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Diffusionaggregation processes with monostable reaction terms
1.  Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OX1 3LB Oxford 
2.  Dept. of Engineering Sciences and Methods, University of Modena and Reggio Emilia, I42100, Italy 
3.  Dept. of Mathematical Sciences, Polytechnic University of Marche, Ancona, I60131, Italy 
4.  Dept. of Electronic and Telecommunications, University of Florence, Florence, I50139, Italy 
$\upsilon_{\tau}=[D(\upsilon)\upsilon_{x}]_{x}+f(\upsilon) \tau\ge 0, x\in R,$
where $f$ is a monostable (i.e. Fishertype) nonlinear reaction term and $D(\upsilon)$ changes its sign once, from positive to negative values, in the interval $\upsilon \in [0, 1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $\upsilon$ at time $\tau$ and position $x$. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states $\upsilon \equiv 0$ and $\upsilon \equiv 1$. In the degenerate case, i.e. when $D(0) = 0$ and/or $D(1) = 0$, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.
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