| Abstract: |
| In this talk we consider initial-Neumann boundary value problem of nonlinear evolution equations
with strong dissipation and proliferation arising from mathematical biology and physics formulated as
\par
\medskip
$\mbox{(NE)}\left\{\begin{array}{l}
\ u_{tt}=D\triangle u_{t}+\nabla\cdot(\chi(u_{t},e^{-u})\nabla u)+\mu u_t(1-u_t), \ \ \ (x,t)\in\Omega\times(0,\infty) \
\ \displaystyle\partial_{\nu}u|_{\partial\mathbf{\Omega}}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}(x)\ \ \ \ \ \ \
\end{array}\right.$
\par
\parindent=0cm
where constants $D, \mu$ are positive, $\Omega$ is a bounded domain in $R^{n}$\
with a smooth boundary $\partial\mathbf{\Omega}$\ and $\nu$\ is the outer unit normal vector.
We show the global existence in time
and asymptotic behavior of the solution. Then we apply the result of (NE) to a chemotaxis model proposed by J.I. Tello and A. Kubo, which is a competitive system of a parabolic equation with a logistic term and ODE describing the behavior of two biological species, and discuss the property of the solution. |
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