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 < subset of>RN(N=2,3) with zero-flux boundary conditions. Although the L-norm of the extracellular matrix density is easy to be obtained, the re-establishment mechanism still cause essential difficulty due to the deficiency of regularity for . We use some iterative techniques to establish the W1, 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 Equations263 (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 Equations263 (2017) 1269-1292], the global existence and boundedness is established only for large . 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.