Related papers: Threshold for blowup for equivariant wave maps in …
In this paper, we show that for certain initial values, the (extrinsic) biharmonic map flow in dimension four must blow up in finite time.
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite…
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…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also…
It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a…
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be…
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on…
We study the semilinear wave equation in Schwarzschild metric (3+1 dimensional space--time). First, we establish that the problem is locally well--posed in $\cs H^\sigma$ for any $\sigma \geq 1$; then we prove the blow up of the solution…
This paper is concerned with the wave breaking phenomena for a coupled periodic Camassa-Holm system. We establish a new blowup criterion for strong solutions by the method of characteristic and convolution estimates, and also give the…
We consider the $SO(d)$-equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions $d>10.$ We construct a family of…
We verify the critical case $p=p_0(n)$ of Strauss' conjecture (1981) concerning the blow-up of solutions to semilinear wave equations with variable coefficients in $\mathbf{R}^n$, where $n\geq 2$. The perturbations of Laplace operator are…
In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(\delta)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound…
We consider the quadratic nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} i\partial_t u +\Delta u =v \overline{u},\\ i\partial_t v +\kappa \Delta v =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq…
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space…
We study energy critical one-equivariant wave maps taking values in the two-sphere. It is known that any finite energy wave map that develops a singularity does so by concentrating the energy of (possibly) several copies of the ground state…
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - \Delta u = u^{p}$ in three space dimensions. We show that for generic large initial…
We study the instability of standing wave solutions for nonlinear Schr\"{o}dinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension…
We consider wave maps from $(1+d)$-dimensional Minkowski space into the $d$-sphere. For every $d \geq 3$, there exists an explicit self-similar solution that exhibits finite time blowup. This solution is corotational and its mode stability…
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up…