Global dynamics of a tumor invasion model with/without logistic source

作者：Chu, Jiawei; Jin, Hai-Yang^*; Xiong, Linjie

来源：Zeitschrift für Angewandte Mathematik und Physik, 2021, 72(5): 181.

DOI：10.1007/s00033-021-01611-w

• Boundedness
• Global dynamics
• Chemotaxis
• Tumor invasion

In this paper, we shall study the following tumor invasion model @@@ {ut = del.(gamma(omega)del u) - chi del.(u del v) + au - bu(2), x is an element of Omega, t > 0, @@@ u(t) = Delta v + wz, x is an element of Omega, t > 0, @@@ omega(t) = -wz, x is an element of Omega, t > 0, @@@ partial derivative u/partial derivative v = partial derivative v/partial derivative v = partial derivative z/partial derivative v = 0, x is an element of partial derivative Omega, t > 0, @@@ (u, v, w, z) (x,0) = (u(0), v(0), w(0), z(0)) (x), x is an element of Omega, @@@ where Omega subset of R-n(n >= 1) is a bounded domain with smooth boundary. The function gamma(w) satisfies the hypothesis: gamma(w) is an element of C-2([0,infinity)) with gamma(w) > 0 for all w = 0 and gamma'(w) <= 0. Based on energy estimates, we first establish the existence of global bounded solution for 1 <= n <= 3 if a = b = 0. If the logistic source is included (i.e., a > 0, b > 0), we prove that the global classical solution exists with uniform-in-time bound for all dimensions (i.e., n = 1), which solves an open problem left in Fujie (Discrete Contin Dyn Syst Ser S 13(2):203-209, 2020). Moreover, we also show that all the global bounded solution will converge to the non-trivial constant steady state exponentially.