Unlimited growth in logarithmic Keller-Segel systems
A parabolic system describing chemotaxis with logarithmic sensitivity function is considered in balls of \(\mathbb R^n\) \[\varepsilon u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v),\] \[v_t=\Delta v-v+u.\] The singular limit \(\varepsilon\searrow 0\) is studied, and the phenomenon of unboundedness of some radially symmetric solutions is shown for small \(\varepsilon>0\), under the assumption \(\chi>\frac{n}{n-2}\). Thus, the property of unboundedness of \(u\) known for \(\varepsilon=0\) is shared by a parabolically perturbed problem in the same range of the parameter \(\chi\).
scientific article; zbMATH DE number 7450432
scientific article