Related papers: Nonradial Quenching Profile for a MEMS Model

In this paper, we prove the existence of non-radial solutions to the problem $-\triangle u=f(z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb C : |z|<1\}$ with $u(z)\in \mathbb R^k$, where $f$ is a sub-linear continuous…

We approximate the solution $u$ of the Cauchy problem $$ \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d, $$ $$ u(0,x)=u_0(x),\quad x\in\bR^d $$ by splitting the equation into the system $$…

We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll}…

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and…

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive…

This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-\Delta u=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We…

We prove that any positive solution of $ \prt_tu-\Delta u+u^q=0$ ($q>1$) in $\BBR^N\ti(0,\infty)$ with initial trace $(F,0)$, where $F$ is a closed subset of $\BBR^N$ can be estimated from above and below and up to two universal…

This work is devoted to study unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: Find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, such…

We consider wave equations with a special type of log-fractional damping. We study the Cauchy problem for this model in the whole space, and we obtain an asymptotic profile and optimal estimates of solutions as time goes to infinity in…

We study quench dynamics of the Bose-Hubbard model by exact diagonalization. Initially the system is at thermal equilibrium and of a finite temperature. The system is then quenched by changing the on-site interaction strength $U$ suddenly.…

We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability…

In this paper, we are concerned with the asymptotic behavior of solutions of M1 model proposed in the radiative transfer fields. Starting from this model, combined with the compressible Euler equation with damping, we introduce a more…

Nonlocal MEMS equations exhibit finite-time quenching, or touchdown, which is difficult to capture numerically. We study a stagewise rescaling algorithm for a two-dimensional nonlocal MEMS equation in an asymptotically constant-feedback…

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p<5$. We generalize inward/outward energy theory and weighted…

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1\Delta u+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2\Delta v+u^q\quad & \quad\mbox{in}\quad{\bf…

In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this…

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\partial\_t u(t, x) = \left(a(x) -- \int\_{\Omega} k(x, y)u(t, y) dy\right ) u(t, x) +…

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a…

In this article we address the regularity of stable solutions to semilinear elliptic equations $-\Delta u = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $\Omega \subset \mathbb{R}^n$ and…

We consider the parabolic chemotaxis model \[ u_t=\Delta u - \chi \nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=\Delta v - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions…