Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction

The parabolic problemJt u(t) = delta(u phi(v)),v(t) = delta v -uv,is considered in smoothly bounded subdomains of Rn with arbitrary n > 1. Under the assumptions that phi E C0([0, oo)) n C3((0, oo)) is positive on (0, oo) and satisfieslim inf xi? 0 phi(xi)/xi alpha > 0 and lim sup xi? 0 {xi beta phi'(xi) | } < oo with some alpha > 0 and beta > 0, for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices alpha = 0 and alpha E (0, 2), or were restricted to low-dimensional settings by requiring that n < 2.Mathematics Subject Classification. 35K65 (primary), 35K51, 35Q92, 92C17, 35K57 (secondary).

Automated summary

This study considers a parabolic problem in smoothly bounded subdomains of Rn with arbitrary n > 1. Under certain assumptions, it is shown that the associated initial-boundary value problem has a global solution in a generalized sense. This extends previous solution theories for this type of problem.