On a nonlocal aggregation model with nonlinear diffusion

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.


Introduction and main results
In this paper we consider the following evolution equation with nonlinear diffusion: where K is an even function of x and has a Lipschitz point at the origin, e.g. K(x) = e −|x| . Here * denotes the usual spatial convolution on R. The parameter r > 0 measures the strength of the nonlinear diffusion term. The function u = u(t, x) represents population density in biology or particle density in material science. (see, for example, [ [51]). This equation was used in the study of biological aggregation such as insect swarms, fish schools and bacterial colonies. It is first derived by Bertozzi, Lewis and Topaz [56] as a modification (more precisely, the addition of the density-dependent term on the RHS of (1.1)) of an earlier classical model of Kawasaki [35]. According to [56], this modification gives rise to biologically meaningful clumping solutions (i.e. densities with compact support and sharp edges). For other similar one-dimensional models and their biological applications we refer the readers to [46] [50] and the references therein for more extensive background and reviews.
To understand the biological meaning of each term, one can rewrite (1.1) as the classical continuity equation ∂ t u + ∂ x (uv) = 0, (1.2) where the velocity v is related to the density u by Here v a is called the attractive velocity since as explained in [56] individuals aggregate by climbing gradients of the sensing function s = K * u. Due to the spatial convolution v a is a nonlocal transformation of the density u. The second term v d is called the dispersal(repulsive) velocity and is a spatially local function of both the population and the population gradient. Biologically v d represents the anti-crowding mechanism which operates in the opposite direction of population gradient and decreases as the population density drops. These constitutive relations of v a , v d , u are natural in view of the basic biological assumption that aggregation occurs on a longer spatial scale than dispersal.
In the mathematics literature, the aggregation equations which have similar forms to (1.1) have been studied extensively [11] [12] [13] [14] [15] [38] [57]. The case of (1.1) with r = 0 and general choices of the kernel K was considered by Bodnar and Velazquez [13]. There by an ODE argument the authors proved the local well-posedness of (1.1) without the density-dependent term for C 1 initial data. For a generic class of choices of the kernel K and initial data, they proved by comparing with a Burgers-like dynamics, the finite time blowup of the L ∞ x -norm of the solution. Burger and Di Francesco [14] studied a class of one-dimensional aggregation equations of the form where V : R → R is a given external potential and the nonlinear diffusion term a(ρ) is assumed to be either 0 or a strictly increasing function of ρ. In the case of no diffusion (a ≡ 0) they proved the existence of stationary solutions and investigated the weak convergence of solutions toward the steady state. In the case of sufficiently small diffusion (a(ρ) = ǫρ 2 ) they proved the existence of stationary solutions with small support. Burger, Capasso and Morale [15] studied the wellposedness of an equation similar to (1.1) but with a different diffusion term: , they proved the existence of a weak solution by using the standard Schauder's method. Moreover the uniqueness of entropy solutions was also proved there. In connection with the problem we study here, Laurent [38] has studied in detail the case of (1.1) without density-dependent diffusion (i.e. r = 0 ) and proved local and global existence results for a class of kernels K with H s x (s ≥ 1) initial data. More recently Bertozzi and Laurent [11] have obtained finite-time blowup of solutions for the case of (1.1) without diffusion (i.e. r = 0) in R d (d ≥ 2) assuming compactly supported radial initial data with highly localized support. Li and Rodrigo [41] [42] [43] studied the case of (1.1) with fractional dissipation and proved finite-time blowup or global wellposedness in various situations. We refer to [44] [45] for the cases with singular kernels and further detailed studies concerning sharp asymptotics and regularity of solutions. We also mention that Bertozzi and Brandman [7] recently constructed L 1 x ∩ L ∞ x weak solutions to (1.1) in R d (d ≥ 2) and with no dissipation (r = 0) by following Yudovich's work on incompressible Euler equations [61]. We refer the interested readers to [55]  From the analysis point of view, equation (1.1) is also connected with a general class of degenerate parabolic equations known as porous medium equations, which takes the form where m is a real number. These equations describe the ideal gas flow through a homogeneous porous medium and other physical phenomena in gas dynamics and plasma physics [48] [6] showed that v becomes C ∞ outside of the free boundary after some waiting time. For derivative estimates of the solution u, a local solution of (1.3) in W 1,∞ x (R) is constructed by Otani and Sugiyama [53]. In the case of (1.3) with m being an even natural number, Otani and Sugiyama [54] proved the existence of smooth solutions. We refer the interested readers to [1]  Our starting point of the analysis of equation (1.1) is to treat it as a degenerate parabolic problem with a nonlocal flux term. We focus on constructing and analyzing classical solutions of (1.1). The bulk of this paper is devoted to proving the existence and uniqueness of classical solutions to (1.1), which is done in section 2 and part of section 3. In the final part of section 3, we prove the continuation and blowup criteria of solutions. In the last section we prove that any smooth initial data with compact support will lead to blowup of the gradient in finite time. The analysis developed in this work can be extended to treat the d-dimensional (d ≥ 2) case of (1.1) which we will address in a future publication.
We now state more precisely our main results. The first theorem establishes the existence of smooth local solutions for initial data which is not necessarily nonnegative. Theorem 1.1 (Existence and uniqueness of smooth local solution).
Remark 1.2. The assumptions on the initial data u 0 in Theorem 1.1 can be weakened significantly (see for example Theorem 3.2). However in order to simplify the presentation, we do not state our theorems here in its most general form.
Our second theorem gives the blowup or continuation criteria of solutions. Roughly speaking, it says that all the L p x -norm of the solution cannot blow up and we can continue the solution as long as we have a control of the gradient of the solution. For any 2 ≤ p < ∞, there exists a generic constant C such that If in addition u 0 ≥ 0, then u(t) ≥ 0 for all t ∈ [0, T ), and we also have the p-independent estimate for all 2 ≤ p ≤ +∞: In particular if p = +∞, then The next theorem states that if we assume the initial data has a little bit more integrability, then the local solution will inherit this property. Note in particular that the L 1 x -norm of the solution is preserved for all time if the initial data is nonnegative and in L 1 x (R).
Then the local solution obtained in Theorem 1.1 also satisfies u ∈ C([0, T ], L p x ). If in addition u 0 ≥ 0, then u(t) ≥ 0 for any t ∈ [0, T ]. If also p = 1, then u(t) 1 = u 0 1 , i.e. L 1 x -norm of the solution is preserved. The last theorem states that any solution with smooth nonnegative initial data will blow up in finite time.
(1.5) Remark 1.6. In the case of (1.1) without diffusion (i.e. r = 0 in (1.1)), if we take the kernel K(x) = e −|x| , then the result of Bodnar and Velazquez [13] says that any solution with initial data u 0 satisfying a slope condition will blow up in finite time in the sense that u(t) ∞ blows up. This is highly in contrast with our result here when the diffusion term does not vanish. In this case the solution will blow up at the level of the gradient, i.e. we have ∇u(t) ∞ tends to infinity while all the other L p x -norm remain finite as t → T , where T is the blowup time.
Notations. Throughout the paper we denote L p When p = 2, we denote H m x = H m x (R) = W 2,p x (R) and · H m as its norm. Occasionally we shall use the Sobolev space of fractional power H s x (R) whose norm can be defined via Fourier transform: For any two quantities X and Y , we use X Y or Y X whenever X ≤ CY for some constant C > 0. A constant C with subscripts implies the dependence on these parameters. We write A = A(B 1 , · · · , B k ) when we want to stress that a quantity A depends on the quantities B 1 , · · · , B k .
From now on we assume r = 1 in (1.1) without loss of generality. Same results hold for any r > 0.

