamics. This is joint work with Manuel del Pino and Monica Musso.

Existence of solutions to a fractional semilinear heat equation in uniformly local weak Zygmund type spaces

Tatsuki Kawakami Ryukoku University Japan Co-Author(s):    Norisuke Ioku, Ryo Takada, and Kazuhiro Ishige

Abstract: The study of the solvability of the Cauchy problem for semilinear (fractional) heat equations is divided into the three cases with respect to the nonlinear exponent: $1

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

Seunghyeok Kim Hanyang University Korea Co-Author(s):    Monica Musso

Abstract: In this talk, we will examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold $M$ exists for all time $t$ and converges uniformly to a solution to the Yamabe problem on $M$ as $t \to \infty$. We will observe that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on $M$ in the infinite time. We are also concerned about the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points. This is joint work with Monica Musso (University of Bath, UK).

Liouville theorems and universal estimates for superlinear parabolic problems

Pavol Quittner Comenius University, Bratislava Slovak Rep Co-Author(s):    Pavol Quittner

Abstract: It is known that Liouville-type theorems guarantee universal estimates of solutions to various superlinear elliptic and parabolic problems which are scale-invariant. We discuss several recent related results, particularly for parabolic problems without scale invariance. Part of these results is a joint work with Philippe Souplet.

Critical norm blow-up rates for the energy supercritical nonlinear heat equation

Jin Takahashi Institute of Science Tokyo Japan Co-Author(s):

Abstract: We study the behavior of the scaling critical Lebesgue norm for blow-up solutions to a nonlinear heat equation (the Fujita equation). For the Sobolev supercritical nonlinearity, we show that the critical norm also blows up. Moreover, we give estimates of the blow-up rate for the critical norm. This talk is based on a joint work with Tobias Barker (University of Bath) and Hideyuki Miura (Institute of Science Tokyo).