Kanazawa Analysis Seminar
Coupled nonlinear degenerate systems arise in various applications of societal relevance, such as biofilm growth, wildfires, and reactive transport in porous media. One of the equations in the system, describing the evolution of a biomass, exhibits degenerate and singular diffusion behaviour. The other equations are either of advection-reaction-diffusion type or ordinary differential equations modelling nutrient dispersion. In our analysis, we first propose a backward Euler time-discretization of the problem where the reactive terms coupling the equations are estimated semi-implicitly, and thus, the equations can be solved sequentially. Properties such as well-posedness, boundedness, and positivity of the time-discrete solutions are proven, and the existence of the time-continuous solutions is shown by passing the time-step size to zero. Global- in-time well-posedness is established for Dirichlet and mixed boundary conditions, whereas only local well-posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems we formulate sufficient conditions for finite-time blow-up or uniform boundedness of solutions. Assuming additional structural assumptions we also prove the uniqueness of solutions. The time-discretization method serves as an efficient way to solve such problems and numerical experiments are provided that support this.
Then, we show the existence of traveling wave (TW) solutions for a special case of PDE-ODE coupled models arising in the growth of cellulolytic biofilms. TW solutions for such systems have previously been observed numerically as well as in experiments. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. We present numerical results that exhibit the existence and stability of the TWs, along with corroborating predictions on parametric dependence.
References
[1] K. Mitra, J.M. Hughes, S. Sonner, H.J. Eberl, & J.D. Dockery
(2023). Travelling Waves in a PDE– ODE Coupled Model of Cellulolytic
Biofilms with Nonlinear Diffusion. Journal of Dynamics and Differential
Equations, 1-35.
[2] K. Mitra, & S. Sonner (2023). Well-posedness and properties of
nonlinear coupled evolution problems modelling biofilm growth.
arXiv:2304.00175.
2013年4月,金沢大学の偏微分方程式研究者有志が集まり本セミナーを企画しました。各回の話題は,偏微分方程式の理論的な側面を中心に,セミナー幹事の関心に従い大らかに選択しています。参加者がセミナーを十分楽しみ,勉強し,新しい発見を得られるように,各回の最初の20分から30分程度,講演者の方にはその話題への導入となるような解説をお願いしています。ご関心がある方はどなたでもご自由にご参加ください。 どうぞよろしくお願いいたします。
今村 悠里・Patrick van Meurs・大塚 浩史・小俣 正朗・蚊戸 宣幸・木村 正人・榊原 航也・Thomas Geert De Jong・野津 裕史・橋本 伊都子・Norbert Pozar・Julius Fergy Tiongson Rabago・和田 啓吾
npozar (at) se.kanazawa-u.ac.jp