Global classical solution and boundedness to a chemotaxis‐haptotaxis model with re‐establishment mechanisms

In this paper, we deal with a chemotaxis‐haptotaxis model with re‐establishment effect. We consider this problem in a bounded domain Ω ⊂ R N ( N = 2 , 3 )with zero‐flux boundary conditions. Although the L ∞ ‐norm of the extracellular matrix density ω is easy to be obtained, the re‐establishment mechanism still cause essential difficulty due to the deficiency of regularity for ω . We use some iterative techniques to establish the W 1 , ∞bound of uPA protease concentration v , and further obtained the L ∞ estimate of the cancer cell density u . Using these a prior estimates, we finally established the existence of global‐in‐time classical solution, which is bounded uniformly. The result of this paper fills the gap of [Pang and Wang, J. Differential Equations  263 (2017) 1269–1292; Tao and Winkler, J. Differential Equations 257 (2014) 784–815] in dimension 2 with q = 1 , in [Tao and Winkler, J. Differential Equations 257 (2014) 784–815], the boundedness of the solution is left open; and in [Pang and Wang, J. Differential Equations  263 (2017) 1269–1292], the global existence and boundedness is established only for large μ . In particular, the global solvability and boundedness of smooth solutions in dimension 3 has never been touched before, this paper is the first attempt to solve this problem.