Related papers: Infinite time blow-up for the 3-dimensional energy…

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^5,~&\mbox{ in } \Omega \times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial \Omega \times \mathbb{R}^+,\\…

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3.…

We consider positive solutions for the fractional heat equation with critical exponent \begin{equation*} \begin{cases} u_t = -(-\Delta)^{s}u + u^{\frac{n+2s}{n-2s}}\text{ in } \Omega\times (0, \infty), u = 0\text{ on }…

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.…

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…

We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…

We consider the nonlinear heat equations with Neumann boundary conditions $$ \begin{cases} u_{t}=\Delta u & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\…

We study blowup solutions of the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Vel\'azquez…

This paper is concerned with blow-up solutions of the five dimensional energy critical heat equation $u_t=\Delta u+|u|^\frac{4}{3}u$. A goal of this paper is to show the existence of type II blowup solutions which behave as…

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…

We address the critical norm blow-up problem for the nonlinear heat equation $u_t-\Delta u=|u|^{p-1}u$ in $\mathbf{R}^n\times(0,T)$. In the supercritical range $p>(n+2)/(n-2)$, we prove that if the maximal existence time $T$ is finite, then…

We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions…

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain…

We prove that solutions to the critical wave equation below can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous blow up conjecture about critical semilinear wave…

We consider the Cauchy problem for the energy critical heat equation $$ u_t = \Delta u + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$.…

We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction…

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We construct for this…

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical…

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear…

In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N $, where $\alpha >0$, $\gamma \in \R$ and $-\pi /2<\theta <\pi /2$. By convexity arguments we prove…