This paper consists of two parts. The first part concerns a question raised by Véron on the symmetry property of positive solutions of the semilinear elliptic equation $$ \Delta u+\frac{c}{|x|^2} u +u^{(n+2)/(n-2)}=0 \quad \mbox{in } \mathbb {R}^n\setminus \{0\}. $$ The second part concerns some nonlinear elliptic equations on the unit sphere ${\mathbb S}^n$. By the method of moving spheres and the global bifurcation theory, we obtain various symmetry, asymmetry, and non-existence results.

A countable set of self-similar solutions of the fourth-order porous medium equation (the PME$-4$), \begin{equation} u_t = - (|u|^n u)_{xxxx} \quad \mbox{in} \,\,\, {{\bf R}} \times {{\bf R}}_+ \quad (n>0), \tag*{(0.1)} \end{equation} is described. The similarity solutions under consideration have the standard form $$ u_l(x,t) = t^{-{\alpha}_l} f_l(y), \quad y= x/{t^{{\beta}_l}}, \quad {\beta}_l = \tfrac {1- {\alpha}_l n}2, \quad l=0,1,2,... \, , $$ where ${\alpha}_l$ and $f_l(y)$ are eigenvalues and eigenfunctions of a nonlinear eigenvalue problem (actually, for a linear pencil of ordinary differential operators) written as $$ {\bf B}_n(f) \equiv -(|f|^n f)^{(4)} + \tfrac {1-{\alpha} n}4 \, y \, f' + {\alpha} f=0 \quad \mbox{in} \quad {{\bf R}}, \quad f \not =0, \,\,\, f \in C_0({{\bf R}}). $$ First four nonlinear eigenfunctions were obtained by Bernis and McLeod in the 1980s. In order to identify the full countable set of eigenfunctions $\{f_l\}$, we check their appearance at the branching point $n=0$ from the eigenfunctions $\{\psi_l\}$ of the non self-adjoint operator $ {\bf B}_0 = - D_y^4 + \frac 14 \, y D_y + \frac 14 \, I, \quad \mbox{with the point spectrum} \,\,\, {\sigma}({\bf B}_0)= \{{\lambda}_l=- \frac l4, \, l \ge 0\}, $ where $I$ is the identity. These eigenfunctions give the solutions $$ u_l(x,t)={t}^{- \frac N4 + {\lambda}_l} \psi_l(y), \quad y=x/t^{\frac 14}, \quad l=0,1,2,... \, , $$ of the linear bi-harmonic equation $ u_t = -u_{xxxx}, $ which is (0.1) for $n=0$. The results extend to the PME$-4$ posed in ${{\bf R}^N}$. A similar classification is performed for the PME$-6$ \begin{equation} \tag*{(0.2)} u_t = (|u|^n u)_{xxxxxx} \quad \mbox{in} \,\,\, {{\bf R}} \times {{\bf R}}_+ \quad (u_t=u_{xxxxxx} \,\,\, \mbox{for} \,\,\, n=0). \end{equation} The general methodology of the study of (0.1) and (0.2) is associated with that developed for the classic second-order PME $ u_t = (|u|^n u)_{xx}$ in $ {{\bf R}} \times {{\bf R}}_+, $ which systematic study was began by the discovery of the first ZKB source-type solution by Zel'dovich, Kompaneetz, and Barenblatt in 1950-52.

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of the Camassa-Holm equation.

We study the asymptotic behavior of Timoshenko systems with memory, where the memory is given by a non-dissipative kernel and is acting only on one equation of the system. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a nice relationship between the coefficients of the system. Moreover, with full-memory effect in the system, we will show exponential stability in the general case.

We study the structure of solutions of the boundary-value problem \begin{equation} \tag*{(0.1)} \Delta^2 u=\frac{\lambda}{(1-u)^2} \;\; \mbox{in $B$}, \;\;\; u=\Delta u=0 \;\; \mbox{on $\partial B$} , \end{equation} where $\Delta^2$ is the biharmonic operator and $B \subset \mathbb R^3$ is the unit ball. We show that there are infinitely many turning points of the branch of the radial solutions of (0.1). The structure of solutions depends on the classification of the radial solutions of the equation \begin{equation} \tag*{(0.2)} -\Delta^2 u=u^{-2} \;\;\; \mbox{in $\mathbb R^3$}. \;\; \end{equation} This is in sharp contrast with the corresponding result in $\mathbb R^2$.

Consider a model of flow of two compressible or incompressible and immiscible phases in a three dimensional porous media. The existence of a weak solution is obtained for two compressible immiscible fluids when the porosity depends on the global pressure and on the space variable.