American Institute of Mathematical Sciences

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

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We deal with a class of solutions having finite energy

$E(t)=\int_{\mathbb R^n} \rho(x)u(x,t) dx$

for all $t\ge 0$. We assume that $m> 1$ (slow diffusion) and the density $\rho(x)$ is positive, bounded and smooth. We prove existence of weak solutions starting from data $u_0\ge 0$ with finite energy. We show that uniqueness takes place if $\rho$ has a moderate decay as $|x|\to\infty$ that essentially amounts to the condition $\rho\notin L^1(\mathbb R^n)$. We also identify conditions on the density that guarantee finite speed of propagation and energy conservation, $E(t)=$const. Our results are based on a new a priori estimate of the solutions.

$A_t=\varepsilon^2 \Delta A-A+\frac{A^2}{H(1+kA^2)},\ A>0,\ $ in $\Omega\times (0,\infty), $

$\tau H_t=D\Delta H-H+A^2,\ H>0,\ $ in $\Omega \times (0,\infty),$

$\frac{\partial A}{\partial \nu}=\frac{\partial H}{\partial \nu}=0,\ $ on $\partial \Omega\times (0,\infty),$

where $\varepsilon>0$, $\tau \geq 0$, $k>0$. The unknowns $A=A(x,t)$, $H=H(x,t)$ represent the concentrations of the activator and the inhibitor at a point $x\in \Omega \subset R^N$ and at a time $t>0$. Here $\Delta$ := $\sum_{j=1}^N\frac{\partial^2}{\partial x^2_j}$ is the Laplace operator in $R^N$, $\Omega$ is a bounded smooth domain in $R^N$, and $\nu=\nu(x)$ is the outer unit normal at $x\in \partial \Omega$. When $\Omega$ is an $x_N$-axially symmetric domain and $2\leq N\leq 5$, for sufficiently small $\varepsilon>0$ and sufficiently large $D>0$ we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of $x^N$-axis and $\partial \Omega$, under the condition that $4k\varepsilon^{-2N}|\Omega|^2$ converges to some $k_0\in[0,\infty)$ as $\varepsilon\to 0$.