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| In this talk, we first discuss large time behavior of the homogeneous Dirichlet problem to the degenerate parabolic equation $u_t = g(u) \D u + f(u)$ in a bounded domain
$\Omega \subset R^n$ with smooth boundary $\partial \Omega$. Under suitable conditions on $f(u)$ and $g(u)$, we show that all solutions will converge to the steady state exponentially. Next, we study the degenerate parabolic system $u_t = u \D u + u(a_1-b_1 u +c_1 v)$ and
$v_t = v \D v + v(a_2+b_2 u -c_2 v)$ with the same boundary condition. We show that any positive solutions converge to a unique steady state exponentially if the coefficients satisfy some conditions. |
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