On a parabolic–elliptic system with gradient dependent chemotactic coefficient

By Mihaela Negreanu, J. Ignacio Tello. Journal of Differential Equations, 2018

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Abstract We consider a second order PDEs system of Parabolic–Elliptic type with chemotactic terms. The system describes the evolution of a biological species “ u ” moving towards a higher concentration of a chemical stimuli “ v ” in a bounded and open domain of R N . In the system considered, the chemotaxis sensitivity depends on the gradient of v , i.e., the chemotaxis term has the following expression − d i v ( χ u | ∇ v | p − 2 ∇ v ) , where χ is a positive constant and p satisfies p ∈ ( 1 , ∞ ) ,  if  N = 1  and  p ∈ ( 1 , N N − 1 ) ,  if  N ≥ 2 . We obtain uniform bounds in L ∞ ( Ω ) and the existence of global in time solutions. For the one-dimensional case we prove the existence of infinitely many non-constant steady-states for p ∈ ( 1 , 2 ) for any χ positive and a given positive mass.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Oncology

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