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- Advances in Mathematics of Communications
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### Open Access Journals

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In order to understand the problems occurring in these areas of application, one should not only apply known methods, but also develop novel mathematical tools. Because of this, many results corresponding to new approaches are given in the main topics of this CPAA Special Volume, including demography and travelling waves in epidemics modelling, structured populations growth, propagation in inhomogeneous media, ecology and dry land vegetation, formation of stationary spatio-temporal patterns in reaction-diffusion systems both from a mathematical and an experimental view point, spatio-temporal dynamics of cooperation, cell migration and bacterial suspensions. This issue also includes more mathematically oriented topics such as interface dynamics, stability of non-constant stationary solutions, heterogeneity-induced spot dynamics, boundary spikes, appearance of anomalous singularities in parabolic equations, finite time blow-up, a multi-parameter inverse problem, and the numerical approximation of parabolic equations and chemotactic systems. We hope these advanced results will be useful to the community of researchers working in the domain of partial differential equations, and that they will serve as examples of mathematical modelling to those working in the different areas of application mentioned above.

$u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$

Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.

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