Boundedness and weak stabilization in a degenerate chemotaxis model arising from tumor invasion

This paper is concerned with the degenerate system ⎧ ⎨⎪⎪⎪ ⎪⎪⎪⎩ ut = backward difference & BULL; (f (u, w) backward difference u - g(u) backward difference v), x & ISIN; Q, t > 0, vt = Av+ wz, x & ISIN; Q, t > 0, wt= -wz, x & ISIN; Q, t > 0, zt = Az - z + u, x & ISIN; Q, t > 0 in a bounded domain Q & SUB; RN (N & GE; 2) under the no-flux boundary condition for u and the homogeneous Neumann boundary condition for v, z with non-negative initial data u0, v0, w0, z0. Here, the diffusivity f and the sensitivity g are assumed to fulfill f (u, w) & GE; um-1 (m > 1), 0 & LE; g(u) & LE; u & alpha; (& alpha; & ISIN; R). It is shown that if & alpha; + 1 < m + 4 N (N & GE; 2) or & alpha; + 1 = m + 4N (N & GE; 3) with small mass of u0, then the system possesses a global bounded weak solution which converges to the constant equilibrium in the weak* topology in L & INFIN;(Q) as t & RARR; & INFIN;. & COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the problem of global bounded weak solutions for a degenerate system in a bounded domain, which converges to a constant equilibrium in the weak* topology in L∞(Q) as t→∞ under certain conditions.