On Some Parabolic Type Problems from Thin Film Theory and Chemical Reaction-Diffusion Networks

This dissertation considers some parabolic type problems from thin film theory and chemical reaction-diffusion networks. The dissertation consists of two parts:;In the first part, we study the evolution of a thin film of fluid modeled by the lubrication approximation for thin viscous films. We prove an existence of (dissipative) strong solutions for the Cauchy problem when the sub-diffusive exponent ranges between 3/8 and 2; then we show that these solutions tend to zero at rates matching the decay of the source-type self-similar solutions with zero contact angle. We introduce the weaker concept of dissipative mild solutions and we show that, in this case, the surface-tension energy dissipation is the mechanism responsible for the H1--norm decay to zero of the thickness of the film at an explicit rate. Relaxed problems, with second-order nonlinear terms of porous media type, are also successfully treated by the same means.;[special characters omitted].;In the second part, we are concerned with the convergence of a certain space-discretization scheme --the so-called method of lines-- for mass-action reaction-diffusion systems. First, we start with a toy model, namely.;[special characters omitted].;and prove convergence of method of lines for this linear case. Here weak convergence in L2(0,1) is enough to prove convergence of the method of lines. Then we adopt the framework for convergence analysis introduced in [23] and concentrate on the proof-of-concept reaction.;within 1D space, while at the same time noting that our techniques are readily generalizable to other reaction-diffusion networks and to more than one space dimension. Indeed, it will be obvious how to extend our proofs to the multi-dimensional case; we only note that the proof of the comparison principle (the continuous and the discrete versions; see chapter 6) imposes a limitation on the spatial dimension (should be at most five; see [24] for details). The Method of Lines (MOL) is not a mainstream numerical tool and the specialized literature is rather scarce. The method amounts to discretizing evolutionary PDE\u27s in space only, so it produces a semi-discrete numerical scheme which consists of a system of ODE\u27s (in the time variable). To prove convergence of the semi-discrete MOL scheme to the original PDE one needs to perform some more or less traditional analysis: it is necessary to show that the scheme is consistent with the continuous problem and that the discretized version of the spatial differential operator retains sufficient dissipative properties in order to allow an application of Gronwall\u27s Lemma to the error term. As shown in [23], a uniform (in time) consistency estimate is sufficient to obtain convergence; however, the consistency estimate we proved is not uniform for a small time, so we cannot directly employ the results in [23] to prove convergence in our case. Instead, we prove all the required estimates from the scratch , then we use their exact quantitative form in order to conclude convergence