Special Session 56: Local and nonlocal diffusion in mathematical biology

Stability of solutions of the porous medium equation with growth with respect to the diffusion exponent

Piotr Gwiazda University of Warsaw Poland Co-Author(s):

Abstract: We consider a macroscopic model for the growth of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Assuming a power-law relation between the mechanical pressure and the cell density, the model can be expressed as the porous medium equation with a growth term. We prove Lipschitz continuous dependence of the solutions of the model on the diffusion exponent. The main difficulty lies in the degeneracy of the porous medium equations at vacuum.

Asymptotic behaviors of solutions to a reaction-diffusion equation with free boundaries

Yuki Kaneko Kanto Gakuin University Japan Co-Author(s):

Abstract: This talk is concerned with a reaction-diffusion equation in a one-dimensional interval whose boundaries are unknown and determined together with the density function. We impose one-phase Stefan conditions to the free boundaries with different coefficients as parameters. Then we can observe that a spreading solution converges to some different kinds of propagating terrace, depending on the parameters, as time tends to infinity. We will also discuss future problems.

Spreading phenomenon in a nonlinear Stefan problem with a certain class of multistable nonlinearity

Hiroshi Matsuzawa Kanagawa University Japan Co-Author(s):    Yuki Kaneko, Yoshio Yamada

Abstract: In this talk, we discuss a spreading phenomenon described by a nonlinear Stefan problem of a reaction-diffusion equation. In particular, we assume that the reaction term is a positive bistable type nonlinearity class of multi-stable type nonlinearity. We first discuss the classification of asymptotic behaviors of solutions. We next discuss the expanding speeds of the free boundary and the level set of the solution in the spreading case.

A Navier-Stokes-Cahn-Hilliard system in 3D: well-posedness and nonlocal-to-local rates of convergence

Andrea Poiatti University of Vienna Austria Co-Author(s):    Christoph Hurm, Patrik Knopf

Abstract: In this talk I would like to present some results concerning a Navier-Stokes-Cahn-Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H, in three (and two) dimensional bounded domains. I will first concentrate on some new results of local-in-time strong well-posedness for the nonlocal version of the model H with singular potential. Namely, I will also discuss the validity of the instantaneous strict separation property of the concentration variable from pure phases, by adapting the recent result for the nonlocal Cahn-Hilliard equation in 3D by Poiatti (Anal. PDE, to appear). I will then present the nonlocal-to-local convergence of strong solutions to the model H. This means that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. To this aim, I will show some uniform bounds on the strong solutions to the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results. The novelty of this approach is that we are able to find precise convergence rates, in suitable norms, of the strong solutions to nonlocal model H to the strong solution to local model H.

Biological aggregations from spatial memory and nonlocal advection

Junping Shi College of William & Mary USA Co-Author(s):    Di Liu, Yurij Salmaniw, Jonathan Potts, Junping Shi and Hao Wang

Abstract: Spatial memory is a key feature driving the movement of mobile organisms. A key tool for modelling movement in response to remembered space use is via an advection term in a partial differential equation (PDE). We use a reaction-diffusion-advection model to describe the movement of an animal species, with a non-local advection term driven by a cognitive map representing memory of past animal locations embedded in the environment. The global existence and boundedness of solutions are shown, and the existence of spatial patterns formed in the model are rigorously proved using spectral analysis and bifurcation methods.

Boundary-layer problem for the singular Keller-Segel model

Zhi-An Wang The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Jose Carrillo, Jingyu Li, Wen Yang

Abstract: In this talk, we shall discuss the boundary layer problem of the singular Keller-Segel model with physical boundary conditions in any dimensions. First, we obtain the existence and uniqueness of boundary-layer solution to the steady-state problem and identify the boundary-layer profile and thickness near the boundary. Then we find the asymptotic expansion of boundary-layer profile in terms of the radius for the radially symmetric domain, which can assert how the boundary curvature affects the boundary-layer thickness. Finally, we establish the nonlinear stability of the unique boundary-layer steady state solution with exponential convergence rate for the radially symmetric domain.