Boundedness to a logistic chemotaxis system with singular sensitivity

Abstract: In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source:

u_{t} = D e l t a u - c h i a b l a c d o t (f r a c u v a b l a v) + r u - m u u^{k}

,

0 = D e l t a v - v + u

under the non-flux boundary conditions in a smooth bounded convex domain

O m e g a s u b s e t m a t h b b R^{n}

,

c h i, r, m u > 0

,

k > 1

and

n g e 2

. It is shown that the system possesses a globally bounded classical solution if

k > f r a c 3 n - 2 n

, and

r > f r a c c h i^{2} 4

for

0 < c h i l e 2

, or

r > c h i - 1

for

c h i > 2

. In addition, under the same condition for

r, c h i

, the system admits a global generalized solution when

k i n (2 - f r a c 1 n, f r a c 3 n - 2 n]

, moreover this global generalized solution should be globally bounded provided

f r a c r m u

and the initial data

u_{0}

suitably small.

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