American Institute of Mathematical Sciences

(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow diffusion) and $\rho(x)$ is positive, bounded and behaves like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with $0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative and such that $\int \rho(x)u_0 dx< \infty$.
Our asymptotic analysis leads to the associated singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which admits a one-parameter family of selfsimilar solutions $ U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty $ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

$ \Delta u+\lambda u+u^{\frac{N+2}{N-2}} =0 \quad$ in$\quad B$

$ u>0 \quad$ in $\quad B $

$ u=0 \quad$ on $\quad \partial B$

where $\lambda$ is a constant. It is proven in [3] that this problem has a classical solution if and only if $\underline \lambda < \lambda < \lambda _1$ where $\underline \lambda = 0$ if $N\ge 4$, $\underline \lambda = \frac{\lambda _1}4$ if $N=3$. This solution is found to be unique in [17]. We prove that there is a number $\lambda_*$ and a continuous function $a(\lambda)\ge 0$ decreasing in $(\underline \lambda, \lambda_*]$, increasing in $[\lambda_*, \lambda_1)$ such that for each $\lambda$ in this range and each $\mu\in (a(\lambda),\infty)$ there exist a $\mu$-periodic function $w_\mu(t)$ and two distinct radial solutions $u_{\mu j}$, $j=1,2$, singular at the origin, with $u_{\mu j}(x)$~$|x|^{-\frac{N-2}2}w_\mu$( log $|x|$) as $x\to 0$. They approach respectively zero and the classical solution as $\mu\to +\infty$. At $\lambda =\lambda_*$ there is in addition to those above a solution ~$ c_N|x|^{-\frac{N-2}2}$. This clarifies a previous result by Benguria, Dolbeault and Esteban in [2], where a existence of a continuum of singular solutions for each $\lambda\in (\underline\lambda, \lambda_1)$ was found.