Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term
Frederic Heihoff
Abstract
Abstract We consider the system with $$\rho \in {\mathbb {R}}, \mu> 0, \chi > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>∈</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> in a bounded domain $$\Omega \subseteq {\mathbb {R}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> with smooth boundary. While very similar to chemotaxis models from biology, this system is in fact inspired by recent modeling approaches in criminology to analyze the formation of crime hot spots in cities. The key addition here in comparison with similar models is the logistic source term. The central complication this system then presents us with, apart from us allowing for arbitrary $$\chi > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>χ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> , is the nonlinear growth term uv in the second equation as it makes obtaining a priori information for v rather difficult. Fortunately, it is somewhat tempered by its negative counterpart and the logistic source term in the first equation. It is this interplay that still gives us enough access to a priori information to achieve the main result of this paper, namely the construction of certain generalized solutions to ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ). To illustrate how close the interaction of the uv term in the second equation and the $$-\mu u^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> term in the first equation is to granting us classical global solvability, we further give a short argument showing that strengthening the $$-\mu u^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> term to $$-\mu u^{2+\gamma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> with $$\gamma > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>γ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> in the first equation directly leads to global classical solutions.