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

Abstract

In this paper, we deal with a chemotaxis‐haptotaxis model with re‐establishment effect. We consider this problem in a bounded domain Ω⊂double-struckRNfalse(N=2,3false) 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 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 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.

Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence

“…1+vε ≤ c 1 . The last term in (40) converges to zero as ε 0 like in the proof of [83,Theorem 3.2]. To see this, let C 4 (T ) denote the constant C(T ) from Proposition 3.6 and let η > 0 be arbitrary.…”

“…Most chemotaxis-haptotaxis models investigated from the above mentioned viewpoint of mathematical analysis are versions of the Chaplain-Lolas models [6,7]. Of these, the vast majority involves linear diffusion, with constant [3,10,40,41,74,75,88,90,92,93,101,105] or solution-dependent [76,89] taxis sensitivity functions. Cells migrate, however, with finite speed, thus models with nonlinear cell diffusion were considered, whereby the diffusion coefficient may [55,91,107] or may not [36,39,56,57,99,100] infer some (mild) form of degeneracy.…”

“…Results about global boundedness are available under various scenarios for some of the above models [3,10,36,39,40,41,55,56,57,75,76,93,99,100,101,105,107]. Whether or not the above type of models feature remodeling of the haptoattractant seems to essentially influence the analytical handling of the problem, as reviewed in [2]; see also [76] for a result on asymptotics of a chemotaxis-haptotaxis system for tumor angiogenesis.…”

A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion

“…As for the global boundedness of the full parabolic model (2) with η > 0, thanks to L q − L p estimates for the Neumann heat semigroup, the authors of [28] can deal with the chemotaxis-related integral term t 0 Ω e −(p+1)(t−s) u p |∇v| 2 dxds, and thereby derive the global boundedness of the first component of solution (u, v, w) when µ is sufficiently large. It is noticed that very recently, the corresponding results of [28,30] have been improved in [24,42], which can be stated as follows.…”

“…In fact, the considerable difficulty in the context of the rigorous analysis stems from the lack of smoothing action on the spatial regularity of ODE. To the best of our knowledge, the mathematical well-posedness of various models of cancer invasion has been receiving increased interest in the literature [1,[22][23][24][25][26][27][28][29][30][31][32][33]. Without the pretension of exhaustiveness, this paper provides a short review of the global bounded results on some cancer invasion models and sketches necessary proofs thereof.…”