Abstract
AbstractWe consider a degenerate chemotaxis model with two-species and two-stimuli in dimensiond≥ 3 and find two critical curves intersecting at one point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time on and below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.
Highlights
The interaction motion of two cell populations in breast cancer cell invasion models in Rd (d ≥ ) has been described by the following chemotaxis system with two chemicals and nonlinear di usion ut = ∆um− ∇ · (u∇v), x ∈ Rd, t >,−∆v = w, x ∈ Rd, t >, wt = ∆wm − ∇ · (w∇z), x ∈ Rd, t >, (1.1)−∆z = u, x ∈ Rd, t >, u(x, ) = u (x), w(x, ) = w (x), x ∈ Rd, where m, m > are constants
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ and nd two critical curves intersecting at one point which separate the global existence and blow up of weak solutions to the problem
Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in nite time on and below these curves
Summary
The interaction motion of two cell populations in breast cancer cell invasion models in Rd (d ≥ ) has been described by the following chemotaxis system with two chemicals and nonlinear di usion (cf [20, 30])
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