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• W2480934485 abstract "In the current paper, we consider the following parabolic-elliptic semilinear Keller-Segel model on $mathbb{R}^{N}$, begin{equation*} begin{cases} u_{t}=nablacdot (nabla u-chi unabla v)+a u -b u^2, quad xinmathbb{R}^N,,, t>0cr 0=(Delta- I)v+ u, quad xinmathbb{R}^N,,, t>0, end{cases} end{equation*} where $ chi >0, age 0, b> 0$ are constant real numbers and $N$ is a positive integer. We first prove the local existence and uniqueness of classical solutions $(u(x,t;u_0),v(x,t;u_0))$ with $u(x,0;u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the constants $a, b, chi$ and the dimension $N$, we prove the global existence and boundedness of classical solution $(u(x,t;u_0),v(x,t;u_0))$ for given initial functions $u_0(x)$. Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants $a, b, chi$ and the dimension $N$, we show that for every strictly positive initial function $u_0(cdot)$, $$lim_{ttoinfty} sup_{xinmathbb{R}^N} big[|u(x,t;u_0)-frac{a}{b}|+|v(x,t;u_0)-frac{a}{b}|big]=0,$$ and that for every nonnegative initial function $u_0(cdot)$ with non-empty and compact support ${rm supp}(u_0)$, there are $0 c_{rm up}^*(u_0).$$" @default.
• W2480934485 created "2016-08-23" @default.
• W2480934485 creator A5007609139 @default.
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• W2480934485 date "2016-08-05" @default.
• W2480934485 modified "2023-09-27" @default.
• W2480934485 title "Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $mathbb{R}^{N}$" @default.
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