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We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree., Comment: 86 pages; comments are welcome

We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{equation*} \begin{cases} \pp_t u=\Delta u+|u|^{\frac{4}{3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in }~ R^5. \end{cases} \end{equation*} For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0(x)\sim |x|^{-\gamma}$ as $|x|\rightarrow \infty$ with $\gamma>\frac32$ such that the global solutions behave asymptotically \begin{equation*} \| u(\cdot,t) \|_{L^\infty (\R^5)} \sim \begin{cases} t^{-\frac{3(2-\gamma)}{2}} ~&\mbox{ if }~ \frac32<\gamma<2 (\ln t)^{-3} ~&\mbox{ if }~ \gamma=2 1 ~&\mbox{ if }~ \gamma>2 \end{cases} \mbox{ \ for \ } t >t_0, \end{equation*} which is slower than the self-similar time decay $t^{-\frac{3}{4}}$. These rates are inspired by Fila-King \cite[Conjecture 1.1]{FilaKing12}., Comment: 19 pages; comments welcome

We consider a nematic liquid crystal flow with partially free boundary in a smooth bounded domain in $\mathbb{R}^2$. We prove regularity estimates and the global existence of weak solutions enjoying partial regularity properties, and a uniqueness result., Comment: 36 pages

For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=\pi ~\mbox{ and }~ v_0(r)\sim r^{1-\gamma} ~\mbox{ as }~ r\to+\infty, \quad \gamma>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\infty ([0,\infty))} \sim \begin{cases} t^{\frac{\gamma-2}{2}}\ln t ~&\mbox{ if }~ 1<\gamma<2,\\ 1 ~&\mbox{ if }~ \gamma=2,\\ \ln t ~&\mbox{ if }~ \gamma>2,\\ \end{cases} ~\mbox{ as }~ t\to +\infty. \end{equation*}, Comment: 30 pages; comments welcome

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(\Delta u+|\nabla u|^2u) -b u\wedge \Delta u &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where $a^2+b^2=1,~a > 0,~ b\in {\mathbb R}$. Given any prescribed $N$ points in $\mathbb{R}^2$ and small $T>0$, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time $t=T$, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} for Harmonic Map Flow (HMF). However, a direct consequence of the presence of dispersion is the {\em lack of maximum principle} for suitable quantities, which makes the analysis more delicate even at the linearized level. To overcome this difficulty, we make use of two key technical ingredients: first, for the inner problem we employ the tool of {\em distorted Fourier transform}, as developed by Krieger, Miao, Schlag and Tataru \cite{Krieger09Duke,KMS20WM}. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space ($\mathsf{DMO_x}$), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently., Comment: 141pages; comments welcome

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}., Comment: 61 pages; any comment welcome

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass., Comment: A new Lemma 2.9 added. Section 4 added. The proof of a priori bound of u revised

Multivariate time series forecasting is widely used in various fields. Reasonable prediction results can assist people in planning and decision-making, generate benefits and avoid risks. Normally, there are two characteristics of time series, that is, long-term trend and short-term fluctuation. For example, stock prices will have a long-term upward trend with the market, but there may be a small decline in the short term. These two characteristics are often relatively independent of each other. However, the existing prediction methods often do not distinguish between them, which reduces the accuracy of the prediction model. In this paper, a MTS forecasting framework that can capture the long-term trends and short-term fluctuations of time series in parallel is proposed. This method uses the original time series and its first difference to characterize long-term trends and short-term fluctuations. Three prediction sub-networks are constructed to predict long-term trends, short-term fluctuations and the final value to be predicted. In the overall optimization goal, the idea of multi-task learning is used for reference, which is to make the prediction results of long-term trends and short-term fluctuations as close to the real values as possible while requiring to approximate the values to be predicted. In this way, the proposed method uses more supervision information and can more accurately capture the changing trend of the time series, thereby improving the forecasting performance.

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=\Delta u+u^{3},~&\mbox{ in } \Omega\times (0,T),\\ u(x,t)=u|_{\partial\Omega},~&\mbox{ on } \partial\Omega\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \Omega, \end{cases} \end{equation*} where $5\leq n\leq 7$, $\Omega=\R^n$ or $\Omega \subset \R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $\Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(\xi_r(t),\xi_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\in\Omega$. Moreover, the singularity location $$(\xi_r(t),\xi_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\la(t)\sim \frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely new phenomenon in the parabolic setting., Comment: 64 pages; comments are welcome

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $\Omega \subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up., Comment: 48pages; any comment is welcome