Uniform boundedness and eventual Hölder continuity to a cancer invasion model with remodeling of ECM and nonlinear diffusion

Abstract This paper is concerned with a class of cancer invasion model with nonlinear diffusion and remodeling of ECM. The primary difficulty that arises here is that due to nonlinear diffusion, the good coupling structure between the diffusion term and haptotactic term is destroyed, rendering the effective methods used in the linear diffusion model no longer applicable. Fortunately, we have found a new effective combination between the diffusion term and the haptotactic term, which allows the diffusion term to dominate the haptotactic term, thus preliminarily improving the regularity of the weak solution. Based on these results, we can prove part of long‐time asymptotic behavior of the solution, thereby finally proving the uniform boundedness of the weak solution. Subsequently, by improving the convergence of cancer cells u from ‐norm to ‐norm, it is also proved that after a long time, the weak solution will eventually be Hölder continuous for some slow diffusion cases.