December 2, :0 WSPC/INSTRUCTION FILE LocExtDraftFinal LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PDE MODEL FOR CRIMINAL BEHAVIOR

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1 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PDE MODEL FOR CRIMINAL BEHAVIOR NANCY RODRIGUEZ Department of Mathematics, UCLA, Los Angeles, CA 94, USA nrodriguez@mathuclaedu ANDREA BERTOZZI Department of Mathematics, UCLA, Los Angeles, CA 94, USA bertozzi@mathuclaedu Received (Day Month Year Revised (Day Month Year Communicated by (xxxxxxxxxx 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 33 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 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 vs blow-up of the solutions in finite time Keywords: Crime Models; local existence; reaction diffusion equation; chemotaxis AMS Subject Classification: E46, 53C35, 57S Introduction The study of crime hardly needs motivation, it is a phenomena that affects all individuals The city of Los Angeles, nicknamed the Gang Capital of the Nation, is of particular interest Violent and non-violent crimes from burglaries to drive-byshootings have affected the citizens of this city since the beginning of the th century One of the most frequently occurring crimes is residential burglaries, a crime which will affect most people at some point The observation that residential burglaries are not spatially homogeneously distributed and that certain neighborhoods have more propensity to crime than others led Short et al to study the dynamics of

2 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Nancy Rodriguez, Andrea Bertozzi residential burglary hotspots 33 A hotspot is a spatio-temporal aggregation of criminal occurrences and the understanding how they evolve can be extremely useful For example, it can help the police force mobilize their resources optimally This would ideally lead to the reduction or even eliminations of these crime hotspots A theoretical understanding of dynamics of hotspots would help predict how these hotpots will change and thus aid law enforcement agencies fight crime Short et al modeled the dynamics of hotspots using an agent-based statistical model based on the broken window sociological effect 37 The idea of the broken window effect is that crime in an area leads to more crime It has been observed in the residential burglary data that houses which are burglarized have an increased probability of being burglarized again for some period of time after the initial burglary This increased probability of burglary also affects neighboring houses and is referred to as the repeat near-repeat effect, 4, 5, 6 The model is based on the assumption that criminal agents are walking randomly on a two-dimensional lattice and committing burglaries when encountering an opportunity Furthermore, there is an attractiveness value assigned to every house, which refers to how easily the house can be burgled without negative consequences for the criminal agent The criminal agents, in addition to walking randomly, have a biased movement toward areas of high attractiveness values and move with a speed inversely proportional to the value in their current position Let A(x, t and ρ(x, t be the attractiveness value and the criminal density at position x and time t respectively, then the continuum limit of the agent-based model gives the following PDE model: A t = η A A + Aρ + Ao, ρ = ρ [ρ χ(a] + B Aρ; t (b (a where χ(a = log(a A formal derivation of this model can be found in 33 From ( we observe that criminal agents are being created at a constant rate B and are removed from the model when a burglary is committed In essence, the number of burglaries being committed at time t and location x is given by the A(x, tρ(x, t Furthermore, the attractiveness value increases with each burglary As we will discuss later the system ( can be seen as a nonlinear version of the Keller-Segel model for chemotaxis with growth and decay The Keller-Segel model is a reactiondiffusion system that models the movement of some mobile species which is being influenced by an external chemo-attractant 4, 8,,, 36, 34 In the Keller-Segel model literature the function χ(a is referred to as the sensitivity function Various forms of the sensitivity function have been analyzed including log(a and A 9,3 For these cases global existence has been proved in one-dimension 7,3 Furthermore, in two-dimension global existence has been proved for small enough initial mass of the cell density 3,5 It is important to note that these models do not include growth or decay Although the logarithmic sensitivity function has been analyzed

3 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 3 most of the research done on the Keller-Segel model has been for χ(a = A Recall that the model ( is the continuum limit of an discrete agent-based model In the discrete model the probability of an agent moving from node s to node n is given by the ratio of the attractiveness value at node n over the sum of attractiveness values of the neighboring nodes of node s This gives the logarithmic sensitivity function we see in ( Therefore, it makes sense for us to analyze the more complicated sensitivity function In fact, we will see later that the term /A in the advective term helps prevent blow-up From the numerical analysis performed in 33 this model seems to have appropriate qualitative properties, ie existence of hotspots However, to show that this model is truly robust the unique existence of a solution, which does not blow up in finite time, is essential The main result of this paper is the local existence of a solution in addition to a continuation argument, which gives a necessary and sufficient conditions for global existence Assuming that the criminals entering the city and the criminals leaving are approximately the same we consider no-flux boundary condition in a bounded domain Ω R : A ν ( ρ Ω = and ν A A Ω = ; ( where ν is the outer normal vector The initial conditions are given by: A(, x = A (x, ρ(, x = ρ (x (3 Outline In Section we state notation and existing theory to be used for the proof of the main result Simultaneously, we give an outline of the proof In Section 3 we prove the existence and uniqueness of a family of solutions to a regularized version of the original model Following, in Section 4 we look at a-priori higher-order energy estimates that enable us to pass to the limit and prove the final result Section 5 is devoted to proving a continuation argument In Section 6 we look at a generalized version of the Keller-Segel model for chemotaxis and compare it with the PDE model for residential burglaries In this section we also prove a blow-up argument for the modified residential burglary model We conclude this paper in Section 7 with a final discussion Notation and Proof Outline of Main Result We begin this section by establishing the notation that will be used throughout the paper The proof of the main result follows the techniques used in 3 for the Navier- Stokes Equation in 3-D (see also 35 Taylor for symmetric hyperbolic systems In the Keller-Segel literature there are two principal methods used to prove global existence of solutions to various versions of the model The first one involves finding L estimates for the advection term The second method involves finding a Lyapunov function Both of these methods use fixed point theory to obtain local

