The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1

Abstract

Abstract In bounded n-dimensional domains Ω, the Neumann problem for the parabolic equation $$\begin{array}{} \displaystyle u_t = \nabla \cdot \Big( A(x,t)\cdot\nabla u\Big) + \nabla \cdot \Big(b(x,t)u\Big) - f(x,t,u)+g(x,t) \end{array}$$(*) is considered for sufficiently regular matrix-valued A, vector-valued b and real valued g, and with f representing superlinear absorption in generalizing the prototypical choice given by f(⋅, ⋅, s) = sα with α > 1. Problems of this form arise in a natural manner as sub-problems in several applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto type in mathematical biology, and accordingly a natural space for initial data appears to be L1(Ω). The main objective thus consists in examining how far solutions can be constructed for initial data merely assumed to be integrable, with major challenges potentially resulting from the interplay between nonlinear degradation on the one hand, and the possibly destabilizing drift-type action on the other in such contexts. Especially, the applicability of well-established methods such as techniques relying on entropy-like structures available in some particular cases, for instance, seems quite limited in the present setting, as these typically rely on higher initial regularity properties. The first of the main results shows that in the general framework of (*), nevertheless certain global very weak solutions can be constructed through a limit process involving smooth solutions to approximate variants thereof, provided that the ingredients of the latter satisfy appropriate assumptions with regard to their stabilization behavior. The second and seemingly most substantial part of the paper develops a method by which it can be shown, under suitably stregthened hypotheses on the integrability of b and the degradation parameter α, that the solutions obtained above in fact form genuine weak solutions in a naturally defined sense. This is achieved by properly exploiting a weak integral inequality, as satisfied by the very weak solution at hand, through a testing procedure that appears to be novel and of potentially independent interest. To underline the strength of this approach, both these general results are thereafter applied to two specific cross-diffusion systems. Inter alia, this leads to a statement on global solvability in a logistic Keller-Segel system under the assumption α > $\begin{array}{} \frac{2n+4}{n+4} \end{array}$ on the respective degradation rate which seems substantially milder than any previously found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto system some apparently first results on global solvability for L1 initial data are derived.