2022/03/17 by Piotr Biler, Biler, Piotr, Alexandre Boritchev +3
Mathematics · Biochemistry, Genetics and Molecular Biology · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Microtubule and mitosis dynamics
paper · pdf · doi:10.48550/arxiv.2203.09130
We study the global existence of the parabolic-parabolic Keller-Segel system\nin Rd , d \≥ 2. We prove that initial data of arbitrary size give rise to\nglobal solutions provided the diffusion parameter \τ is large enough in the\nequation for the chemoattractant. This fact was observed before in the\ntwo-dimensional case by Biler, Guerra & Karch (2015) and Corrias, Escobedo &\nMatos (2014). Our analysis improves earlier results and extends them to any\ndimension d \≥ 3. Our size conditions on the initial data for the global\nexistence of solutions seem to be optimal, up to a logarithmic factor in\n\τ, when \τ>>1: we illustrate this fact by introducing two toy models,\nboth consisting of systems of two parabolic equations, obtained after a slight\nmodification of the nonlinearity of the usual Keller-Segel system. For these\ntoy models, we establish in a companion paper [4] finite time blowup for a\nclass of large solutions.\n