4 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 4 Nancy Rodriguez, Andrea Bertozzi solutions Since we do not know of the existence of a Lyapunov function for ( our method is more closely related to the first method mentioned We use an abstract version of Picard s Theorem for ODEs to obtain a local solution to ( We will see that global existence will end up being dependent on some L estimates Notation We have a initial-boundary value problem with no-flux boundary conditions For simplicity assume that our domain is a square This problem can be mapped into the periodic problem with symmetry on a domain four times the size of the original domain This is true provided A o and the initial data satisfy reflection symmetry, in which case the model preserves symmetry Hence, from now on we work with periodic boundary conditions and Ω = T unless otherwise specified It is useful to define the following notation: vdx = vdx, Ω v = v dx Furthermore, for a multi-index α = (α, α,, α N, α i Z + {}, we define the H m (Ω-norm as follows: v m = D α v α m Finally, we define the spaces with their corresponding norms to be used: Ω For X a Banach Space with norm X, C ([, T ] ; X is the space of continuous functions mapping [, T ] into X This space has the following norm: v C([,T ];X := sup v X t T L (, T ; X is the space of functions such that v(t X for ae t (, T has finite norm: v L(,T ;X := ess sup t (,T v(t X L (, T ; X is the space of functions such that v(t X for ae t (, T with finite norm: v L (,T ;X := ( T v(t X dt Definition The space C weak ([, T ]; H s (Ω denotes continuity on the interval [, T ] with values in the weak topology of H s In other words, for any fixed Φ H s,

5 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 5 (Φ, u(t s is a continuous scalar function on [, T ] The inner-product of H s is given by: (u, v s = D α u D α vdx ( α s The Hilbert Spaces we will be working on for most of the time is: V m = {(u, v H m (Ω H m (Ω} ( Since we are working extensively with different bounds and the constants are not always important, we introduce the notation A B to mean that there exists a positive constant c such that A cb This notation will be used when the constants are irrelevant and become tedious Main Result and Outline of its Proof Our main contribution is to prove local existence and uniqueness of solutions to the system ( More precisely, we prove the following theorem Theorem (Local Existence of Solutions to the PDE Residential Burglaries Model Given initial conditions (A (x, ρ (x V m for m > 3 such that A (x > A o there exists a positive time, T >, such that A, ρ C([, T ]; C (Ω C ([, T ]; C(Ω form a unique solution to ( on the time interval [, T ] We first modify the system ( by regularizing it, for the purpose of bounding differential operators in Sobolev Spaces This is useful because finding a family of solutions to the regularized system is straightforward Given v L p (T for p we define the mollification of v by J ɛ v(x = k Z ˆv(ke ɛ k +πik x, (3 where ˆv(k = Ω v(xe πik x dx The mollified function, J ɛ v ɛ, has many useful properties, some of which are summarized in the following lemma A proof can be found in 7 We note that this is analogous to mollification by convolution with smooth functions in R The interested reader is referred to 5 Lemma (Properties of Mollifiers Let J ɛ be a mollifier defined in (3 Then J ɛ v C and has the following properties: ( v C (Ω J ɛ v v uniformly and J ɛ v v ( Mollifiers commute with distribution derivatives, D α J ɛ v = J ɛ D α v α m, v H m

6 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 6 Nancy Rodriguez, Andrea Bertozzi (3 u, v L (Ω, (J ɛ uvdx = (J ɛ vudx Ω (4 v H s (Ω, J ɛ v converges to v in H s and the rate of convergence in the H s norm is linear in ɛ: Ω lim J ɛv v s =, ɛ J ɛ v v s Cɛ v s (5 v H m (Ω, γ, k Z +, and ɛ : J ɛ v m+γ c mγ ɛ γ v m, J ɛ D k v c k ɛ v N/+γ k k Once the original system has been regularized it is easy to show that the assumptions of the Picard Theorem on a Banach Space are satisfied by the regularized model for any fixed ɛ > We now state this theorem along with a natural continuation theorem, since autonomous ODEs on a Banach Space have a natural continuation theorem A proof of the following two theorems can be found in 8 Theorem (Picard Theorem on a Banach Space Let O B be an open subset of a Banach Space B, and let F : O B be a mapping satisfying: ( F(x maps O to B ( F is locally Lipschitz continuous ie for any x O there exists L > and an open neighborhood U x O of x such that for all x, ˆx U x we have F (x F (ˆx B L x ˆx B Then for any x o O, there exist a time T such that the ODE dx dt = F (x, has a unique local solution x C (( T, T ; O x = x( O Theorem 3 (Continuation on a Banach Space Let O B be an open subset of a Banach Space B, and let F : O B be a locally Lipschitz-continuous map Then the unique solution X C ([, T ; O to the autonomous ODE dx dt = F (x, x = x( O, either exists globally in time, or T < and X(t leaves the open set O as t T We will see that the above theorem can be applied provided an appropriate functional framework is chosen We use some calculus inequalities in the Sobolev Spaces to show that this theorem can be used to obtain a family of solutions which depend

7 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 7 on the regularizing parameter ɛ Refer to 3 for a proof of the following lemma in the case when Ω = R N The proof for the case when Ω is the torus follows exactly Lemma (Calculus Inequalities in the Sobolev Spaces ( m Z +, there exists c such that for all u, v L (Ω H m (Ω: α m uv m c { u D m v + D m u v }, D α (uv ud α v c { u D m v + D m u v } ( s > N, Hs (Ω is a Banach algebra That is, there exists c > such that for all u, v H s (Ω: uv s c u s v s The next step is to pass to the limit as ɛ Energy estimates, which are independent of the regularizing parameter, are essential for this purpose 3 Local Existence and Uniqueness of Solution to a Regularized Version of the Crime Model We consider the following regularization of (: A ɛ = ηjɛ A ɛ A ɛ + ρ ɛ A ɛ + A o, t (3a ρ ɛ t = J ɛ(j ɛ ρ ɛ J ɛ [ ( ρɛ A ɛ J ɛ A ɛ ] ρ ɛ A ɛ + B (3b This choice of regularization will become clear when we perform the energy estimate calculations The goal of this section is to prove the local existence and uniqueness of solutions to the system (3 for fixed ɛ Consider the function space for the solution to (3 to be the Banach Space V, m = in (, with norm (A, ρ V := A + ρ Theorem 3 (Local Existence of solutions to the Regularized Residential Burglary Model For any ɛ > and initial conditions (A (x, ρ (x V such that A (x > A o there exists a solution, (A ɛ, ρ ɛ C ([, T ɛ ; V, for some T ɛ >, to the regularized system (3 Furthermore, the following energy estimate is satisfied, d dt (Aɛ, ρ ɛ V c 3 (A ɛ, ρ ɛ 3 V + c (A ɛ, ρ ɛ V + c (A ɛ, ρ ɛ V ; (3 where c, c, and c 3 are constants that depend only on A o, ɛ and η Proof Define the map F ɛ = [F ɛ, F ɛ ] : O V X To use Theorem we need a suitable set O such that F ɛ maps O to V, (ie X = V Defined the function

