It is known that an increased potassium concentration in brain tissue can trigger CSD. Thus, it would be of interest, how variations of the ion concentration of the glial bath affect system dynamics. In detail, it could be analyzed by numerical trials, or better still, in a bifurcation analysis, whether increasing the potassium concentration of the bath changes system dynamics, e.g., as in the FHN model, from excitable to oscillatory. To analyze whether lateral **diffusion** of ions in the extracellular space and through glial and neuronal gap junctions contributes to ionic homeostasis, we composed a spatially one-dimensional **reaction**-**diffusion** model consisting of three-compartmental neuronal elements, Chapt. 3. Thereby, we paid attention to electroneutral lateral **diffusion**. We investigated the stable propagating solutions of the system, i.e., excitation pattern, that propagate with constant shape and velocity. The system has, depending on the pa- rameter values of the local elements and on the **diffusion** strength within the respective compartments, propagating front solutions, i.e., the buffer fails completely, or propagat- ing pulse solutions, i.e., the system recovers after a transient period of depolarization. In addition, we identified a parameter regime, where no propagating solution exists. This behaviour is expected in healthy brain, where the buffer works regularly.

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Fig. 14. Comparison between experimental and computed values of secondary electron emission yield see( t ) versus trapped charge Q p ( t ).
5.1 Mathematical Modelling
We present the modelling composed of a set of two, one dimensional **reaction**-**diffusion** equations for electrons/holes coupled with Gauss equation for the electric field and an equation for trapped electrons/holes evolution.

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This thesis investigates the optimal control of **reaction**-**diffusion**-systems that exhibit various interesting patterns such as traveling wave fronts or spiral waves. In particular, we consider a model that consists of a semilinear parabolic equation which is linearly coupled with finitely many linear parabolic equations and covers a number of well-known systems such as the one component Schlögl-equation, the two component FitzHugh-Nagumo equations, or a three component system that admits localized spot solutions. The well-posedness of the model equation is discussed. We prove the existence of a well-defined control-to-state operator for the model equation, where the control is applied linearly to the first component of the system as distributed and/or boundary control. Under certain assumptions, this mapping is twice continuously Fréchet-differentiable. This circumstance al- lows us to deduce necessary optimality conditions of first order for associated optimal control problems with a tracking-type objective functional and Tikhonov regularization. Let us remark that those conditions are essential for employing (conjugate) gradient-type algorithms for numerical computations of an optimal control and therefore, of great importance.

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Robust spatial patterning was crucial just from the beginning of cellular evolu- tion, and is key to the development of multicellular organisms. The oscillatory pole-to-pole dynamics of MinCDE proteins prevent improper cell divisions apart from midcell [24, 31]. Due to it’s critical role for the cell cycle, a robust regulation of Min oscillations is of fundamental importance. As origin of ro- bustness, an efficient mechanism, only depending on a few central molecular processes seems most likely. Indeed, experimental evidence supports a mecha- nism based on nonlinear **reaction**-**diffusion** dynamics. The Min-proteins diffuse through the cytoplasm and the ATPase MinD attaches in its ATP-bound form to the cell membrane, where it recruits MinE, MinC and MinD-ATP from the cytosol [13]. MinC inhibits cell division, but plays no role in establishing os- cillations [24, 31]. MinE, which is present as a dimer [9, 21, 30, 34], hydrolyses MinD on the membrane and thereby initiates detachment. As consequence, pole-to-pole oscillations arise in wild type cells, and striped oscillations in fil- amentous cells [31], revealing the presence of an intrinsic spatial wavelength. Experiments indicate that the temporal and spatial properties of patterns are established independently of each other, as temperature variations strongly af- fect the oscillation frequency, while leaving the spatial wavelength unchanged [38]. Thereby, proper cell division is ensured in a wide temperature range. In nearly spherical mutant cells one observes predominantly pole-to-pole os- cillations along the major or an irregularly wandering axis, as well as circular waves on the membrane [36].

