Partial Differential Equations for Social and Biological Events
【 日時 】
10月7日 10:00-11:40, 13:10-16:50 （講演者６名）
10月8日 10:00-11:40, 13:10-14:50 （講演者4名）
【 講演者 】
Inkyung Ahn (Korea Univ.), 石渡通徳（阪大), Philippe Laurençot (Toulouse), Frédéric Marbach (Renne), 太田家健佑（阪大), Michel Pierre (Renne), Clair Poignard（Inria Bordeaux), Baudouin Denis de Senneville(Inria Bordeaux), 鈴木貴（阪大), Manwai Yuen (EUHK)
【 プログラム 】
≪ October 7, 2018 ≫
1000-1050 Clair Poignard (Inria Bordeaux) “The potential impact of mathematics in clinical oncology: the example of electroporation ablation”
1050-1140 Baudouin Denis de Senneville (Inria Bordeaux)“On-line guidance of non-invasive therapies in mobile organs”
1310-1400 Inkyung Ahn (Korea Univ.)“ Non-uniform dispersal on population models under free boundary in a spatially heterogeneous environment”
1400-1450 Frédéric Marbach (Renne)“Obstructions to controllability for PDEs”
1510-1600 Michel Pierre (Renne)“Old and new results on global existence in reaction-diﬀusion systems”
1600-1650 Takashi Suzuki (Osaka) “Classical solutions to reaction diffusion systems”
≪ October 8, 2018 ≫
1000-1050 Manwai Yuen (EU Hong Kong)“ Blowup for the Compressible Euler Equations in R^N”
1050-1140 Philippe Laurençot (Toulouse) “ Global bounded and unbounded solutions to a chemotaxis system with indirect signal production”
1310-1400 Kensuke Ohtake (Osaka) “ A system of nonlinear integral differential equations in economic geography”
1400-1450 Michinori Ishiwata (Osaka)“ On the soliton decomposition associated with the energy critical heat equation”
【 Presentation Information 】
Non-uniform dispersal on population models under free boundary in a spatially heterogeneous environment
In many cases, the movement of species within a region depends on the availability of food and other resources necessary for its survival. Starvation driven diﬀusion (SDD) is a dispersal strategy that increases the motility of biological organisms in unfavorable environments i.e., a species moves more frequently in search of food if resources are insuﬃcient (Cho and Kim, 2013). In this study, the proposed model represents the dispersion of an invasive species undergoing SDD, where the free boundary represents the expanding front. We observe that the spreading-vanishing dichotomy, which holds in the linear dispersal model (Zhou and Xiao, 2013), also holds in the model undergoing SDD. We also provide the estimates for the spreading speed of the free boundary during the spreading process. Finally, our results are compared with the results of the linear dispersal model to investigate the advantages of this strategic dispersal with respect to survival in new environments.
On the soliton decomposition associated with the energy critical heat equation
In this talk, we are concerned with the asymptotic behavior of a semilinear parabolic equation with critical Sobolev exponent. We give a soliton decomposition type result for time-global solution and discuss the asymptotic behavior of time-global solution. We also give the asymptotic behavior of finite-time blow-up solutions with bounded energy.
Global bounded and unbounded solutions to a chemotaxis system with indirect signal production
Qualitative properties of a chemotaxis model describing the space and time evolution of the densities of two species and the concentration of a chemoattractant are studied. In contrast to the classical Keller-Segel chemotaxis system which involves only one species producing its own chemoattractant, the species which is influenced by the chemoattractant in the model under study is related to another species producing the chemoattractant. As already observed by Tao & Winkler (2017) in a particular case and for radially symmetric solutions, this process has far reaching consequences and shifts finite time blowup to infinite time blowup. The approach in Tao & Winkler
(2017) relies on the reduction of the system to a single equation by exploiting both the structure of the equation and the radial symmetry of the solutions, this transformation allowing one to use comparison arguments. We here construct a Liapunov functional and exploit its properties to show the existence of global bounded and unbounded solutions. This construction does not require radial symmetry and extends to other models as well.
Obstructions to controllability for PDEs
Controllability is the question of whether one can act on the state of a system by means of a time-dependent input. For example, for a social or biological system whose evolution is modeled by a PDE, one might wish to drive an initially perturbed state back to an equilibrium.
We will present obstructions to controllability for some very simple non-linear diffusive models, which generalize obstructions which can be encountered on ODE models.
A system of nonlinear integral differential equations in economic geography
In this talk, we consider a mathematical model which describes geographical population movement driven by economic incentive. The model was introduced by Krugman et al. in new economic geography, which explains geographical phenomena such as urbanization by economic theory.
We begin with explaining economic meanings of the model. Next, mathematical formulation, global existence and uniqueness of solutions, and some analytical results for asymptotic behavior of the solution are presented. Numerical computation is also carried out to explore details about the asymptotic behavior of the solution. This talk is based on collaborative research with Professor Emeritus Atsushi Yagi of Osaka University.
Old and new results on global existence in reaction-diﬀusion systems Michel PIERRE, Ecole Normale Sup´erieure de Rennes (ENS Rennes) and Institut de Recherche Math´ematique de Rennes (IRMAR), France
We will give a survey on global existence of solutions to reactiondiﬀusion systems where two main properties hold which often occur in applications, particularly in biochemistry, namely : positivity of solutions is preserved and the total mass of components is controlled for all time. Old and recent results will be described together with open problems.
The potential impact of mathematics in clinical oncology: the example of electroporation ablation
Electroporation-based therapies (EPT) consist in applying high voltage short pulses to cells (typically several hundred volts per centimeter during about one hundred microseconds) in order to create defects in the plasma membrane. They provide interesting alternatives to standard ablative techniques, in particular for deep seated tumors (located near vital organs or important vessels). In this talk we present the rationale of electroporation and its modeling at different scales. We will also show that combining well suited clinical workflow with mathematical models can help physicians.
Baudouin Denis de Senneville
On-line guidance of non-invasive therapies in mobile organs
Non-invasive interventional procedures show a high potential in oncology as an alternative to classical surgery. Their objective is to precisely control on-line an energy deposition within a pathological area in order to achieve an effective treatment, with a reduced duration and an increased level of safety for the patient. These new types of non-invasive interventional procedures are very interesting for the treatment of vital organs (such as the kidney, liver and pancreas). However, the treatment of those organs has so far been hampered by the complications arising from their physiological motion. As a consequence, real-time organ motion estimation is rapidly gaining importance for the on-line guidance of such interventional procedures. Modern Magnetic Resonance Imaging (MRI), Cone beam computed tomography (CBCT) or Echography methods now allow a fast acquisition of images with an excellent tissue contrast and high spatial resolution, which opens great perspectives to estimate complex organ deformations. This talk address mathematical issues designed to estimate organ deformations with short latency during the therapy, using real-time image registration techniques applied to anatomical images acquired on-the-fly.
Classical solutions to reaction diffusion systems
We study global-in-time existence of the classical solution to the reaction diffusion system with mass dissipation, where some results on the entropy dissipation system are not available. A small assumption, however, assures it beyond the critical growth of the nonlinearity. Among them is the Lotka-Volterra system in three space dimension.
Blowup for the Compressible Euler Equations in R^N
The compressible Euler equations are fundamental models in the fluid dynamics. In this talk, we present rotational and self-similar solutions for the compressible Euler equations in R^N using the separation method and the Cartesian matrix method for the free boundary problems. Based on the analytical solutions, some blowup phenomena and global existences of the responding solutions can be easily determined. After that, we discuss the new blowup phenomena with the functional energy methods for the solutions of the Euler equations in R^N for the initial value problems.
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