Related papers: Blowup dynamics for equivariant critical Landau--L…
This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and…
We are concerned with blow-up solutions of the 5-dimensional energy critical heat equation $u_t=\Delta u + | u |^{\frac{4}{3}}u$. Our main result is to show that the existence of type 2 solutions blows up at 2 points with 2 different…
In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of…
For the 2D cubic (mass-critical) Zakharov-Kuznetsov equation, \begin{equation*} \partial_t\phi+\partial_{x_1}(\Delta \phi+\phi^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}, \end{equation*} we prove that there exist no finite/infinite…
We consider the focusing $L^2$-critical half-wave equation in one space dimension $$ i \partial_t u = D u - |u|^2 u, $$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_*…
We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ \alpha \in \mathbb{R}$. Unlike the standard case…
We consider the focusing inhomogeneous mass critical half-wave equation in one dimension. Under the mild conditions of the inhomogeneous factor, we show that the existence of the radial blowup solutions with ground state mass…
We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first…
We prove that the finite time blow up solutions of type II character constructed by Krieger-Schlag-Tataru as well as Krieger-Schlag are unstable in the energy topology, in that there exist open data sets in the energy topology containing…
We consider the semilinear heat equation $u_t - \Delta u = f(u)$ in $\Omega = B_R(0) \subset \mathbb{R}^n$ with super-exponential nonlinearities $f(u) = e^{u^p}u^q$ ($p>1$, $q \in \{0\}\cup [1,\infty)$), nonnegative bounded radially…
We establish the uniqueness of the positive solution for equations of the form $-\Delta u=au-b(x)f(u)$ in $\Omega$, $u|\_{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is…
We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = \Delta u + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For…
Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v =…
In this work, we consider the following focusing inhomogeneous nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\Delta u +|x|^{-b}|u|^p u=0,\quad (t, x)\in\mathbb{R}\times\mathbb{R}^N \end{align*} with $0<b<\mbox{min}\{2, N\}$…
We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical…
The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth…
This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition, \begin{equation*} \left\{\begin{array}{ll}\ds u_{t}+u_{xxxx}=|u|^{p-1}u-\frac{1}{a}\int_{0}^a|u|^{p-1}u\…
The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^{\beta}}\Delta u_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $\beta$ in the blow-up…
The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of…
We investigate the finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential and power nonlinearities in the oscillatory regime. For smooth, compactly supported, nonnegative initial data, we prove…