Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source

Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity

“…Remark 1. Recall from [16,17] that the problem (1) with = 1 admits a globally bounded classical solution provided the chemotactic sensitivity coefficient χ suitably small with respect to r > 0, and k ≥ 2 for n = 2 or k > 3(n+2) n+4 for n ≥ 3. Theorem 1.1 says that the system (1) possesses a globally bounded classical solution provided χ + ∈ (0, 1), without the restriction on the dampening exponent k > 1 in the logistic source ru − µu k .…”

“…Remark 2. By [16,17] it was known that the uniformly lower bound estimate for v is the main step to establish the global boundedness of the solution to system (1) with = 1. Differently, without considering this crucial estimate (v may tend to 0), it is proved in Theorem 1.1 that the system admits a globally bounded classical solution via the transformation uv − χ…”

“…If n, k ≥ 2, the global solvability of classical or generalized solutions has been discussed in [5,17]. We refer to [8] on the related parabolic-elliptic case.…”

“…However, this a priori uniform estimate on v will be false for the singular chemotaxis model with logistic-type source ru − µu k , i.e., the singularity in the chemotactic sensitivity function χ v may happen. By means of a weighted integral Ω u −p v −q dx, we can firstly obtain the desired uniformly lower bound estimate for v, and then study the global solvability of solutions to (1) for = 1 [16,17]. In this paper, with a transformation w = uv − χ…”

“…A simple calculation shows that pq2m p−q2 < nq1 n−2q1 . Hence, it is shown from (14) with (9) and (17) that…”