Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

Processes influencing the nitrate concentration in an aquifer for drinking water production in Bremen-Nord - Northern Germany

Groundwater represents the most important raw material. Germany struggles to maintain the best water quality possible by providing advanced monitoring systems and legal measures to prevent further pollution. In areas involved in the intensive growing of plantations, one of the major contamination factors derives from nitrate. The aim of this master thesis is the characterisation of the Water Protection Area of Bremen (Germany). Denitrification is a natural process, representing the best means of natural reduction of the hazardous nitrate ion, which is dangerous both for human health and for the development of eutrophication. The study has been possible thanks to the collaboration with the University of Bremen, the Geological Service of Bremen (GDfB) and Peter Spiedt (Water Supply Company of Bremen). It will be defined whether nitrate amounts in the groundwater still overcome the threshold legally imposed, and state if the denitrification process takes place, thanks to new samples collected in 2015 and their integration with historical data. Gas samples have been gathered to test them with the “N2/Ar method”, which is able to estimate the denitrification rate quantitatively. Analyses stated the effective occurrence of the reaction, nevertheless showing that it only affects the chemical of the deep aquifers and not shallow ones. Temporal trends concentrations of nitrate have shown that no real improvement took place in the past years. It will be commented that despite the denitrification being responsible for an efficacious lowering in the nitrate ion, it needs reactive materials to take place. Since the latter are finite elements, it is not an endless process. It is thus believed that is clearly necessary to adopt a better attitude in order to maintain the best chemical qualities possible in such an important area, providing drinking water.