相关论文: The blow-up problem for a semilinear parabolic equ…

We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - \Delta u= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. Given any compact set $ E \subset {\mathbb R}^N $, we construct finite energy solutions which blow…

For differential inequalities of the form $$ \sum_{|\alpha| = m} (- 1)^m \partial^\alpha a_\alpha (x, u) \ge b (x) |u|^\lambda \quad \mbox{in } {\mathbb R}^n, \: n \ge 1, $$ where $a_\alpha$ and $b$ are some functions, we obtain conditions…

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small…

We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is…

We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…

In this paper, we obtain upper bounds for the critical time $T^*$ of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data;…

We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - \Delta u = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous…

We investigate the following repulsion-consumption system with flux limitation \begin{align}\tag{$\star$} \left\{ \begin{array}{ll} u_t=\Delta u+\nabla \cdot(uf(|\nabla v|^2) \nabla v), & x \in \Omega, t>0, \tau v_t=\Delta v-u v, & x \in…

Let $\Omega$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |\zeta|^p \le \zeta A (x, \zeta), \quad |A (x, \zeta)| \le C_2 |\zeta|^{p-1} $$…

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…

In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial…

We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations \[ i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $\alpha>0$. We prove the global…

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 study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive…

For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region…

This paper investigates the initial boundary value problem of a finitely degenerate semilinear pseudo-parabolic equation associated with H\"{o}rmander's operator. Based on the global existence of solutions in previous literature, the…

We consider the problem v_t & = \Delta v+ |v|^{p-1}v \quad\hbox{in }\ \Omega\times (0, T), v & =0 \quad\hbox{on } \partial \Omega\times (0, T ) , v& >0 \quad\hbox{in }\ \Omega\times (0, T) . In a domain $\Omega\subset \mathbb R^d$, $d\ge 7$…

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 $n=5$.…