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 dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $\Delta u = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.

In this paper, we consider the parabolic equation $w_t=\Delta w+|x|^{l} w^p$, $x \in {\bf R}^n$, $t>0$ with $w(x, 0)=f(x)$ and show the existence of global solution if $1+(2+l)/n < p <(n+2+2l)/(n-2)$ for each $n \geq 3$ and $l \in (-2, l^*]$, where $l^*=0$ if $n \geq 4$ and $l^*=\sqrt3-1$ if $n=3$. In order to prove this result, we need an upper solution for this Cauchy problem. If $f(x)$ satisfies some condition, then we can show the existence of upper solution by investigating the structure of positive radial solutions for related elliptic equation which has a gradient term.

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schrödinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second-order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies, in particular, the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.

We use the maximum principle type estimate and interpolation inequality on Besov spaces to show some regularity criteria for the generalized Navier-Stokes equations, the quasi-geostrophic equations, and the harmonic heat flow.

We consider the following nonlinear parabolic equation \begin{equation} \left\{ \begin{aligned} \partial_tu-\Delta u+\frac{u}{\varepsilon}(|\nabla u|^2-1) & =0, \quad(t,x)\in(0,\infty)\times{\mathbb{R}}^n, \\ u(0,x) & =u_0(x),\quad x\in{\mathbb{R}}^n, \end{aligned} \right. \label{eq:16} \tag{#} \end{equation} which is derived by Goto-K. Ishii-Ogawa [6] to show the convergence of some numerical algorithms for the motion by mean curvature. They assumed that the solution of (#) is sufficiently regular. In this paper, we study the regularity of solutions of (#) from the Harnack estimate. We show the explicit dependence of a constant in the Harnack inequality using the De Giorgi-Nash-Moser method. We employ the Cole-Hopf transform to treat the nonlinear term.

We are concerned with the Green's function associated with the non-linear degenerate elliptic operator of divergence form. We present several theorems on local estimates and comparisons with the p-Laplacian for the Green's function under certain conditions in the sense of distributions. As an application of these estimates of the Green's function, the regularity in Morrey spaces of the so-called inhomogeneous A-harmonic equation is derived.

We study the local and global well posedness of the initial-value problem for the non-linear Schrödinger-Boussinesq System. Local existence results are proved for three initial data in Sobolev spaces of negative indices. Global results are proved using the arguments of Colliander Holmer and Tzirakis (2006 Arxiv preprint math.AP/0603595).

We study the Gevrey regularity down to $t=0$ of solutions to the initial-value problem for the semilinear heat equation $\partial_tu-\Delta u+F[u]=0$ with polynomial non-linearities. The approach is based on suitable iterative fixed point methods in $L^p$-based Banach spaces with anisotropic Gevrey norms with respect to the time and space variables. We also construct explicit solutions uniformly analytic in $t\geq 0$ and $x\in {\mathbb R}^n$ for some conservative non-linear terms with symmetries.