Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over RN\mathbb R^N

We study the propagation profile of the solution $u(x,t)$ to the nonlinear diffusion problem $u_t-\Delta u=f(u)\; (x\in \mathbb R^N,\;t>0)$, $u(x,0)=u_0(x) \; (x\in\mathbb R^N)$, where $f(u)$ is of multistable type: $f(0)=f(p)=0$, $f'(0)<0$, $f'(p)<0$, where $p$ is a positive constant, and $f$ may have finitely many nondegenerate zeros in the interval $(0, p)$. The class of initial functions $u_0$ includes in particular those which are nonnegative and decay to 0 at infinity. We show that, if $u(\cdot, t)$ converges to $p$ as $t\to\infty$ in $L^\infty_{loc}(\mathbb R^N)$, then the long-time dynamical behavior of $u$ is determined by the one dimensional propagating terraces introduced by Ducrot, Giletti and Matano [DGM]. For example, we will show that in such a case, in any given direction $\nu\in\mathbb{S}^{N-1}$, $u(x\cdot \nu, t)$ converges to a pair of one dimensional propagating terraces, one moving in the direction of $x\cdot \nu>0$, and the other is its reflection moving in the opposite direction $x\cdot\nu<0$. Our approach relies on the introduction of the notion "radial terrace solution", by which we mean a special solution $V(|x|, t)$ of $V_t-\Delta V=f(V)$ such that, as $t\to\infty$, $V(r,t)$ converges to the corresponding one dimensional propagating terrace of [DGM]. We show that such radial terrace solutions exist in our setting, and the general solution $u(x,t)$ can be well approximated by a suitablly shifted radial terrace solution $V(|x|, t)$. These will enable us to obtain better convergence result for $u(x,t)$. We stress that $u(x,t)$ is a high dimensional solution without any symmetry. Our results indicate that the one dimensional propagating terrace is a rather fundamental concept; it provides the basic structure and ingredients for the long-time profile of solutions in all space dimensions

Dissipation of information in channels with input constraints

One of the basic tenets in information theory, the data processing inequality states that output divergence does not exceed the input divergence for any channel. For channels without input constraints, various estimates on the amount of such contraction are known, Dobrushin's coefficient for the total variation being perhaps the most well-known. This work investigates channels with average input cost constraint. It is found that while the contraction coefficient typically equals one (no contraction), the information nevertheless dissipates. A certain non-linear function, the \emph{Dobrushin curve} of the channel, is proposed to quantify the amount of dissipation. Tools for evaluating the Dobrushin curve of additive-noise channels are developed based on coupling arguments. Some basic applications in stochastic control, uniqueness of Gibbs measures and fundamental limits of noisy circuits are discussed. As an application, it shown that in the chain of $n$ power-constrained relays and Gaussian channels the end-to-end mutual information and maximal squared correlation decay as $\Theta(\frac{\log\log n}{\log n})$, which is in stark contrast with the exponential decay in chains of discrete channels. Similarly, the behavior of noisy circuits (composed of gates with bounded fan-in) and broadcasting of information on trees (of bounded degree) does not experience threshold behavior in the signal-to-noise ratio (SNR). Namely, unlike the case of discrete channels, the probability of bit error stays bounded away from $1\over 2$ regardless of the SNR.Comment: revised; include appendix B on contraction coefficient for mutual information on general alphabet

Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq 2)$}, \leqno(1) where $p>1$, $\theta, l\in\R^1$ with $N+\theta>2$, $l-\theta>-2$, and $\Omega$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^\sigma u(x)$, equation (1) can be rewritten as a nonlinear Schr\"odinger equation with Hardy potential -\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $\Omega \subset \R^N \;\; (N \geq 2)$}, \leqno{(2)} where $p>1$, $\alpha \in (-\infty, \infty)$ and $\ell \in (-\infty,(N-2)^2/4)$. We show that under our chosen setting for the finite Morse index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent $p$ in (1) that divide the behavior of finite Morse index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2)