We study the stationary state of the energy transport problem for self-gravitating particles and show that radially symmetric negative energy solutions exist if and only if $3\leq n\leq 9$, where $n$ denotes the space dimension.

This paper is dealing with entire solutions of a bistable reaction-diffusion equation with Nagumo type nonlinearity, so called the Allen--Cahn equation. Here the entire solutions are meant by the solutions defined for all $(x,t)\in \mathbb{R}\times\mathbb{R}$. In this article we first show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of $x$-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. We also show the existence of an entire solution emanating from the unstable standing pulse solution and converges to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle we prove a rather general result on the existence of an unstable set of an unstable equilibrium to apply to the present case.

We establish local and global existence results for Dirac-Klein- Gordon equations in one space dimension, employing a null form estimate and a fixed point argument.

We study the forward self-similar solutions to a parabolic-elliptic system $$ u_t = \Delta u - \nabla \cdot (u\nabla v),\quad 0 = \Delta v + u $$ in the whole space $\bf R^2$. First it is proved that self-similar solutions $(u, v)$ must be radially symmetric about the origin. Then the structure of the set of self-similar solutions is investigated. As a consequence, it is shown that there exists a self-similar solution $(u, v)$ if and only if $\|u\|_{L^1({\bf R}^2)} \lt 8\pi$, and that the profile function of $u$ forms a delta function singularity as $\|u\|_{L^1({\bf R}^2)} \to 8\pi$.

We establish that if $n\geq3$ and $p\gt 1$ are large enough, then for each $\alpha\gt 0$ the elliptic equation $\Delta u+\frac12x\cdot\nabla u+\frac m2u+|x|^lu^p=0$ in $\bf R^n$ with $l\gt -2$ and $m=\frac{l+2}{p-1}$ possesses a positive radial solution $u_\alpha$ with $u_\alpha(0)=\alpha$ such that (i) $u_\beta\gt u_\alpha$ for $\beta\gt \alpha\gt 0$; (ii) for every $\alpha\gt 0$, $r^mu_\alpha(r)\rightarrow \ell$ as $r\rightarrow\infty$ for some $0\lt \ell=\ell(\alpha) \lt l =l(\alpha) \lt L$; (iii) $l(\alpha)$ is a one-to-one and onto increasing map from $(0, \infty)$ to $(0,L)$, where $L=[m(n-2-m)]^{1/(p-1)}$

We prove the existence of classical positive solutions for a class of indefinite weight semilinear elliptic partial defferential equations on the homogeneous Dirichlet boundary conditions and with that the growth of the perturbation is critical Soboler exponent.

The $g$-Navier-Stokes equations in spatial dimension 2 are the following equations introuduced in [3] $$ \frac{\partial \mathbf u}{\partial t}-\nu\Delta {\mathbf u} + (\mathbf u \cdot\nabla)\mathbf u +\nabla p =\mathbf f, $$ with the continuity equation $$ \frac{1}{g}\nabla\cdot (g {\bf u})= 0. $$ Here, we show the existence and uniqueness of solutions of $g$-Navier-Stokes equations on $\mathbf R^n$ for $n=2, 3$.

In this paper we consider the solvability for a new class of doubly nonlinear evolution equations. The motivation of this work comes from a transmission problem of two degenerate parabolic equations with convection term, in which the transmission boundary is time-dependent. We give an abstract existence result, and show that the weak variational formulation for the transmission problem can be solved by applying this abstract result. In our existence proof, the abstract theory of pseudo-monotone operators is useful.

In this note we classify positive solutions of an equation$$\Delta u + \frac 1 2 x \cdot \nabla u + \frac 1 {p-1} u - |u|^{p-1} u =0\quad\text{in} \quad R^N,$$where $1\lt p\lt (N+2)/N$.

Under the assumption that $|x|^{2/(p-1)} u(x)$ is uniformlybounded in $ R^N$, we show that as $r=|x|$ tends to $\infty$, $r^{2/(p-1)} u(r \sigma)$converges uniformly to a continuous function $A(\sigma)$ on$S^{N-1}$. Conversely we also show that given any nonnegative continuous function$A(\sigma)$ on $S^{N-1}$,there exists a unique positive solution with that property.

We study the Cauchy problem for the nonlinear dissipative equations \begin{equation} \left\{ \begin{array}{c} \partial _{t}u+\alpha \left( -\Delta \right) ^{\frac{\rho }{2}}u+\beta \left\vert u\right\vert ^{\sigma }u+\gamma \left\vert u\right\vert ^{\varkappa }u=0,\text{ }x\in {\mathbf{R}}^{n},t\gt 0, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{n}, \end{array} \right . \end{equation} where $\alpha ,\beta ,\gamma \in {\mathbf{C}},$ Re $\alpha \gt 0$, $\rho \gt 0$, $\varkappa \gt \sigma \gt 0.$ We are interested in the critical case, $\sigma =% \frac{\rho }{n}$ and sub critical cases $0\lt \sigma \lt \frac{\rho }{n}$. We assume that the initial data $u_{0}$ are sufficiently small in a suiatble norm, $\left\vert \int u_{0}\left( x\right) dx\right\vert \gt 0$ and Re$\beta \delta (\alpha ,\rho ,\sigma )\gt 0$, where $$ \delta (\alpha ,\rho ,\sigma )=\int \left\vert G\left( x\right) \right\vert ^{\sigma }G\left( x\right) dx $$ and $G\left( x\right) ={\mathcal{F}}^{-1}e^{-\alpha \left\vert \xi \right\vert ^{\rho }}.$ In the sub critical case we assume that $\sigma $ is close to $\frac{\rho }{n}.$ Then we prove global existence in time of solutions to the Cauchy problem (1) satisfying the time decay estimate $$ \left\Vert u\left( t\right) \right\Vert _{\mathbf{L}^{\infty }}\leq \left\{ \begin{array}{c} C\left( 1+t\right) ^{-\frac{1}{\sigma }}\left( \log \left( 2+t\right) \right) ^{-\frac{1}{\sigma }}\text{ if }\sigma =\frac{\rho }{n}, \\ C\left( 1+t\right) ^{-\frac{1}{\sigma }}\text{ if }\sigma \in \left( 0,\frac{% \rho }{n}\right) . \end{array} \right. $$