In this thesis we study the existence of global classical solutions to a weighted nonlocal reaction-diffusion equation on the whole space R
m
. In this equation we consider the evolution as a result of diffusion, nonlinear local growth and a weighted nonlocal inhibition term that considers the current state of the entire domain. These equations are particularly useful for models in biology, chemistry and evolution theory.

The main objective is to extend an existing global existence result for an unweighted nonlocal reaction-diffusion equation to a weighted one for a positive bounded weight. Unlike the original result, showing only the boundedness of solutions, we show the entire process of proving the global existence of classical solutions.

After introducing the necessary analytical tools and notations we go about proving the main result in five steps. Firstly we prove the existence of a unique mild solution using the well-known Banach fixed-point method. Standard regularity results are then used to obtain a local classical solution. With a classical solution in hand we can study various properties of the solution. First, it is shown that non-negative initial data results in non-negative solutions by means of bounding the size of the negative part of the solution by 0. Next, uniform a priori estimates are derived in L
k
(R
m
) for β≤k≤∞, where β comes from the nonlinearity inside the nonlocal inhibition term. These estimates hold under some restrictions concerning the exponents of the equation and the dimension of the x-domain. The most important aspect of these estimates is that they are independent of the local existence time. The proof for these estimates follows that of the unweighted nonlocal reaction-diffusion equation.

Finally, a continuation argument is employed, using the a priori estimate and the fact that it is independent of the local existence time, to show that every unique local classical solution of an equation satisfying the restrictions needed for the a priori estimate can be extended to a unique global classical solution. This concludes the extension of the global existence of solutions to the weighted problem.

As a last note, we also discuss the possibility of further study into the existence of global solutions for less restricted weights. ... Read more

Graph burning models the transmission of viruses and information. Graph burning is an iterative process in which vertices go from unburned to burned. In every round, first, all neighbors of burned vertices are also burned, second, one unburned vertex is selected as a source and also burned. These rounds are repeated until every vertex is burned. The burning number of a graph is the minimum number of rounds needed to burn every vertex in that graph.

Finding the burning number for a general graph is an NP-complete problem. Therefore, computing the burning number for relatively big graphs takes a long time. One way to reduce this time is by finding lower and upper bounds for the burning number that are close to the actual burning number.

In this thesis, we find lower and upper bounds for the burning number of Cartesian and strong products of multiple path graphs. These graphs are also known as higher-dimensional grid and king's graphs. The technique used to find these bounds is based on the technique Mitsche et al. used to asymptotically determine the burning number of grid graphs and king's graphs. A grid graph is the Cartesian product of two path graphs and a king's graph is the strong product of two path graphs.

For the burning number of higher-dimensional grid graphs, we found an implicit lower bound. The lower bound for the burning number of an i-dimensional grid graph can be calculated using the lower bound for the burning number of the (i-1)-dimensional grid graph.

We found that the burning number of the strong product of i paths of length n is larger than 0.5n^i/(i+1) and smaller than ⌈0.5(n+1)⌉. For the strong product of i paths of lengths m₁,m₂,…,m_i, we proved the burning number is larger than 0.5∏ⁱ_j=1m_j^1/(i+1). ... Read more

Due to climate change, low river discharges are expected more frequently. As a result, salt intrusion in the Rotterdam Waterway will be critical at least five times more frequently, negatively impacting freshwater availability. A possible new measure to reduce salt intrusion is providing additional mixing by internal wave generation, which reduces the estuarine circulation. The potential to reduce salt intrusion in the Rotterdam Waterway by internal wave generation is investigated. To determine this the irreversible mixing generated by internal waves is quantified and related to salt intrusion length. Numerical 2DV simulations of a decelerating flow over an undular bed were executed for an idealized section of the Rotterdam Waterway. The mixing was quantified by determining the background potential energy in a chosen control volume. This method can be applied if disturbances in the velocity or density profile do not reach the boundaries of the control volume. The mixing observed in the control volume is related to salt intrusion by the buoyancy frequency of the background state. The net salt flux imported by estuarine circulation was found to be proportional to the squared buoyancy frequency. Estuarine circulation is the main forcing mechanism of salt intrusion in the seaward part of the Rotterdam Waterway. For the small idealized section of the Rotterdam Waterway, internal waves generated over geometrical features are capable of reducing the salt intrusion caused by estuarine circulation by approximately 1.7% of the initial value, if bed waves are present over 27.5% of the control volume. ... Read more

In dit verslag behandelen we de Extrapolatie Stelling van Yano: een stelling die voor operators van de vorm T : f(x) -> T(f)(x) in bepaalde omstandigheden een afschatting geeft van de absolute integraal over T(f)(x) in termen van f(x) zelf. We zullen zien dat deze afschatting, en meer afschattingen van soortgelijke vorm, onder de juiste voorwaarden continuïteit impliceert voor de operator. Om deze stelling te kunnen behandelen zullen we wat theorie over vectorruimtes opgebouwd uit functies, en enkele voorbeelden relevant voor de stelling, introduceren. Ook behandelen we wat theorie omtrent operators, en passen we de Extrapolatie Stelling van Yano toe op enkele operators. ... Read more