We consider a nonlinear parabolic system modelling chemotaxis , in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.
We consider the boundary value problem involving the one dimensional -Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.
We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.
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