The paper is concerned with some chemotaxis model u /t = D(uln(u/w)) + u ( a- bu ), w / t = f (u , w ).To study the behavior of the solution, some function transformations are intro- duced, and the main tool is sup-sub-solution method. The result shows that, whether the solution blows up in finite time depends on the parameters and the initial data. As the chemical substance w has linear growth, f (u , w) = β u -δ w,where β> 0, δ>0, and a +δ>0, wherein the solution exists globally. However, as w possesses ex- ponential growth, f (u , w) = ( β u - δ) w, wherein both u and w share the same blowup point and time if the solution blows up in finite time.
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