Attractors for reaction-diffusion equations on arbitrary unbounded domains

We prove existence of global attractors for parabolic equations of the form $$u_t+\beta(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)=f(x,u)$$ with Dirichlet boundary condition on an arbitrary unbounded domain $\Omega$ in $\R^3$, without smoothness assumptions on $a_{ij}(\cdot)$ and $\partial\Omega$.Comment: 26 page

Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitely computed and can be exploited to detect the existence of recurrent dynamics in the equation.Comment: 19 page

Lithium and Alzheimer’s Disease: Experimental, Epidemiological, and Clinical Findings

Alzheimer’s disease (AD) represents one of the greatest health-care challenges of the twenty-first century. Besides known pathologies such as intracellular accumulation of neurofibrillary tangles and extracellular deposition of amyloid-beta plaques, other factors, such as dysregulated GSK-3 activity, mitochondrial dysfunction, inflammation, and oxidative stress, have been shown to play a role in the pathogenesis of AD. Over the last two decades, the evidence accumulated for a neuroprotective effect of lithium, as an important mechanism of this ion in mood disorders, reflected by an increase in cerebral gray matter volume in lithium-treated subjects. Neurobiological mechanisms of lithium neuroprotective actions may also be relevant to the pathogenesis and treatment of AD, and they will be delineated. In most epidemiological studies, a negative association between lithium use and dementia has been shown, including two most recent papers regarding a concentration of lithium in drinking water. In this article, the results of initial studies using lithium in the treatment of dementia and showing some promise will also be presented. Therefore, considering the current paucity of treatments for the AD, further testing of lithium as a disease-modifying treatment in this illness may be warranted

Nested sequences of index filtrations and continuation of the connection matrix

AbstractIn this paper, we prove the existence of nested sequences of index filtrations for convergent sequences of (admissible) semiflows on a metric space. This result is new even in the context of flows on a locally compact space. The nested index filtration theorem implies the continuation of homology index braids which, in turn, implies the continuation of connection matrices in the infinite-dimensional Conley index theory