8 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 8 Nancy Rodriguez, Andrea Bertozzi by: F ɛ (A ɛ, ρ ɛ = ηj ɛ A ɛ A ɛ + ρ ɛ A ɛ + A o, F ɛ (A ɛ, ρ ɛ = J ɛ ρ ɛ J ɛ [ ( ρɛ A ɛ J ɛ A ɛ ] ρ ɛ A ɛ + B Hence, if v ɛ = (A ɛ, ρ ɛ V, the original model reduces to an ODE in V dv ɛ dt = F ɛ (v, v ɛ ( = (A (x, ρ (x (33a (33b (34a (34b With this framework we can prove that the conditions of Theorem are satisfied Let v ɛ i = (Aɛ i, ρɛ i V (i =,, we drop ɛ for notational convenience By definition of the V -norm and F we have: F (v F (v V = F (v F (v + F (v F (v After substituting (33 above and using (5 of Lemma and ( of Lemma we obtain a suitable bound for F Initially we have: F (v F (v η J ɛ (A A + A A + ρ A A ρ The last term in the above inequality will appear repeatedly and can be bounded using ( of Lemma by: ρ A A ρ ρ A A + A ρ ρ (35 Using (35 we easily obtain the final estimate for F : F (v F (v ( η ɛ + + ρ A A + A ρ ρ (36 For F we only state the final bound, refer to Appendix A for more detailed computations If we define the open set { } O = (u, v V : u < K, u < L, v < L, we obtain similar estimates for F In particular, if v, v O then where, F (v F (v C A A + C ρ ρ ; (37 C = K ɛ 3 ( ρ + K A ρ + K A ρ + K A ρ + K ɛ 3 A A ρ + K ɛ ρ + ρ, C = ɛ + A + C ɛ 3 A A ( + K A + K A

9 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 9 The important thing to note is that C and C depend only on A i, ρ i, ɛ, and K for i =, Combining (36 and (37 gives: F (v F (v V C(η, L, L, K, ɛ A A +C(L, L, K, ɛ ρ ρ (38 Setting A = and ρ = we see that F does map O to V Furthermore, F : O V is locally Lipschitz therefore the conditions of Theorem are satisfied for fixed ɛ Consequently, we obtain a family of unique local solutions to (3, {(A ɛ, ρ ɛ } ɛ>, such that (A ɛ, ρ ɛ C ([, T ɛ ; V O A careful look at the computations performed (see A enables us to see that the constants in the above inequality are at most cubic in (A, ρ V Once again, setting A = and ρ = in (38 from (34 we obtain the desired inequality (3 Note that the constants c, c and c 3 depend solely on C, ɛ, and η We by taking K = A we obtain the o dependence on A o 4 Local Existence and Uniqueness of Solution to Original Residential Burglary Model In the previous section we successfully showed the unique existence of a solution to (3 on [, T ɛ for fixed ɛ The next step is to show that a subsequence of these solutions converge to a solution of the original system ( To do this we need estimates that are independent of ɛ The following section is devoted for this purpose 4 Energy Estimates From Theorem (3 we see that the time interval on which the solutions to (3 exist depend on ɛ To be able to pass to the limit it is essential that we find a uniform time interval of existence To obtain such an interval we look at higherorder energy estimates which are essential to show that the solution to (3 is in C ([, T ; V m We will see that provided m is chosen large enough then we obtain that the solution is classical For simplicity from now on we denote C = A o Proposition 4 (Higher-Order Energy Estimates Let (A ɛ, ρ ɛ be a solution to the regularized system (3 with initial conditions (A ɛ (, ρ ɛ ( V m, where V m is defined by ( for m 3, such that A (x > A o If M is chosen large enough then Em(t ɛ = M Aɛ m + ρɛ m satisfies the following differential inequalities: For m = 3: d dt Eɛ 3(t C (M, C (E ɛ 3 + C(A o, B, M For m > 3: d dt Eɛ m(t C (M, C, A, ρ, ρ ɛ, A ɛ E ɛ m(t + C(A o, B, M

10 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Nancy Rodriguez, Andrea Bertozzi The proof of this proposition requires a sequence of lemmas For these lemmas we let A ɛ and ρ ɛ be as in Proposition 4 Lemma 4 If M is an arbitrary constant then the following holds: M d dt Aɛ m Mη J ɛ A ɛ m + M Ao + M ( Aɛ + ρ ɛ + A A ɛ m + M ( A ɛ + A ρ ɛ m (4 Proof Following standard procedure we first look at the time evolution equation of A m We drop ɛ for notational simplicity Recalling the multi-index notation from Section and using the chain rule we obtain: d dt A m = (D α A(D α A t dx α m For fixed α substitute in (3a and obtain: (D α A(D α A t dx = (D α AD α (ηjɛ A A + Aρ + A o dx = η J ɛ D α A Dα A + (D α A(D α A o dx + (D α A(D α (Aρdx Note that the third term of the last equality will only contribute when α = For now consider the case α The Cauchy-Schwarz inequality gives: (D α A(D α A t dx η J ɛ D α A Dα A + Dα A D α (Aρ (4 To simplify the computations we first look at the following claim The derivation can be found in Appendix A and uses part ( of Lemma Claim : D α u D α (uv ( u + u + v u m + ( u + u v m α m Adding (4 over α m: M d dt A m Mη J ɛ A m M A m + M Ao A +M D α A D α (Aρ α m Applying Cauchy-Schwarz Inequality to M A o A and Claim to the summation term gives the final result Since the computations for ρ are more complicated we first look at the advection term

11 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior Lemma 4 For I α = { ( } D α (J ɛ ρ ɛ D α ρ ɛ A J ɛ A ɛ dx the following estimate ɛ holds for any < δ < : I α δ J ɛ ρ ɛ m + (C C J ɛ A ɛ m δ + δ (C A ɛ ρ ɛ m α m ( + m C ρ ɛ δ + C ρ ɛ A ɛ + ρ ɛ C k C k+ A ɛ k+ A ɛ The proof can be found in the Appendix A We note that the power in the energy inequality for the case when m = 3 in Proposition 4 is comes from that fact that we are taking multiple derivatives of /A Lemma 43 d dt ρɛ m ( δ J ɛ ρ ɛ m + B + ρɛ + β A ɛ m + β ρ ɛ m (43 + (C C J ɛ A ɛ m δ where, β = ρ + ρ + C δ β = ρ ɛ + A ɛ + ρ + δ C A k= ( ρ ɛ + C ρ ɛ A ɛ + ρ ɛ m, k= C kc k+ A ɛ k+ Proof For fixed α substitute in (3b: ( ( ρ (D α ρ(d α ρ t dx = (D α ρd α Jɛ ρ J ɛ A J ɛ A Aρ + B dx m J ɛ D α ρ + Dα ρ D α B + D α ρ D α (Aρ ( + D α (J ɛ ρ D α ρ A J ɛ A dx }{{} I α Simply using Lemma 4 and Claim we obtain the final estimate for ρ given by (43 Combining Lemma 4 and Lemma 43 gives the proof of Proposition 4 Proof (Proposition 4 Recalling that we have the estimate ρ c ρ then ρ (x cl =: C Combine (4 and (43 by first fixing δ < and then choosing M > ηδ (C C In fact, if δ = ( δ > and δ = Mηδ (C C > then: d dt E m(t+δ J ɛ ρ ɛ m +δ J ɛ A ɛ m D A ɛ m +D ρ ɛ m +C(Ao, B, M (44 where,

