L~? Uniform Boundedness Of Solution To Model With An Attractive-repulsive Potential

In this thesis, we consider the following attractive-repulsive model where n ? 3, the diffusion exponent m > 1 - 2/n, the initial value ?0(x) ? L+1(Rn) ?L?(Rn), ?(x,t) represents the density of bacteria, U(x,t) represents attractive-repulsive potential function. In this thesis, we divide several sections to prove the L? uniform bound of solutions. If attractive potential dominates this model and its singularity is weaker (2 - n ? B < A ? 2), we utilize the Sobolev inequality, the Young inequality and the differential iterative inequality etc. to prove L? uniform bound. If attractive potential dominates this model and its singularity is stronger (-n < B <2 - n = A),we obtain the L? uniform bound under some assumptions on initial data and the diffusion exponent m.If repulsive potential dominates this model (-n ? A<2 - n?B?2), we prove that there is a constant C such that ||?}}L? ? C by utilizing the property of the fractional exponent power Laplace operator Ls = (-?)s, 0 < s < 1 and Stroock-Varopoulos inequality.