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We consider co-rotational wave maps from (3+1) Minkowski space into the three-sphere. 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…
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
We extend and apply a recently developed approach to the study of dynamic bifurcations in PDEs based on the geometric blow-up method. We show that this approach, which has so far only been applied to study a dynamic Turing bifurcation in a…
We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for $x\in\real$,…
For the 3d cubic nonlinear Schr\"odinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time…
We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schr\"{o}dinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence…
In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in…
In this note, we investigate the stability of self-similar blow-up solutions for superconformal semilinear wave equations in all dimensions. A central aspect of our analysis is the spectral equivalence of the linearized operators under…
The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$, restricted to…
We consider the focusing nonlinear Schr\"odinger equation in three spatial dimensions with powers close to three and prove the existence of a self-similar solution. This generalizes a previous result on the cubic case and shows that…
A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blast wave, propagating in a powerlaw density profile of the form $\rho \propto z^{-k}$. Self-similar solutions of the first kind can be found for…
We analyze a one-dimensional two-scalar fields reaction advection diffusion model for the globally subcritical transition to turbulence. In this model, the homogeneous turbulent state is disconnected from the laminar one and disappears in a…
For the prescribed scalar curvature equation on $S^n$ ($n \ge 6$), we consider the situation where the number of bubbles tends to infinity in the Lyapunov-Schmidt (finite dimension) reduction method. In an outstanding paper by Wei and Yan,…
We consider wave maps from $(1+d)$-dimensional Minkowski space, $d\geq3$, into rotationally symmetric manifolds which arise from small perturbations of the sphere $\mathbb S^d$. We prove the existence of co-rotational self-similar finite…
We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can…
While finite-time blowup solutions have been studied in depth for the Keller-Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To…
This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of…
Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical…
The susceptibility of timestepping algorithms to numerical instabilities is an important consideration when simulating partial differential equations (PDEs). Here we identify and analyze a pernicious numerical instability arising in…