12 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Nancy Rodriguez, Andrea Bertozzi C(A o, B, M = M Ao + B, D = β + M ( A ɛ + ρ ɛ + A, D = β + M ( A ɛ + A Observe that the coefficients of ρ ɛ m and Aɛ m depend only on Aɛ, ρ ɛ, A ɛ, ρ ɛ, and C From Sobolev embedding estimates we have u c u 3 ; hence, it is natural to first consider the case m = 3 This case is useful to get an initial estimate of T from (44 Indeed, we obtain the desired result for this case: d dt E 3(t C (M, C (E 3 + C(A o, B, M (45 The power ten on E 3 in (45 comes from Lemma 4 Fortunately, this estimate is independent of the regularizing parameter ɛ Hence, there exists a positive time, T, such that the H 3 -norms of A and ρ are bounded on [, T ] Considering the case where m > 3 gives the second desired inequality: d dt E m(t + δ J ɛ ρ ɛ m + δ J ɛ A ɛ m CE m(t + C(A o, B, M, (46 with C = C (M, C, A, ρ, ρ ɛ, A ɛ Remark 4 Note that for the above argument we needed ρ < C Due to the Sobolev Embedding Theorem the L -norm is controlled by the H -norm From Theorem 3 each ɛ > we know that ρ ɛ < L for t [, T ɛ However, we know that [, T ] [, T ɛ The bound on the higher-norms of the regularized solutions prove to be extremely useful in multiple ways To begin with, all higher-order norms are bounded on [, T ] Moreover, we know that there exists some τ > such that A ɛ (x, t A o for all (x, t Ω [, τ] if A ɛ (x, > A o Indeed, if we define A ɛ = min x Ω A ɛ (x, t then we have a point-wise bound on its time derivative thanks Proposition 4 In fact, we know that: da ɛ dt η Aɛ + A ɛ + A ɛ ρ ɛ + A o η A ɛ + A ɛ + A ɛ ρ ɛ + A o, where we need A ɛ H m for m > 4 to use of Lemma and then Sobolev embedding estimates Since A ɛ 4 is bounded independent of ɛ then A ɛ > A o on [, τ] for some τ [, T ] For simplicity let T = min{t, τ}, from now on we consider the time interval [, T ] to be the interval on which the higher-order norms are bounded and A ɛ A o Now that we have a non-trivial interval on which all the higher-order norms are bounded we show that the family of solutions to the regularized system (3, {(A ɛ, ρ ɛ } ɛ>, form a Cauchy sequence in the L -norm This enables us to obtain the necessary limiting functions A, ρ, which are a solutions to ( Lemma 44 The family of solutions {(A ɛ, ρ ɛ } ɛ> to (3 form a Cauchy sequence

13 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 3 in C([, T ]; L (Ω L (Ω In particular, there exists a constant C and a time T > such that for all ɛ and ɛ { A ɛ A ɛ + ρ ɛ ρ ɛ } C max(ɛ, ɛ sup t T Proof Let (A ɛ, ρ ɛ and (A ɛ, ρ ɛ solve their respective regularized systems (3 and satisfy the conditions of the lemma Take the inner-product of A ɛ A ɛ and A ɛ t A ɛ t d ( A ɛ A ɛ ( A ɛ dt A ɛ A ɛ = t A ɛ t dx ( = A ɛ A ɛ ( ηjɛ A ɛ ηjɛ A ɛ dx A ɛ A ɛ + ( A ɛ A ɛ ( A ɛ ρ ɛ A ɛ ρ ɛ dx = I + I + I 3 Since I has a negative sign it is not problematic The other two terms can be easily dealt with using (4 of Lemma I = η J ɛ (A ɛ A ɛ + η (Jɛ Jɛ Aɛ (A ɛ A ɛ dx η J ɛ (A ɛ A ɛ + η max(ɛ, ɛ A A ɛ 3 A ɛ For the last term, I 3 = ρ ɛ (A ɛ A ɛ dx + ( ρ ɛ + Aɛ A ɛ A ɛ A ɛ ( A ɛ A ɛ ( ρ ɛ ρ ɛ dx + ρ Aɛ ɛ ρ ɛ Combine these inequalities and return to the initial estimate to obtain: d A ɛ A ɛ ( ρ ɛ dt + A Aɛ ɛ A ɛ + η max(ɛ, ɛ A A ɛ 3 A ɛ + ρ Aɛ ɛ ρ ɛ (47 Perform a similar computation for ρ: d ρ ɛ ρ ɛ ( = ρ ɛ ρ ɛ ( ρ ɛ t ρ ɛ t dx dt ( ( = ρ ɛ ρ ɛ ( Jɛ ρ ɛ Jɛ ρ ɛ dx+ ρ ɛ ρ ɛ ( A ɛ ρ ɛ A ɛ ρ ɛ dx ( ( + ρ ɛ ρ ɛ J ɛ ( ρɛ A ɛ J ɛ A ɛ J ɛ ( ρɛ J ɛ A ɛ dx A ɛ = F + F + F 3

