Hybrid

• 自然科学５号館数学棟４階コロキウム
  NST Hall 5, Math building, 4th floor, Colloqium room
  
• Zoom registration: https://forms.gle/g3nkMFBRb2UhpwJG9

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.

In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This problem, known as the inverse conductivity problem or Calderon problem, is notoriously ill-posed, especially in real applications with only partial boundary data available.

We discuss various approaches to modeling the inverse conductivity problem, the characteristics of these models, and the known regularity results. We then focus on the variational approach by considering the minimization of an energy regularized by the Mumford-Shah functional. As this minimization poses a significant challenge, we take the classical Ambrosio-Tortorelli approximation approach, showing its approximation properties for the inverse conductivity problem. Finally, we take a look at a few numerical examples.

A retarded functional differential equation (RFDE) is a differential equation describing the dependence of the time-derivative \dot{x}(t) of an unknown function x on the history segment x_t. The RFDEs gives a mathematical model of various delay differential equations (DDEs), and it determines an important class of infinite-dimensional dynamical systems as the time evolution of the history segment x_t in a space of continuous functions. In this talk, we develop a theory of “mild solutions” to autonomous linear RFDEs in order to resolve difficulties about discontinuous history functions in the theory of RFDEs. Furthermore, we apply it to an extension of the Poincaré–Lyapunov theorem to RFDEs.

参考文献 (References)：
[1] J. Nishiguchi, Mild solutions, variation of constants formula, and linearized stability for delay differential equations, Electron. J. Qual. Theory Differ. Equ. 2023, No.~32, 1–77. https://doi.org/10.14232/ejqtde.2023.1.32.
[2] J. Nishiguchi, On regularity of mild solutions for linear delay differential equations, submitted.