Global well-posedness and uniform boundedness of urban crime models: one-dimensional case. (English) Zbl 1440.35329
Summary: In this paper, we study a PDE system which was proposed by M. B. Short et al. [Math. Models Methods Appl. Sci. 18, 1249–1267 (2008; Zbl 1180.35530)] to describe the spatial-temporal dynamics in urban criminal activity. By establishing a user-friendly integral inequality, we prove the global existence, uniqueness and uniform boundedness of the classical solution to this system in one-dimension space. Our result extends the local well-posedness by N. Rodriguez and A. Bertozzi [Math. Models Methods Appl. Sci. 20, 1425–1457 (2010; Zbl 1200.35308)] to global.
MSC:
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 91D10 | Models of societies, social and urban evolution |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B45 | A priori estimates in context of PDEs |
| 35B50 | Maximum principles in context of PDEs |
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