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The goal of this thesis was to design a numerical method which can be applied to a special form of partial differential equations called **reaction**-**diffusion** equations. The systems of **reaction**-**diffusion** equations we consider in our work is based on the article ‘The Chemical Basis of Morphogenesis’ by Alan Turing from 1952 [Tur52]. In his work Turing proposed that **diffusion** does not always has to have a stabilising effect but can rather be destabilising for an already stable steady state. Several years after Turing’s early death in 1954 there were some suggestions for proper **reaction** kinetics. The **reaction** kinetics proposed by Schnakenberg [Sch79] and by Gierer and Meinhardt [Gie72; Koc94] are of theoretical nature whereas the kinetics proposed by Thomas [Tho75] come from experimental results. Based on these three kinetics we will analyse the performance of our newly developed method. To solve the Turing mechanisms numerically we choose the spectral methods as a basis. These methods were proposed in the middle of the 20th century [Sil54] and further elaborated in the 70s and 80s [Eli70; Ors70; Ors72]. The advantage of the spectral methods is that they provide spectral convergence but with the big disadvantage that they can only be applied to simple geometries. To avoid this geometrical problem we are using the spectral element methods [Pat84] for our algorithm. In particular we decided to use

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This work is focused on time optimal control of the monodomain equations. This **reaction** **diffusion** equation is a simplified version of the bidomain equations which were developed in the late 1970’s and which, in conjunction with different ionic models, provide the description of the electrophysiological activity of the heart [ 32 ]. The monodomain equations are a **reaction** **diffusion** system consisting of a partial differential equation for the electrical potential coupled with an ordinary differential equation describing the ionic variables. They allow for challenging wave phenomena which physiologically correspond to undesired arrhythmias. We introduce a control mechanism which models an external stimulus exerted by means of an electrode, with the goal of dampening the undesired waves. Due to the dynamical properties of the underlying equations, which include, for example, that excited cells need a certain amount of time before they return to rest, the formulation of the optimal control problem as a time optimal problem offers itself as particularly useful one.

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forcing given by W onto the dominant constant modes, i.e., the direct impact of the noise on the average. This is the somewhat expected result, and in a slightly different setting was already derived in [6] (see also [17] for a special case), where the **reaction**- **diffusion** equation under fast **diffusion** is well approximated by the **reaction** ODE. As the main work is in the first case, we only give a short proof of the second case here using the technical tools developed for the first case.

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We emphasize that, although in the deterministic case, the regularity of attrac- tors for **reaction**-**diffusion** equations (or their generalized forms) is well understood (see e.g. [2, 15, 18, 32]), in the stochastic case, since random pertubations are taken into account, the regularity of random attractors is less known. Up to the best of our knowledge, there are only three results in this direction [1, 29, 30], and all of them dealt with bounded domains and autonomous external forces. The main contribu- tion of this paper is showing the higher regularity of random attractors for (1.1) in unbounded domains without restriction on the growth of nonlinearities. Another interesting feature of the present paper is that we consider stochastic **reaction**- **diffusion** equation not only with random perturbations but also non-autonomous deterministic terms.

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The second application problem concerns the adaptive numerical simulation of intra- cellular calcium dynamics. The modeling of **diffusion**, binding and membrane transport of calcium ions in cells leads to a system of **reaction**-**diffusion** equations. The strongly local- ized temporal behavior of calcium concentration due to opening and closing of channels as well as their spatial localization are effectively treated by an adaptive finite element method. The discrete approximation of deterministic equations produces a system of stiff ordinary differential equations with multiple time scales. The time scales are handled using linearly implicit time stepping methods with an adaptive step size control. The opening and closing of channels is typically a stochastic process. A hybrid method is adopted to couple the deterministic and stochastic equations. The adaptive numerical convergence of solutions is studied with different cluster arrangements. The deterministic equations are solved with parallel numerical methods to reduce the computational time using domain decomposition methods. A good parallel efficiency is achieved with different numbers of processors.

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The concept of Turing instability is an important concept in the field of pattern formation. It explains the property of some **reaction**-**diffusion** systems to exhibit stationary solutions which are heterogeneous in space. This heterogeneity, given that the observed solution is stable, corresponds to the final pattern. Basically, according to Turing [59], the pattern is caused by those modes of the solution which are stable without **diffusion**, but became unstable after **diffusion** is introduced into the system. This idea of Turing has been quite innovative, since before establishing this concept, **diffusion** had been understood by scientists only as a smoothing factor.

