Boundedness and large time behavior of solutions of a higher-dimensional haptotactic system modeling oncolytic virotherapy

Abstract

This paper is concerned with the higher-dimensional haptotactic system modeling oncolytic virotherapy, which was initially proposed by Alzahrani–Eftimie–Trucu [Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci. 310 (2019) 76–95] (see also the survey Bellomo–Outada et al. [Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]) to model the process of oncolytic viral therapy. We consider this problem in a bounded domain [Formula: see text] with zero-flux boundary conditions. Although the [Formula: see text]-norm of the extracellular matrix density [Formula: see text] is easily obtainable, the remodeling process still causes difficulty due to the deficiency of regularity for [Formula: see text]. Relying on some [Formula: see text]-estimate techniques, in this paper, under the mild condition on parameters, we finally established the existence of global-in-time classical solution, which is bounded uniformly. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when [Formula: see text], the corresponding solution of the system decays to [Formula: see text] algebraically. To the best of our knowledge, these are the first results on boundedness and asymptotic behavior of the system in three-dimensional space.