Stability of deterministic and stochastic nonlinear PDEs of parabolic type with degeneracies

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We are investigating (stochastic) partial differential equations of second and fourth order.One focus is placed on the investigation of free boundary problems for viscous thin films, described by a degenerate-parabolic partial differential equation of fourth order (thin-film equation). This is a continuation of existing research of the applicant, for which the respective publications have been prepared during the applicant's Ph.D. studies at the MPI Leipzig and postdoctoral stays in Canada and the United States. Specific interest lies in the understanding of the regularity of solutions at the contact line, where the phases liquid, gas, and solid coalesce. Sufficient regularity makes the proof of uniqueness and qualitative properties of solutions possible. The proposed projects aim at improving existing research regarding the extension from two to three dimensions and the generalization to a larger class of equations, corresponding to different assumptions at the interface between liquid and solid. The temporal and spatial asymptotics of solutions are of interest as well. Parts of these projects will be carried out jointly with Dr. Christian Seis (University of Bonn) and Dr. Mircea Petrache (MPI Leipzig/University of Bonn).In addition to a continuation of existing projects, we would further like to extend the analysis to a stochastic version of the thin-film equation. This equation has been proposed roughly ten years ago in the physical literature and has up to now not been addressed in mathematical journals. The focus will be placed on the proof of existence of mild solutions, where we expect that the analysis for the deterministic case can be applied. Furthermore, we would like to investigate the time asymptotics of this equation. The aim is to rigorously prove the change of the time asymptotics in the stochastic case. This question is connected to the stability of self-similar solutions, which the applicant has investigated in detail in the deterministic case.A further focus is placed on the analysis of stability of traveling waves for stochastic reaction-diffusion equations. Different notions of stability shall be investigated numerically and analytically, and we would further like to consider the asymptotics in time. This will be partially in collaboration with Prof. Christian Kuehn, Ph.D. The applicant expects synergies between this and the previously mentioned research projects.In the third year, the applicant would like to start working on long-term projects such as the description of the coarsening dynamics of droplets using systems of stochastic ordinary differential equations and the investigation of traveling waves for stochastic partial differential equations using the Evans function.