14 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 4 Nancy Rodriguez, Andrea Bertozzi The terms F and F are dealt with exactly as was done for the attractiveness value F 3 is not as straight forward but it can be simplified using Cauchy-Schwarz inequality: F 3 ( J ɛ ( ρɛ A ɛ J ɛa ɛ + J ɛ ( ρɛ J ɛ A ɛ A ɛ ρ ɛ ρ ɛ We can extract an ɛ at the expense of a higher-order norm and the loss of a mollifier For example we have: J ɛ ( ρɛ A ɛ J ɛ A ɛ ɛ ρ ɛ A ɛ J ɛ A ɛ { ρ ɛ ɛ D A ɛ ( ρ + A ɛ D ɛ A ɛ } A ɛ From the proof of Lemma 4, refer to the inequality (A3, the above inequality has a bound that depends only on ρ ɛ, A ɛ 3, and C Define v = A ɛ A ɛ + ρ ɛ ρ ɛ Since A ɛ 3 and ρ ɛ are bounded on [, T ] then we have the following differential inequality: d dt v C(max(ɛ, ɛ + v Notice that the constant depends on C, ρ ɛ and A ɛ 3 The above differential inequality gives v(t e Ct (v( + max(ɛ, ɛ max(ɛ, ɛ Since (A ɛ, ρ ɛ and (A ɛ, ρ ɛ satisfy the same initial conditions we have that v( =, which implies: sup v(t C max(ɛ, ɛ t<t 4 Existence and Uniqueness of Solutions to the Original Residential Burglary Model We have all the tools to prove Theorem (; however, we first state and prove the result for uniqueness of solutions More precisely, if we assume that we have existence of a smooth enough solution to ( then this solution must be unique Lemma 45 (Uniqueness of Smooth Solutions Let (A, ρ, (A, ρ be localin-time solutions, with a common interval of existence [, T ], to the system ( Furthermore, suppose these solutions are smooth enough and with the same initial data in V m, for m 3, which satisfy the conditions stated in Theorem then A = A and ρ = ρ on [, T ] Proof We consider the difference of both variables u = A A and v = ρ ρ

15 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 5 From ( we can see that u and v satisfy the following system: u t = η u u + ρ u + A v, ( ρ v t = v A ρ A ρ u A v A A (48a (48b The time evolution of the L -norm of u multiplied by a constant M (the same M used in Lemma 4 satisfies the following inequality: ( d M dt u Mη u + M ρ + A u + M A v (49 The above inequality can be seen simply by taking the L -inner product of u t and u Substituting (48a for u t into this inner product and integrating by parts gives (49 The same is done for v The following inequality holds: d dt v C C u +C ( ρ, A u +C( ρ, A, A v (4 For detailed computations of the upper bound given by (4 refer to Appendix A3 Define F (t = M u(t + v(t, again choosing M > η (C C then from (49 and (4 we see that F (t satisfies the following ode: df (t dt C M F (t (4 In (4 the constant C M = C M (M, ρ, A, A, A, C We are set to apply a Grönwall s lemma 3 Applying this lemma to (4 gives that sup t T {F (t} F (e C M T All terms that compose C M are bounded on the interval [, T ] Since the two solutions satisfy the same initial conditions then F ( =, which implies uniqueness of the solution We now progress to the proof of the main result: Theorem Proof From Theorem 3 we have that given the initial conditions in the hypothesis of Theorem, there exists a family of solutions {(A ɛ, ρ ɛ } ɛ> to the regularized problem (3 These solutions exist on the time interval [, T ɛ The interval of existence depends on the regularizing parameter; however, from Lemma 4 we know that the V -norm of the solutions are bounded independent of ɛ This gives a uniform interval of existence [, T ] Furthermore, from Lemma 44 we conclude that there exist A, ρ C([, T ]; L (Ω such that: sup { A ɛ A + ρ ɛ ρ } Cɛ t T Therefore, the solutions converge strongly in the low-norm We state an interpolation lemma needed to show strong convergence in intermediate norms This lemma

16 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 6 Nancy Rodriguez, Andrea Bertozzi offers a connection between Lemma 4 and Lemma 44 which leads to the desired result Lemma 46 (Interpolation in Sobolev Spaces Given s, there exists a constant C s so that for all v H s (Ω, and < s < s the following inequality holds: v s v s /s v s /s s To use Lemma 46 having strong convergence in the L -norm and some bounds on the higher norms is essential For m > 3 we apply the above lemma to A = A ɛ A and ρ = ρ ɛ ρ sup t T { ( A Am + ρ m } m /m ( A m /m m A m /m + ρ m /m m ρ m /m m ɛ m /m + ρ m /m m ɛ m /m The estimate (46 implies that A ɛ, ρ ɛ are uniformly bounded in H m, for m Therefore, the above inequality implies strong convergence in C([, T ], V m Taking m to be larger than three implies strong convergence in C([, T ], C (Ω due to the Sobolev Embedding Theorem 3 Now, we simply need to verify that the limits A and ρ actually satisfy ( Since (A ɛ, ρ ɛ (A, ρ from (3 we see that A ɛ t converges to η A A+Aρ+A o in C([, T ], C(Ω Correspondingly, ρ ɛ t converge to [ ρ ρ A A] + B Aρ Finally, since A ɛ t A t and ρ ɛ t ρ t then A and ρ are classical solutions of ( Since the solutions satisfy the smoothness requirements of Lemma 45 they are unique 5 Continuation of the Solutions to the Residential Burglary Model In the previous section we proved that if the initial data (A(, x, ρ(, x V m then there exists some positive time T, such that there exists a classical solution (A(x, t, ρ(x, t to ( on [, T ] We are interested in whether this solution can be continued for all time or if there exists a blow-up in finite time A natural subsequent step is to prove a continuation argument which gives necessary and sufficient conditions for global existence Recall that we used the Picard Theorem on a Banach Space to prove local existence, for fixed ɛ, to the regularized system (3 in Lemma 3 This theorem has a natural continuation argument The family of solutions can be extended in time provided /A ɛ, A ɛ m, and ρ ɛ m remain bounded 3 This argument does not directly apply to the solution of the original system and to prove a similar result we need the following theorem Theorem 5 (Continuity in the High Norms Given initial conditions (A (x, ρ (x V m, for m > 3, which satisfy the conditions stated in Theorem Let {(A ɛ, ρ ɛ } ɛ> be the family of solutions to (3 and (A, ρ be the solution described in Theorem The following hold: ( {(A ɛ, ρ ɛ } ɛ> and (A, ρ are uniformly bounded in C weak ([, T ]; V m

