In many applications, one is interested in the nonlinear parabolic problem \[u_t-\Delta u=f(u)\; (x\in R^N, t>0), u(x,0)=u_0(x)\; (x\in R^N),\], where $u_0\in L^\infty(R^N)$ is nonnegative and has compact support, $f$ is a smooth function satisfying $f(0) = 0$. One wants to know how much of the long-time dynamics of this problem is determined by the corresponding elliptic problem \[ -\Delta u=f(u), u\geq 0\; (x\in R^N).\].
We show that, if $u(\cdot, t)$ stays uniformly bounded in $L^\infty(R^N)$ for all $t>0$, then as $t\to\infty$, $u(\cdot, t)$ converges to a stationary solution in $L_{loc}^\infty(R^N)$, provided that all the zeros of $f(u)$ in $[0,\infty)$ are nondegenerate (i.e., $f(u)=0$ and $u\geq 0$ imply $f'(u)\not=0$). Moreover, this stationary solution is either a stable constant solution (hence a stable zero of $f$), or a ground-state solution based on a stable zero of $f$ (namely a solution $v(x)$ of the elliptic problem which is radially symmetric about some point $x_0\in R^N$, and decreases in $|x-x_0|$, with $\lim_{|x|\to\infty} v(x)$ a stable zero of $f$). Thus, viewed in the space $L^\infty_{loc}(R^N)$, the long-time behavior of $u(\cdot, t)$ is determined by two simplest types of solutions of the corresponding elliptic problem. Furthermore, we show that, viewed in the space $L^\infty(R^N)$, the long-time behavior of $u(\cdot, t)$ resembles that of a radial terrace solution $v(|x|, t)$, whose limit as $t\to\infty$ is determined completely by a propagating terrace of 1-space dimension, which by definition is a set of traveling wave solutions $\{w_i\}_{i=1}^k$, with each $w_i$ solving the elliptic equation\[-w_{zz}+c_i w_z=f(w) \; (z\in R^1),\], where $c_i$ is a certain constant, known as the wave speed of $w_i$.
This talk is based on joint works with Peter Polacik and with Hiroshi Matano.
Biography
Yihong Du is Professor of Mathematics in School of Science and Technology at University of New England, Australia. His research concerns mainly with nonlinear elliptic and parabolic differential equations and nonlinear functional analysis. His current research interest focuses on the existence, uniqueness and qualitative properties of solutions of various nonlinear partial differential equations, mostly arising from applied sciences, such as mathematical biology, invasion ecology, and chemical reaction theory.