Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy

Abstract

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system \[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \] with positive parameters $D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$ , $\alpha _u,\alpha _w,\mu _u,\beta$ . When posed under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{2}$ , and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020, J. Differ. Equ. 268, 4973–4997). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate $\beta <1$ , the global classical solution $(u,v,w,z)$ is uniformly bounded and exponentially stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $(L^{\infty }(\Omega ))^{4}$ as $t\rightarrow \infty$ .

Global existence and asymptotic behavior of a triply haptotactic cross-diffusion model for oncolytic virotherapy

“…And (1.4) is the result of ξ z ∇ • (z∇v). Compared with the results in [17], the existence of ξ z ∇ • (z∇) causes lots of difficulties in the proof of global existence and the large time behavior. Therefore, we need (1.4).…”

“…The following lemma shows the basic estimates of (u, v, w, z). We refer readers to Lemma 3.1 in [12] and Lemma 2.2 in [17] for more details.…”

“…We first need to obtain a pointwise lower bound for a. Different from the model in [17], our system has terms µ v v(1 − v) and ξ z ∇(z∇v), which bring difficulties in the proof. So we need some restrictions to cover the problems.…”

“…When ξ w ̸ = 0 and ξ z = 0, Tao and Winkler obtained the global classical solutions to (1.1) by constructing a quasi-Lyapunov functional structure in one-or two-dimensional settings in [12]. Based on this work, Wang and Xu ( [17]) further proved the asymptotic behavior of the solutions with ρ u = ρ w = ρ z , µ v = 0, δ w = 1 and 1 > β > 0. And when ξ z ̸ = 0 too, Tao obtained the global existence of the solution to (1.1) in one-dimensional space ( [8]).…”

Global weak solutions to a doubly haptotactic cross-diffusion system with $ p $-Laplacian diffusion

“…In fact, after the publication of [1], more and more scholars began to study the derivative problem of (1.2). For example, Wang and Xu [24,25] studied the existence of a global classical solution for (1.2) in a smoothly bounded domain Ω ⊆ R n (n = 2, 3) under appropriate initial and parameter conditions. Besides, they proved that the solution of the problem converges to the constant equilibrium (1, 0, 0, 0) in the topology (L ∞ (Ω)) 4 in a large time limit.…”