相关论文: Long-time dynamics for the energy critical heat eq…

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…

Long time dynamics of solutions to the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ on $\R^6\times(0,\infty)$ is investigated. It is shown that there exists a radially symmetric global solution $u(x,t)\in C([0,\infty);\dot…

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 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 examine the energy-critical nonlinear heat equation in critical spaces for any dimension greater or equal than three. The aim of this paper is two-fold. First, we establish a necessary and sufficient condition on initial data at or below…

In this paper we consider the initial value {problem $\partial_{t} u- \Delta u=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under…

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = \Delta u + |u|^\alpha u$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < \alpha <\alpha _0$, where $\alpha _0=\frac {4} {N-4+2\sqrt{…

In this paper, we consider the following nonlocal parabolic equation \begin{equation*} u_{t}-\Delta u=\left( \int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right) |u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty),…

We study the limit, when $k\to\infty$ of solutions of $u_t-\Delta u+f(u)=0$ in $R^N\times(0,\infty)$ with initial data $k\gd$, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible…

In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To…

We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly…

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - \Delta u + |u|^\alpha u =0$, where $u=u(t,x)\in {\mathbb R}, $ $(t,x)\in…

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study…

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 prove the existence of global solutions to the energy-supercritical wave equation in R^{3+1} u_{tt}-\Delta u + |u|^N u = 0, u(0) = u_0, u_t(0) = u_1, 4<N<\infty, for a large class of radially symmetric finite-energy initial data.…

In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In…

Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the…

We consider the critical focusing wave equation $(-\partial_t^2+\Delta)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\frac{\mu}{2}W(t^\mu x)+\eta(t,x)] in the forward lightcone…

We consider the energy critical heat equation in $\mathbb R^n$ for $n\ge 7$ $$\left\{ \begin{aligned} u_t & = \Delta u+ |u|^{\frac 4{n-2}}u \hbox{ in }\ \mathbb R^n \times (0, \infty), \\ u(\cdot,0) & = u_0 \ \hbox{ in }\ \mathbb R^n,…