Local existence and uniqueness of solutions to approximate systems of 1D tumor invasion model

2010 ◽  
Vol 11 (5) ◽  
pp. 3555-3566 ◽  
Author(s):  
Akio Ito ◽  
Maria Gokieli ◽  
Marek Niezgódka ◽  
Zuzanna Szymańska
Keyword(s):  
Existence And Uniqueness ◽  
Tumor Invasion ◽  
Local Existence ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
Invasion Model

LOCAL EXISTENCE AND UNIQUENESS THEORY FOR THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

2012 ◽  
Vol 09 (01) ◽  
pp. 177-193 ◽  
Author(s):  
KEIICHI KATO ◽  
YUUSUKE SUGIYAMA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Local Existence ◽  
Second Sound ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Uniqueness Theory

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.

LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PDE MODEL FOR CRIMINAL BEHAVIOR

2010 ◽  
Vol 20 (supp01) ◽  
pp. 1425-1457 ◽  
Author(s):  
NANCY RODRIGUEZ ◽  
ANDREA BERTOZZI
Keyword(s):  
Existence And Uniqueness ◽  
Criminal Behavior ◽  
Blow Up ◽  
Local Existence ◽  
Coupled System ◽  
Qualitative Behavior ◽  
Residential Burglary ◽  
Uniqueness Of Solutions ◽  
Pde Model ◽  
Local Existence And Uniqueness

The analysis of criminal behavior with mathematical tools is a fairly new idea, but one which can be used to obtain insight on the dynamics of crime. In a recent work,34 Short et al. developed an agent-based stochastic model for the dynamics of residential burglaries. This model produces the right qualitative behavior, that is, the existence of spatio-temporal collections of criminal activities or "hotspots", which have been observed in residential burglary data. In this paper, we prove local existence and uniqueness of solutions to the continuum version of this model, a coupled system of partial differential equations, as well as a continuation argument. Furthermore, we compare this PDE model with a generalized version of the Keller–Segel model for chemotaxis as a first step to understanding possible conditions for global existence versus blow-up of the solutions in finite time.

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