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We study a non-Abelian Chern-Simons system in $\mathbb{R}^2$, including the simple Lie algebras $A_2$ and $B_2$. In a previous work, we proved the existence of radial non-topological solutions with prescribed asymptotic behaviors via the…

Analysis of PDEs · Mathematics 2017-06-13 Zhijie Chen , Chang-Shou Lin

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter $\epsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary…

Analysis of PDEs · Mathematics 2007-05-23 Juan Casado-Diaz , Manuel Luna-Laynez , Francois Murat

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of…

Fluid Dynamics · Physics 2021-10-20 Liam C. Morrow , Michael C. Dallaston , Scott W. McCue

We analyse the asymptotic behaviour of solutions of the Teichm\"uller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the…

Analysis of PDEs · Mathematics 2017-07-26 Melanie Rupflin , Matthew R. I. Schrecker

In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain…

Analysis of PDEs · Mathematics 2021-05-21 Salvador Moll , Ken Shirakawa , Hiroshi Watanabe

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to…

Analysis of PDEs · Mathematics 2020-04-13 Stan Alama , Lia Bronsard , Dmitry Golovaty , Xavier Lamy

For a smooth bounded domain $\Omega \subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_\epsilon$ to $-\Delta u_\epsilon + (a+\epsilon V) u_\epsilon =…

Analysis of PDEs · Mathematics 2025-12-23 Tobias König , Paul Laurain

This paper is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid,…

Analysis of PDEs · Mathematics 2024-08-30 Chengchun Hao , Tao Luo , Siqi Yang

We consider singular quasilinear elliptic systems with homogeneous Dirichlet boundary condition. Using Leray-Schauder topological degree, combined with the sub-supersolutions method and suitable truncation arguments, we establish the…

Analysis of PDEs · Mathematics 2025-10-27 Nouredine Medjoudj , Abdelkrim Moussaoui

The basic stationary buoyant flow in a vertical annular porous passage induced by a boundary temperature difference is investigated. The vertical cylindrical boundaries are considered both isothermal and permeable to external fluid…

Fluid Dynamics · Physics 2023-02-03 A. Barletta , M. Celli , D. A. S. Rees

We study the following Schr\"odinger-Poisson system (P_\lambda){ll} -\Delta u + V(x)u+\lambda \phi (x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -\Delta\phi = u^2, \lim\limits_{|x|\to +\infty}\phi(x)=0, u>0, where $\lambda\geqslant0$ is a…

Analysis of PDEs · Mathematics 2015-02-10 Yongsheng Jiang , Huan-Song Zhou

We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain…

Analysis of PDEs · Mathematics 2012-09-21 Seppo Granlund , Niko Marola

In this paper, we are concerned with the following Schr\"{o}dinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -\Delta u+u+\lambda\phi…

Analysis of PDEs · Mathematics 2022-11-29 Xiao-Ping Chen , Chun-Lei Tang

This dissertation discusses solutions to the nonlinear Klein-Gordon equation with symmetric and asymmetric double-well potentials, focusing on the collapse and collision of bubbles and critical phenomena found therein. A new method is…

High Energy Physics - Phenomenology · Physics 2007-05-23 Ethan Honda

We construct a radially smooth positive ancient solution for energy critical semi-linear heat equation in $\mathbb{R}^n$, $n\geq 7$. It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity.

Analysis of PDEs · Mathematics 2021-09-08 Liming Sun , Juncheng Wei , Qidi Zhang

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

Analysis of PDEs · Mathematics 2020-03-31 Denis Bonheure , Ederson Moreira dos Santos , Enea Parini , Hugo Tavares , Tobias Weth

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega,\\ -\Delta…

Analysis of PDEs · Mathematics 2014-09-25 Zhijie Chen , Chang-Shou Lin , Wenming Zou

In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $\Delta u = -2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. From a physical perspective,…

Analysis of PDEs · Mathematics 2021-09-24 Virginia Agostiniani , Stefano Borghini , Lorenzo Mazzieri

This paper deals with isothermal Euler-Poisson system which is used to model collapse of self-gravitating Newtonian star. Density dependent viscosity term is added on the right-hand side of momentum equation and it has been proved that…

Analysis of PDEs · Mathematics 2025-08-19 Marko Nedeljkov , Sanja Ružičić

In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a…

Analysis of PDEs · Mathematics 2007-05-23 Khalil El Mehdi