Global classical solutions to a triply haptotactic cross-diffusion system modeling oncolytic virotherapy

In this paper, we study a triply haptotactic cross-diffusion system, which is proposed to describe the interaction among uninfected cancer cells, infected cancer cells, extracellular matrix (ECM) and oncolytic virus particles in oncolytic virotherapy. Our main result asserts that, in the two-dimensional domain, an associated initial-boundary value problem has a unique classical solution, which exists globally and is uniformly bounded under suitable assumptions on the parameters and initial data of the system. The key feature of this model is the combination of three simultaneous haptotaxis processes and the nonlinear production term for infected cancer cells, where the tendency towards blow-up of solutions is enhanced by the former and suppressed by the latter. Hence, the main idea of the proof is to make full use of the production term to balance the blow-up effect arising from the haptotaxis processes on the basis of Lyapunov-like functionals, general Young’s inequality and Gagliardo–Nirenberg inequality. This paper improves the work begun in [J. Li and Y. Wang, Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differential Equations 270 (2021) 94–113].