17 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 7 ( (A, ρ C([, T ]; V m C ([, T ]; V m Proof From Lemma 4 we conclude that: sup (A ɛ, ρ ɛ V m K (5 t T Furthermore, automatically from (: t (Aɛ, ρ ɛ K (5 V m sup t T We need to show that the limiting solution is continuous in the weak topology of V m (Ω From Definition in Section it suffices to show that (A, φ m and (ρ, φ m, where these inner-products are defined by (, are continuous scalar functions φ, φ H m Actually, since H m is the dual of H m we simply need to prove that for all ψ H m the following is true: (ψ, A ɛ L (ψ, A L The same needs to hold for ρ Previously we proved that A ɛ A in the intermediate norms, ie in C([, T ; H m, where m < m This implies that A ɛ A Consider the L -inner product of ψ H m and A ɛ A: (ψ, A ɛ A L = (ψ φ j, A ɛ L + (φ j, A ɛ A L + (φ j ψ, A L, (53 where {φ j } j N is a sequence in H m which converges strongly in H m to ψ Such a sequence exists because H m is dense in H m These terms are bounded above on [, T ]: (ψ φ j, A ɛ L ψ φ j m A ɛ m Kδ/3, (φ j, A ɛ A L φ j m A ɛ A m K δ/3, (φ j ψ, A L ψ φ j m A m Kδ/3 These inequalities substituted into (53 gives that (ψ, A ɛ A L The same argument can be made for ρ and this wraps up the proof of part We are left to prove that (A, ρ C([, T ]; V m (Ω C ([, T ]; V m (Ω Thanks to part it suffices to show that A(t m and ρ(t m are continuous functions in time We take advantage of (46 by integrating it on the interval [, T ]: T E m (T + δ J ɛ ρ ɛ m dt + δ T J ɛ A ɛ m dt E m( + T {CE m (t + D } dt Applying Grönwall s Lemma we obtain that E m (T (E m ( D /Ce CT +D /C Taking the limit as T + we see that E m (t is continuous at t = + Furthermore, being that E m (t is bounded on [, T ] and δ, δ > the inequality above implies that (A, ρ L ([, T ]; V m+ (Ω Thus, for ae t [, T ] then (A(t, ρ(t V m+ Indeed, the initial conditions have gained regularity Take an arbitrarily small t and let (A(t, ρ(t to be a new set of initial conditions Running through the same existence and uniqueness arguments we obtain a solution (A, ρ which exist

18 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 8 Nancy Rodriguez, Andrea Bertozzi on an interval [t, T ], (A, ρ C([t, T ]; V m, where now m < m+ In view of the fact that for m > 3, E m and E m+ satisfy the same differential inequality then T T Uniqueness and the arbitrary choice of t implies that (A, ρ C([, T ]; V m Furthermore, by virtue of the equation then (A, ρ C ([, T ]; V m Remark 5 From (46 we know we have control of the V m+ norm as long as we have control A(t, ρ(t, A(t, ρ(t and M Furthermore, control of /A(t implies control of M Fortunately, we find that the terms mentioned in Remark 5 are interdependent and we can obtain a dominating term However, before we discuss this we state and prove a regularity argument Theorem 5 (Regularity The solutions A, ρ of the system ( obtained from Theorem are in the space C ((, T Ω Proof Since (A, ρ C([, T ]; V m C ([, T ]; V m from Sobolev embedding estimates (A, ρ C([, T ]; C m s C ([, T ]; C m s for s > This will give us smoothness in space To obtain smoothness in time we simply look at the timederivates of the system of equations ( and use a bootstrap argument Next we show that if the appropriate initial and boundary data are chosen for A then only control of ρ(t is needed to continue the solution We prove this in the following sequence of lemmas The first one states that ρ and A controls ρ and A respectively This holds because there is a bound for the mass of ρ and A on any finite time interval Lemma 5 Let A and ρ be solutions from Theorem with initial conditions A (x and ρ (x, for p the following estimate holds for A and ρ on [, T ] for any T > : for all t [, T ] u(, t L p c u L p + (B + A o T, (54 Proof Adding both equations in the system ( we obtain that ρ(x, tdx ( B + A o t The same estimate holds for A Since Ω is the unit torus the average value of a function u is given by u = udx Now, by Poincaré inequality u L p c u L p + u L p This gives the final result Furthermore, since there is a max principle for the attractiveness value equation we prove that if A(x, > A o for all x then A(x, t A o during the interval of existence We state this result formally in following lemma Lemma 5 (Lower-Bound of Attractiveness Value Let Ω = T and A, ρ

19 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 9 C([, T ]; C (Ω C ([, T ]; C(Ω be a solutions to ( with initial conditions: Then A(x, t A o in Ω for all t [, T ] A(x, = A (x > A o, ρ(x, = ρ (x Proof We see directly from ( that A, ρ Let w = A o A then w satisfies: w t = η w w + wρ A o ρ Since both ρ and A o are nonnegative then we have: w t η w λw, (55 where λ = sup t T ρ(, t Then w = e λt v satisfies (55 if v satisfies v t η v From the initial data we know that w(x, < for all x Ω and the same is true for v By continuity in time v must remain nonnegative for some nontrivial time interval say < t < t Assume that at t we have that v(x, t = for some x This means that v t (x, t and since we have a maximum then v(x, t which is a contradiction unless v(x, t = for all x Ω Therefore, v(x, t and since w and v have the same sign then w(x, t This proves the result Lemma 5 tells us that if ρ is bounded then A > A o, provided we have appropriate initial and boundary data We also need for the solutions to the regularized model to remain bounded from below However, we know that this is true on [, T ] as was discussed earlier In addition, we prove that if ρ remains bounded then A also remains bounded This will be demonstrated in the following two lemmas Lemma 53 Let (A, ρ satisfy ( in the classical sense and assume that ρ is bounded on [, T ], for T > then A(, t L ( A(, L C e C(η, ρ,ao,bt + C, (56 where C = C(η, ρ, A o, B This holds t [, T ] Proof d A dx = dt A A t dx ( = A (η A A + Aρ + A o dx = η A dx A dx + A (Aρdx = η A dx A dx + A ρdx + A A ρdx Cauchy-Schwarz η A L + ( ρ A L + ρ ( A L + A L (54 C(η, ρ, ρ A L + C( ρ, Ao, B, T

