Model equations for magnetic monopoles have been obtained independently by 't Hooft, Polyakov, and Julia and Zee. This paper analyzes these equations (three simultaneous ordinary differential equations), shows that the corresponding boundary-value problems possesses a solution and discusses its qualitative properties.

Let $\Omega $ be a bounded domain in $\Bbb R^N$ with smooth boundary $\partial \Omega$. In this paper, we consider the existence and energy decay of solutions of the following problem: \begin{align} & u_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma) \Delta u(t,x)+\beta \Delta^2 u(t,x) \nonumber \\ & \quad+\delta|u_t(t,x)|^{\rho}u_t(t,x) =\mu|u(t,x)|^{\alpha}u(t,x), \quad x \in \Omega,\ t \in [0, T], \nonumber \\ & v_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma ) \Delta v(t,x)+\beta \Delta^2 v(t,x) \nonumber \\ & \quad+\delta|v_t(t,x)|^{\rho}v_t(t,x) =\mu|v(t,x)|^{\alpha}v(t,x), \quad x \in \Omega,\ t \in [0, T], \tag*{(1.1)} \\ & u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega, \nonumber \\ & v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega, \nonumber \\ & u|_{\partial \Omega}=v|_{\partial \Omega}=0, \nonumber \end{align} where $T >0$, $\alpha > 0$, $\rho \geq 0$, $\delta >0$, $\mu \in \Bbb R$, $a+b \geq 0$, $b \geq 0$, $\gamma \geq 1 $ and $$ \|\nabla u\|^2_2 =\sum_{i=1}^N \int_\Omega|\frac{\partial u}{\partial x_i}(t,x)|^2dx, \quad u_t=\frac{\partial u}{\partial t}\quad\hbox{and}\quad \Delta u=\sum_{i=1}^N \frac{\partial^2u}{\partial x_i^2}. $$

A detailed study of solutions to the first-order partial differential equation $H(x,y,z_x,z_y)=0$, with special emphasis on the eikonal equation $z_x^2+z_y^2=h(x,y)$, is made near points where the equation becomes singular in the sense that $dH=0$, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function $H$ and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.

In the framework of a classic approach to phase transitions, the standard model of it phase relaxation is generalized on the basis of physical motivations. Convergence to a weak formulation of the Stefan problem is proved by means of $L^1$-type techniques.

We study the existence of positive solutions for singular Dirichlet problems. By the variational method, we show that if $\lambda_1$ is contained in some interval, then a singular Dirichlet problem has a radially symmetric, positive solution on an annulus $\Omega$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with homogeneous Dirichlet boundary condition. Since we consider a problem with singularity, we have difficulties that either the functional $I$ associated with the problem is not differentiable even in the sense of Gâteaux or the strong maximum principle is not applicable. But we show that a function which attains some minimax value for $I$ is a solution. We also treat the case that $\Omega$ is a ball.