Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

In this paper, we are interested in a model derived from the 1‐D Keller‐Segel model on the half line x  >  as follows:u t − l u x − u x x = − β ( u v x ) x , x > 0 , t > 0 ,λ v − v x x = u , x > 0 , t > 0 ,l u ( 0 , t ) + u x ( 0 , t ) = v x ( 0 , t ) = 0 , t > 0 ,u ( x , 0 ) = u 0 ( x ) , x > 0 ,where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l  > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l  < 0. Copyright © 2016 John Wiley & Sons, Ltd.