Global solutions to a haptotaxis system with a potentially degenerate diffusion tensor in two and three dimensions

Abstract

Abstract We consider the potentially degenerate haptotaxis system u t = ∇ ⋅ ( D ∇ u + u ∇ ⋅ D ) − χ ∇ ⋅ ( u D ∇ w ) + μ u ( 1 − u r − 1 ) , w t = − u w in a smooth bounded domain Ω ⊆ R n , n ∈ { 2 , 3 } , with a no-flux boundary condition, positive initial data u 0, w 0 and parameters χ > 0, µ > 0, r ⩾ 2 and D : Ω ‾ → R n × n , D positive semidefinite on Ω ‾ . Our main result regarding the above system is the construction of weak solutions under fairly mild assumptions on D as well as the initial data, encompassing scenarios of degenerate diffusion in the first equation. As a step in this construction as well as a result of potential independent interest, we further construct classical solutions for the same system under a global positivity assumption for D , which ensures the full regularising influence of its associated diffusion operator. In both constructions, we naturally rely on the regularising properties of a sufficiently strong logistic source term in the first equation.