Green’s function of a chemotaxis model in the half space and its applications

In this paper, we are interested in a degenerate Keller–Segel model in the half space xn>ltxn>lt in high dimensions, with a constant ll. Under the conservative boundary condition, we study the global solvability, regularity and long time behavior of solutions. We first construct Green’s function of the half space problem and study its properties. Then by Green’s function, we give the implicit formula of solutions. We prove that when the initial data is small enough, the half space problem is always globally and classically solvable. We also obtain decay estimates of solutions when the parameter ll have different signs. For l<0l<0, the time decay of solutions is (1+t)−n/2(1+t)−n/2. While for l>0l>0, the time decay is (1+t)−(n−1)/2(1+t)−(n−1)/2, and in this case, we also obtain an exponential decay in the spatial space in the xnxn-direction.