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Target waves have received much less attention than, for example, rotating spiral waves. This is partially explained by the fact that for years it was believed that pacemakers in chemical **reaction**-**diffusion** systems necessarily consist of heterogeneities which locally modify the properties of the medium [22]. Indeed, the majority of target patterns observed in the BZ **reaction** are associated with the presence of a local impurity like a dust particle or gas bubble which plays the role of a catalytic particle [54]. By carefully filtering the BZ solution, the number of evolving target patterns can be significantly reduced. The developing patterns have a range of operation frequencies, indicating that the frequency is not uniquely determined by the parameters of the system [132]. Furthermore, in the experiment described in Ref. [133], the activity of a pacemaker generating a target pattern in the BZ **reaction** was suppressed by application of another, high-frequency wave source. When this other source, however, was removed, the initial pacemaker reappeared at the same location with the same frequency.

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The importance of oscillatory and wave phe nomena in developing biological systems was first proposed by Turing [1 ] in a theoretical paper in 1952. With the increasing volume of experimental work on oscillatory phenomena their importance is now fairly generally accepted. The experimental systems studied are very complex and the current upsurge of interest in model **reaction** systems which exhibit oscillatory behaviour is both a recognition of the importance of the subject and an attempt to mir ror the systems studied experimentally. It is hoped that a study of such models will provide some insight into various aspects of monphogenesis.

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Nucleation and dynamics of localized structures in dissipative, bistable systems described by nonlinear partial differential equations are of great experimental and theoretical intere[r]

Bifurca- tion points J| and J 2 occurring at the basic branch a of symmetric solutions give rise to a closed branch j of asymmetric solutions, cf.. The branches of asymmetric solution[r]

− solutions globales faibles pour des systèmes dont les nonlinéaritiés ont une croissance au plus quadratique, pour des coefficients de diffusion non linéaires du type di ci , et pour de[r]

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In a 2-cellular (and hence 4-variable) symmetrical morphogenetic system of Rashevsky-Turing type, a nonperiodic return toward an (almost) undifferentiated state is observed under numer[r]

In this thesis we have investigated parameter identication in general systems of semilinear **reaction**-diusion equations, where the parameters are space and time dependent. The rst thing we noticed is that the solution of a parameter identication problem associated with those systems is not unique in general and in some cases not even locally. To deal with ill-posedness in the sense of non continuous dependence of the parameters on the data, we analyzed regularization properties of the parameter-to-state map associated with the underlying partial dierential equation. We only proposed a variational regularization approach, i.e. Tikhonov regularization, but in principle the introduced concepts should carry over to other regularization methods.

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In this work, we introduced a new notational framework for understanding **reaction**–**diffusion** compartmental models by interpreting them as balance equations similar to those found in continuum mechanics. We first used this system to derive and explain a simple two-compartment Lotka– Volterra model as a simple example. We then examined a more complex compartmental system: the model of COVID- 19 spatiotemporal contagion dynamics introduced in [ 22 ]. We showed that this model may be regarded as a sort of con- servation law, further justifying the continuum-mechanics type interpretation.

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Furthermore, the convection-dominated **diffusion**-convection-**reaction** equation with a variable velocity field w and a variable **reaction** coefficients c is considered, compared to the pure Laplace-Beltrami or **diffusion**-**reaction** equations mainly studied above. Most of the previous studies have been performed for a fixed ε. Here, we are interested, how the error constants depend on the pertubation parameter ε. A crucial issue in the analysis of convection dominated problems is the quasi-uniformity of the error estimates with reference to ε. A negative power of ε, for example, would tend to infinity, if the **diffusion** coefficient tends to zero, and the constant in the estimates becomes unbounded. Additionally, the **diffusion**-convection problem is studied. The challenge here is the missing L 2 -control.

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We were able to measure the water distribution in formerly OH-doped periclase after a rim growth experiment (Fig. 32). In order to estimate the amount of water that was available as a catalyst for nucleation and growth, it is important to have detailed information about the water distribution in every phase involved in the experimental setup. We were not able to measure this because water-containing glue was used to stabilize the product phases inside the **reaction** rim during polishing. Consequently, it is not known if bands in Raman- or IR-spectra, measured on the product phases, are related to water that was present during the rim growth **reaction** itself or to water incorporated in glue that was used for preparation. Therefore, a completely dry preparation method is needed. This will allow quantification of free molecular water and structural OH-defects in quartz and wollastonite starting materials via IR- or Raman-spectroscopy. Product phases in **reaction** rims need, due to their small thickness in the µm-range, to be analyzed using a quantification method with high spatial resolution, which is e.g. provided by synchrotron Infrared-spectroscopy.

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