相关论文: On the characterization of Bow-up and Global Radia…

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=g(|x|,v(x)) &&\quad\mbox{in}\ \Omega, \\ \Delta v&=f(|x|,|\nabla u(x)|)…

The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u=0 &\text{on $(0,\infty)\times \Gamma_0$,}\\…

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll} \Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x…

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…

In this article, we investigate the initial and boundary blow-up problem for the $p$-Laplacian parabolic equation $u_t-\Delta_p u=-b(x,t)f(u)$ over a smooth bounded domain $\Omega$ of $\mathbb{R}^N$ with $N\ge2$, where $\Delta_pu={\rm…

In this paper, we consider the wave equation with variable coefficients and boundary damping and supercritical source terms. The goal of this work is devoted to prove the local and global existence, and classify decay rate of energy…

If $p>1+2/n$ then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions…

In general, solutions $u$ to \[ \Delta u(\mathbf{x})=f(\mathbf{x})\chi_{\{u>\psi\}} \] are not $C^{1,1}$, even for $f$ smooth and $\psi(\mathbf{x})\equiv0$. Points around which $u$ is not $C^{1,1}$ are called singular points, and the set of…

We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in…

This paper investigates the repulsion-consumption system \begin{align}\tag{$\star$} \left\{ \begin{array}{ll} u_t=\Delta u+\nabla \cdot(S(u) \nabla v), \tau v_t=\Delta v-u v, \end{array} \right. \end{align} under no-flux/Dirichlet…

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q}…

As the main problem, the bi-Laplace equation $\Delta^2u=0 (\Delta=D_x^2+D_y^2)$ in a bounded domain $\Omega \subset \re^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In…

We establish the a priori estimates and prove a blow-up criterion for the three-dimensional free boundary incompressible ideal magnetohydrodynamics equations. The fluid occupies a bounded region with a free boundary that is a closed…

In this work we consider the boundary blow-up problem $$ \left\{ \begin{array}{ll} \Delta u = f(u) & \hbox{in } B\\ \ \ u=+\infty & \hbox{on }\partial B \end{array} \right. $$ where $B$ stands for the unit ball of $\mathbb{R}^N$ and $f$ is…

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem$$\text{div}\big(\phi_{N}(\nabla v)\big)+\lambda f(|x|, v)=0 \text{in} B(R), v=0 \text{on} \partial B(R), $$where…

For superlinear heat equations with the Dirichlet boundary condition, the $L^\infty$ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with…

This paper deals with the blow-up properties of positive solutions to a system of two heat equations.

In this note, we consider an evolution coercive Hamilton-Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient…

In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the…