Boundedness and asymptotic behavior of solutions to one-dimensional urban crime system with nonlinear diffusion. (English) Zbl 1521.35045
Summary: This paper deals with a one-dimensional cross-diffusion system
\[
\begin{cases}
u_t = (D(u)u_x)_x - \chi(\frac{u}{v}v_x)_x - uv + B_1(x, t), & x\in\varOmega, t > 0, \\
v_t = v_{xx} - v + uv + B_2(x, t), & x\in\varOmega, t > 0,
\end{cases}
\]
which is proposed by M. B. Short et al. [Math. Models Methods Appl. Sci. 18, 1249–1267 (2008; Zbl 1180.35530)] to describe the dynamics of urban crime. If \(D(u) \geq D_0(u + 1)^{m - 1}\) with \(D_0, m > 0\), it is proved for arbitrary \(\chi > 0\) that the system possesses a globally bounded classical solution provided \(m > \frac{1}{4}\) with some mild assumptions on nonnegative functions \(B_1\), \(B_2\). In addition, if \(B_2 \equiv 0\), the attractiveness value of \(v\) and its derivative \(v_x\) decay to zero in the long time limit.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
Citations:
Zbl 1180.35530References:
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