2020/04/21 by Debabrata Karmakar, Karmakar, Debabrata, Gershon Wolansky +1 · 1 citation
Mathematics · Computer Science · #Mathematical Biology Tumor Growth #Topological and Geometric Data Analysis #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2004.10132
We study the global in time existence and long time asymptotics of solutions\nto the parabolic-elliptic Patlak-Keller-Segel system for the multi-species\npopulations in the whole Euclidean space \ℝ2. We prove that at the\nborderline case of critical mass there exists a global it free energy\nsolution subject to initial data with finite entropy and second moment.\nMoreover, we show that as time t approaches to infinity, all the components\nof the solutions concentrate in the form of a Dirac measure at a single point.\nOur approach utilizes the gradient flow structure in Wasserstein space in the\nspirit of De Giorgi's minimizing movement or the JKO-schemes. Due to the\ncritical mass, the minimization problem in JKO-schemes may not admit a solution\nin general. We find a necessary and sufficient criterion for which any\nminimizing sequence remains uniformly bounded in an appropriate topology to\nensure the existence of a minimizer.\n