Singular limit of the generalized Burgers equation with absorption

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $Rtimes (0,infty)$, $u(x,0)=u_0(x)ge 0$ in $R$, as $mtoinfty$ for any $p>1$ and $u_0in L^1(R)cap L^{infty}(R)$ or as $ptoinfty$ for any $m>1$ and $u_0in L^{infty}(R)$ . We also show that in general $underset{ptoinfty}limunderset{mtoinfty}lim u_{m,p} eunderset{mtoinfty}limunderset{ptoinfty}lim u_{m,p}$.

Existence of Neumann and singular solutions of the fast diffusion equation

Let $Omega$ be a smooth bounded domain in $R^n$, $nge 3$, $0<mlefrac{n-2}{n}$, $a_1,a_2,..., a_{i_0}inOmega$, $delta_0=min_{1le ile i_0}{dist }(a_i,1Omega)$ and let $Omega_{delta}=Omegasetminuscup_{i=1}^{i_0}B_{delta}(a_i)$ and $hat{Omega}=Omegasetminus{a_1,...,a_{i_0}}$. For any $0<delta<delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=Delta u^m$ in $Omega_{delta}times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $hat{Omega}times (0,T)$ for some $T>0$ that blow-up at the points $a_1,..., a_{i_0}$.

Decay rate and radial symmetry of the exponential elliptic equation

Let $ngeq 3$, $alpha$, $betainmathbb{R}$, and let $v$ be a solution $Delta v+alpha e^v+beta xcdot abla e^v=0$ in $mathbb{R}^n$, which satisfies the conditions $lim_{Rtoinfty}frac{1}{log R}int_{1}^{R}rho^{1-n} (int_{B_{rho}}e^v,dx)drhoin (0,infty)$ and $|x|^2e^{v(x)}le A_1$ in $R^n$. We prove that $frac{v(x)}{log |x|}to -2$ as $|x|toinfty$ and $alpha>2beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.