20 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Nancy Rodriguez, Andrea Bertozzi Integrating this and using (54 for p = gives the desired result (56 Lemma 54 Let (A, ρ satisfy ( in the classical sense and assume that ρ is bounded on [, T ], for T > then ( A(, t c 4 max{ A(,, A(, L C e CT + C} (57 t [, T ] The constants C and C are defined as in Lemma 53 The proof of Lemma 54 uses the Moser-Alikakos iteration Proof Let s : d A s dx = A s A t dx s dt ( = A s (η A A + Aρ + A o dx η(s A s A dx A s dx + A s ρdx + A A s ρ dx 4η(s ( Hölder s Ineq s A s/ + ρ L A s L A s L + ( ρ A s L s (54 4η(s ( = s A s/ + c A s L L + c, s where c = c ( ρ, ρ, A o, B and c = c (A o, B Multiplying both sides by s, s gives: d A s dx η ( A s/ dt + sc A s L L + sc s We need to make use of an extended Sobolev inequality: 6 u ɛ L ( u L ɛ + c ɛ u L (58 A derivation of (58 can be found in Appendix A4 Taking u = A s/ gives: d A s η( ɛ dx A s s dt ɛ + c ɛ A s L + c s A s s/ L + sc, s η Choose ɛ = sc noting that s > η [, ] (refer to 33 +η then: d A s L dt dx c s A s s L + c 3s A s s L + sc s/ By multiplying both sides by e cst the above inequality is equivalent to d { e c st A s} ( e cst c 3 s A L s dt + c s/

21 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior Integrating this over [, t] gives: e cst A(, t s L s Therefore, A(, t s L s t A(, s L + sup A(, τ s s L c s/ 3 s e csτ dτ + τ t t c e csτ sdτ A(, s L + sup A(, τ s s L c 4s(e cst + c s/ 5 (e cst τ t ( A(, + c 6 s + c 4 s sup A(, t s Ls/, (59 τ t where c 6 = max{, c 5 } Define M(s = max{ A(, + c 6, sup t T A(, t L s} From (59 we conclude that Let s = k for k N the recursive relation (5 gives: M(s (c 7 s /s M(s/ (5 M( k (c 7 k j= j ( k j= j j M( Since both sums k j= j and k j= j j converge as k taking the limit as s we get: A(, T lim s M(s Applying Lemma 53 gives the final result (c 7 j= j ( j= j j M(, From Theorem 5 and Lemma 5-54 proved above we obtain necessary and sufficient conditions for the continuation of the solution to ( Corollary 5 Given initial conditions (A(x,, ρ(x, V m, m > 3 such that A(x, > A o > and no-flux boundary conditions, there exist a maximal time of existence < T max and a unique solution (A(x, t, ρ(x, t C([, T max ; V m C ([, T max ; V m the the system ( Furthermore, if T max is finite then one of the following must hold: ( lim t Tmax ρ =, ( lim t Tmax A = (3 lim t Tmax ρ =, or (4 lim t Tmax A = In addition, if lim t Tmax A = then necessarily lim t Tmax ρ =

22 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Nancy Rodriguez, Andrea Bertozzi 6 Analysis of a modified PDE model of criminal behavior and its relation to Keller-Segel Model Though we succeeded in proving local existence and uniqueness of solutions to ( the question of whether the solutions can be extended for all time has not been addressed To be confident that we have a robust model, suitable for the target application, we need insight on global existence and/or possible blow up Working with a strongly coupled system of nonlinear PDEs makes it difficult to apply the usual techniques to prove well-posedness Fortunately, as was mentioned before, there is an evident relation between the model for residential burglaries and the Keller-Segel model for chemotaxis, developed in 7 by Keller and Segel in 97 This is not surprising since both processes are usually modeled by a parabolicparabolic system and include motion up gradients of some external field Chemotaxis is the influence of a chemical substance in the environment on the movement of a mobile species This process is key in cellular communications Keller and Segel developed a general model for the chemotaxis phase of aggregation of slime mold, ie Dictyostelium Discoidium in 7 There has been a great deal of analysis on various versions of the Keller-Segel model since it was developed and research is still in progress 6,9,,4,, 8, Thus far the most studied version is: u t = κ u χ (u v, (6a ɛ v t = k c v αv + βu (6b with Neumann boundary conditions In (6 u is the myxamoebae density of slime mold and v the chemo-attractant concentration Comparing this model to ( we can see that the chemo-attractant density is comparable to the attractiveness value It is worth noting that chemotaxis is sometimes modeled by an elliptic-parabolic system; however, in the residential burglaries model the timescale of the change in attractiveness value is similar to the change in criminal density From (6 we see that the myxamoebae move up gradients of chemo-attract concentration like criminals move up gradients of attractiveness value Global existence and finite time blow-up of the (6 is highly dependent on the dimension In one-dimension finite time blow up cannot occur 7 In two-dimensions it has been shown, by Corrias and Calvez 5, that the solution exist globally in time if the initial mass is below the critical quantity 8π If the initial mass is above 8π then aggregation occurs,9 For higher dimensions, d, there exists a similar critical quantity that is governed by the L d/ -norm of the initial myxamoebae density Although the most studied version of the Keller-Segel model is (6 various variations of the model have also been analyzed A comprehensive summary of much of this work can be found in and 3 In a sense, the model given by ( can be thought of as a generalized and more complicated version of (6, which includes growth and decay of the myxamoebae density and the chemo-attractant We want to take advantage the extensive body of work done on (6 as a first step to obtaining insight on the global existence or finite

23 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 3 time blow-up of ( To accomplish this analyze a simplified model of ( This will ease the mathematical analysis while maintaining fundamental assumptions made in 33 The model we propose is: A t = η A A + βρ + Ao (x, ρ = ρ (ρ A + B(x f(aρ t (6b (6a From now on we work in all of R Notice that now A o and B are functions of the space variable and must have sufficient decay as x Model (6 makes three simplifications to ( First, the advection speed is now given simply by A The second modification is that the attractiveness value increases with the number of criminals with constant of proportionality β, ie we replace Aρ with βρ in (a We have no reason to believe that this modification will decrease the accuracy of the model Finally, the criminal density decays with a rate of f(a and we assume that f(a has a lower and upper bound 6 Useful Properties of the Modified Residential Burglaries Model It is not surprising that (6 is the most studied version of the Keller-Segel model since is possesses properties that facilitates mathematical anlysis There are three properties worth noting First, the system (6 conserves mass of the cell density Furthermore, one can express the chemo-attractant concentration as the con- t z e volution of the Bessel Kernel, B η (z = 4ηt t 4ηπ dt, and the cell density In two-dimensions this is especially useful for proving blow-up results given large enough initial mass of the cell density Most importantly, this model, after nondimensionalization, has a Lyapunov functional 5 : F(t = u log(udx uv dx + v dx + αv dx This functional is key in proving global existence The model (6 does not possess these exact properties; however, it does possess ones which are useful enough For v L (Ω let M v (t = v(x, tdx As an example of a useful property Ω if A, ρ are solutions to (6 then M ρ (t is bounded above and below Define f min = min A R + f(a then: M ρ (t e (M fmint ρ ( M B + M B (63 f min f min Replacing f min with f max gives a similar lower bound for M ρ (t Another key property is the explicit expression of the attractiveness value in terms of the criminal density: A(t = B η (βρ + A o (64 We conjecture that solutions to (6 satisfy an energy functional whose upper bound can be controlled with time Being that this is beyond the scope of this paper

