Analysis of traveling fronts for chemotaxis model with the nonlinear degenerate viscosity

In this paper, we are interested in chemotaxis model with nonlinear degenerate viscosity under the assumptions of beta = 0 (without the effect of growth rate) and u+ = 0. We need the weighted function defined in Remark 1 to handle the singularity problem. The higher-order terms of this paper are significant due to the nonlinear degenerate viscosity. Therefore, the following higher-order estimate is introduced to handle the energy estimate:Um-2 = (1/U)(2-m) <= Kw(z) <= Cw(z)/U , if 0 < m < 2,Um-2 <= Lu- <= Cu/U , if m >= 2,a where C = max {K, L} = max {a/m-a, (m + a)(m)} for a > 0 and m > a, and w(z) is the weighted function. Then we show that the traveling waves are stable under the appropriate perturbations. The proof is based on a Cole-Hopf transformation and weighted energy estimates.