Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

Khelghati, Ali; Baghaei, Khadijeh

Comptes Rendus. Mathématique, Tome 353 (2015) no. 10, pp. 913-917.

Nous considérons le système quasi linéaire, parabolique–elliptique, de chimiotaxie

{ut=∇⋅(D(u)∇u−χu∇v)+g(u),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,

avec des conditions au bord homogènes de Neumann, dans un domaine lisse, borné Ω⊂Rn, n≥1. Nous supposons que les fonctions D et

D(s)>0pours≥0,D(s)≥CDsm−1pours>0,

g(0)≥0,g(s)≤a−bsγ,s>0

pour certaines constantes CD>0, m≥1, a≥0, b>0 et γ>2.

Nous démontrons que les solutions classiques du système ci-dessus sont uniformément bornées en temps, sans restriction sur m et b. Ceci étend un résultat récent de Wang et al. (2014) [16], qui borne les solutions pour γ>2 sous la condition b>b⁎, où b⁎=0 si m≥2−2n et b⁎=(2−m)n−2(2−m)nχ si m<2−2n.

We consider the quasilinear parabolic–elliptic chemotaxis system

{ut=∇⋅(D(u)∇u−χu∇v)+g(u),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,

under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn,n≥1. We assume that the functions D and g are smooth and satisfy

D(s)>0fors≥0,D(s)≥CDsm−1fors>0,g(0)≥0,g(s)≤a−bsγ,s>0

with some constants CD>0,m≥1,a≥0,b>0 and γ>2.

We prove that the classical solutions to the above system are uniformly in-time-bounded without any restrictions on m and b. This result extends one of the recent results by Wang et al. (2014) [16], which assert the boundedness of solutions for γ>2 under the condition b>b⁎ with b⁎=0 for m≥2−2n and b⁎=(2−m)n−2(2−m)nχ for m<2−2n.

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