Boundedness in a two-dimensional two-species cancer invasion haptotaxis model without cell proliferation

in a bounded and smooth domain omega subset of R(2 )with homogeneous Neumann conditions, where chi 1, chi(2,) eta > 0, tau is an element of {0, 1}, f(v) is an element of C1 ([0, infinity); [0, infinity)) and f (0) = 0. It is well known that the absence of logistic source aggravates mathematical difficulties, which are overcome by constructing suitable Lyapunov functional. When the remodeling of ECM includes a competition with cancer cells (i.e., alpha(1) = alpha(2) = 1), we prove that the associated initial-boundary value problem of (SIC) with tau = 0 admits a globally bounded classical solution for suitably small eta, which complements the boundedness result on the homogenous Neumann problem of (SIC) with tau = 1 obtained in Dai and Liu (SIAM J Math Anal 54:1-35, 2022). When the competition with cancer cells is taken no account in the re-establishment of ECM (i.e., alpha(1) = alpha(2) = 0), we establish the global boundedness of classical solution to the corresponding initial-boundary value problem of (SIC) with tau is an element of {0,1} for arbitrarily large eta, which is completely new. These results reveal the significant difference on the global boundedness of classical solution for the case whether or not the competition with cancer cells is contained in the remodeling of ECM.