24 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 4 Nancy Rodriguez, Andrea Bertozzi we only mention that proving such an energy functional is important for proving global existence via the Lyapunov functional method discussed in the introduction 6 Blow-up of the Modified Residential Burglaries Model In this section we look at the possibility of blow-up in finite time of the solution to the modified residential burglaries model (6 It turns out, that similar to the Keller-Segel model, if the lower-bound on the mass of the criminal density is large enough there is mass concentration on a set of measure zero Let Mρ min = min {M ρ (, M B /f max } and Mρ max = max {M ρ (, M B /f min } and for a function v we denote the finite second moment by I v = x vdx We state this blow-up result in the following theorem Theorem 6 (Blow-up of Modified Residential Burglaries Model Let (A(x,, ρ(x, L (R be initial data such that ( βmρ min 4π Mρ min > πi B Furthermore, let ρ be the non-negative smooth solution to (6b such that if A is defined by (64, then A, ρ are a non-negative smooth solutions to (6 Then, if the initial second moment is small enough That is if x ρdx [( ] β K π M ρ min 4 Mρ min I B, (65 where I B (x = [ Bdx, K = β π C ( 3/+ ( ] Mρ max a M max / ρ, a = 4 B η (x A o (x and C a constant, then there exists a finite time singularity Proof Consider the time evolution of the second moment of ρ, I(t = x ρdx: R di dt = x ( ρ (ρ A f(aρ + B dx R 4 ρ dx + 4β ρ (x B η ρ dx + 4 ρ (x B η A o (x dx + I B (66 R R R The third term of on the right in the above inequality can be bounded above using Cauchy-Schwarz Inequality and Young s inequality for convolutions 38 : 4 ρ x B η A o (x dx 4 B η A o (x ρ x dx R 4 B η L A o / (x }{{ Mρ (t I(t (67 } a We use the explicit expression of the gradient of the Bessel Kernel, B η (z = z z s π e z 4ηs ds, to bound the second term Let gη (z = da = dxdy then: z s e 4ηs ds and

25 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal Local Existence and Uniqueness of Solutions to a PDE Model for Criminal Behavior 5 4β ρ (x B η ρ dx β (x y ρ(x, tx R π R R x y g η(x yρ(y, tdydx β ρ(x, t[ g η (x y] ρ(y, tda β ρ(x, tρ(y, tda π π = β π M ρ (t + β ρ(x, t [ g η (x y] ρ(y, tda π Observe that g η (z is a positive, radially symmetric, decreasing function with maximum of one This implies that ( g η (z Now, consider the derivative of ( g η (r with respect to r = z : d dr ( g η(r r η r s e s 4ηs ds π η rb ( r η π sup ( rb ( r, η r (, where, r = r η for r η If C = π η max(sup r (, { rb ( r}, then ( g η (z C z Hence, we have: 4β ρ (x B η ρ dx β R π M ρ (t + β π CM ρ(t ρ(x, t x dx R CS β π M ρ (t + β π C (M ρ(t 3/ I(t (68 Substituting (67 and (68 into (66 gives: ( di dt 4 β π M ρ(t ( 4 β π M ρ(t ( 4 β π M ρ min M ρ (t + β 3/ CMρ (t I(t + a Mρ / (t I(t + I π B ( β 3/ I(t M ρ (t + CMρ (t + a Mρ (t / + IB π ( β 3/ ( + a M max / I(t + IB Mρ min + π C ( Mρ max } {{ } K In the last inequality we use the fact that the initial conditions are chosen so that > 4π Integrating on [, t gives the integral inequality: βm min ρ ( where g(i(t = 4 β π M ρ min I(t I( + M min ρ t ρ g(i(sds, (69 +K I(t+I B The function g(i is continuous, increasing and such that g(i(t = for t > such that: I(t = [( ] β K π M ρ min 4 Mρ min I B,

26 December, 9 8: WSPC/INSTRUCTION FILE LocExtDraftFinal 6 Nancy Rodriguez, Andrea Bertozzi where K is defined in the theorem Since I( I(t by continuity of g there exists a t > such that t g(i(sds < Hence, I( t < I( Repeating this process will eventually give that I(t = for some positive t which proves the result From Theorem 6 we see that a logarithmic sensitivity function is more suitable than a linear sensitivity function Moreover, from the maximum principle of the attractiveness value a lower bound on f(a is implicit in the original model Hence, setting f(a = A is only eliminating the upper bound on f(a This would only help prevent blow-up The remaining difference between the two models is less obvious to analyze We conjecture that the nonlinear Aρ aids blow-up more than βρ This is because we expect, and indeed we observe numerically, that A and ρ grow and decay together Hence, we have that Aρ ρ which would aid blow-up more so than βρ would 7 Discussion The overarching goal of the development of model ( by Short et al is to help understand the spatio-temporal dynamics of residential burglary hotspots to aid law-enforcement in the mobilization of their resources In this paper we studied the well-posedness of this model to determine whether it is a suitable model for the target application In particular, we know that certain types of finite time blow-up would invalidate the model For example, blow-up in the L norm of the criminal density would not make any physical sense As a first step to determining whether this model is well-posed we proved local existence of classical solutions Furthermore, we know that with no-flux boundary conditions the attractiveness value will never go below its static component A o From the continuation argument we know that the model has a global solution provided the L bound of the gradient of ρ remains bounded In the case of blow-up we also know that if the L norm of the gradient of A blows up in finite time then the same has to be true for the gradient of ρ In the final part of the paper we explored the connections of the residential burglary model and the Keller-Segel model for chemotaxis, which has been vastly studied Considering a modified residential burglary model we determined that the logarithmic sensitivity function in ( is essential to preventing blow-up In fact, preliminary numerical results show finite time blow-up for ( with χ(a = A with no other modifications in two-dimensions In one-dimension no such blow-up has been observed This serves to confirm the connections between the Keller-Segel model and the residential burglary model