A NEW RESULT FOR GLOBAL SOLVABILITY OF A TWO SPECIES CANCER INVASION HAPTOTAXIS MODEL WITH TISSUE REMODELING

This paper deals with the two species cancer invasion haptotaxis model c(1t) = Delta c(1) - chi(1)del.(c(1)del v) - mu(EMT) c(1) + mu(1)c(1)(r(1) - c(1) - c(2) - v); c(2t) = Delta c(2) - chi(2)del.(c(2)del v) + mu(EMT) c(1) + mu(2)c(2)(r(2) - c(1) - c(2) - v); tau m(t) = Delta m + c(1) + c(2) - m; v(t) = -mv + eta v(1 - c(1) - c(2) - v), x is an element of Omega, t > 0, in a bounded and smooth domain Omega subset of R-N (N = 2, 3) with no-flux boundary conditions, where chi(1), chi(2), r(1), r(2), eta > 0, mu(1), mu(2) >= 0, tau is an element of {0, 1}, and mu(EMT) := mu(EMT) (c(1), c(2), m, v) = f(v)m with f is an element of C-1 ([0, infinity); [0, infinity)). Compared with a previous work [F. Dai and B. Liu, J. Math. Anal. Appl., 483 (2020), 123583], the novelties here are that the function mu(EMT) (c(1), c(2), m, v) is not bounded, and that the logistic source in cell equations is possibly not taken into account. For any appropriately regular initial data, we prove the global existence and uniqueness of the classical solution to the corresponding two-dimensional system when mu(1) >= 0, mu(2) = 0, and tau is an element of {0, 1}, and obtain the boundedness of this solution under an explicit smallness condition on eta when mu(1) = mu(2) = 0 and tau = 1. Moreover, we establish the global existence, uniqueness, and boundedness of the classical solution to the associated three-dimensional system under some explicit smallness conditions on r(1), r(2), and initial dada when mu(1), mu(1) > 0, and tau is an element of {0, 1}. These results significantly improve and extend the result obtained for previously known ones.