| Abstract: |
| We investigate the global existence in time and asymptotic profile of the solution of some nonlinear evolution equations with strong dissipation and the proliferation term:
$$\ w_{tt}= D\Delta w_{t} +\nabla\cdot(\alpha(w_{t})e^{-w}\chi[w])
+ \mu(1-w_{t})w_{t},\ \mbox{in}\ {\Omega}\times(0,T)\,$$
where $D, \mu$ are positive constants, $\alpha(\cdot)$ is an sufficiently smooth function, $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial \Omega$ and $\nu$ is the outer unit normal vector on $\partial \Omega$, $\chi[w]:=\chi[w](x,t)$ is a non-local term.
We will show the existence and asymptotic behaviour of solutions to the initial and zero-Neumann boundary value problem of the equation. We will apply our results to a model of mathematical biology, and we discuss the smoothing effect. |
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