Persistence phenomena of classical solutions to a fractional Keller-Segel model with time-space dependent logistic source

In this paper, we consider the following fractional parabolic-elliptic Keller-Segel system with time-space dependent logistic source on the whole space { u(t) + (-Delta)(s)u+chi del . (u del v) = u(a(t, x) - b(t,x)u), t > 0, x is an element of R-N, 0 = Delta v - lambda v + mu u, t > 0, x is an element of R-N, where s is an element of (0,1), chi, lambda and mu are positive constants and the functions a(t,x) and b(t,x) are positive and bounded. When the order of fractional diffusion s is an element of (1/2,1) and the functions a(t,x) and b(t,x) satisfy some conditions, we first prove the local existence and uniqueness of nonnegative classical solutions (u(t,x; t(0), u(0)), v(t,x; t(0), u(0))) with u(t(0),x; t(0), u(0)) = u(0) for every t(0) is an element of R and every nonnegative bounded and uniformly continuous initial function uo. Next, under some parameter conditions, we prove the global existence and boundedness of classical solutions (u(t,x; t(0), u(0)), v(t,x; t(0), u(0))) for given initial function u(0). Finally, we obtain the pointwise persistence phenomena of solutions; that is, for any strictly positive initial function u(0), there are 0 < m(u(0)) <= M(u(0)) < infinity such that m(u(0)) <= u(t,x; t(0), u(0)) <= M(u(0)), for all t >= 0, x is an element of R-N, and obtain the uniform persistence phenomena of solutions; that is, there are 0 < m < M < infinity such that for any strictly positive initial function u(0), there exists T(u(0)) > 0 such that m <= u(t, x; t(0), u(0)) <= M, for all t >= T(u(0)), x is an element of R-N, In summary, these results are extension to the case of fractional parabolic-elliptic Keller-Segel system.

Automated summary

In this paper, we study the fractional parabolic-elliptic Keller-Segel system with a time-space dependent logistic source. We prove the local existence and uniqueness of nonnegative classical solutions, as well as the global existence and boundedness of classical solutions under certain parameter conditions. We also find the pointwise persistence phenomena and uniform persistence phenomena of the solutions.