The regularized equation and its wellposedness
Since (1.1) is a degenerate parabolic equation, in order to construct a local solution, we have to regularize the equation. To this end, we consider the following regularized version of (1.1) Here ǫ > 0 is a parameter. We are going to prove the following x energy method used by Otani and Sugiyama [54] where they dealt with the porous medium equation (1.3). Denote the set . As a very first step, we shall show the local existence of the solution to (2.1) in B k T . At this point, we need the following lemma from [54]. Lemma 2.4. Consider the initial value problem Proof. See [54].
Based on this lemma, we establish the local existence of solutions of the regularized equation. Note however that the time of existence of the solution depends on the regularization parameter ǫ. We have the following It suffices to show that for some suitable R and T 0 , φ is a contraction from the set This shows that if we take The lemma is proved. Lemma 2.5 is not satisfactory since the time of existence of the solution depends on the regularization parameter ǫ. The following lemma removes this dependence, and at the same time, weakens the dependence on the initial norm.
Proof. By Lemma 2.5, we can continue the solution as long as we can control the H 2k+1 x -norm of u ε . In the following we give the a priori control of u ǫ . As we will see, the H 2k+1 x -norm of u ε will stay bounded on a time interval [0, T 0 ] for a certain small T 0 depending only on the norms ∂ x u 0 ∞ , u 0 2 . We are then left with the task of estimating the various Sobolev norms which we will do in several steps. For simplicity of notations we shall write u ǫ as u throughout this proof.
Step 1: L p x -norm estimate for 2 ≤ p ≤ +∞. Multiply both sides of (2.1) by |u| p−2 u and integrate over x, we have Gronwall's inequality implies that In particular we have To control L ∞ x -norm, we are going to choose T 0 < 1 ∂xK 1 . This implies that u(t) 2 has a uniform bound on any such [0, T 0 ]. Now use again the RHS of (2.4) to get Now we use the fact that K(x) = e −|x| and therefore ∂ xx K(x) = −2δ(x) + e −|x| , and this gives This implies Integrating over [0, t], we obtain Letting p → ∞, we obtain Now it's easy to see that if we choose T 0 sufficiently small depending on u 0 2 , u 0 ∞ , then we have Step 2: Control of ∂ x u(t) ∞ . At this point, one can appeal directly to the maximum principle (see for example Theorem 11.16 of [39]). We give a rather direct estimate here without using the maximum principle. Differentiating both sides of (2.1) w.r.t x and denoting v = u x , we have where Multiplying both sides of (2.8) by |v| p−2 v and integrating over x, we obtain 1 p Integrate by parts and we have Plugging the above estimates into (2.9), we get

Now we compute
Finally we obtain Integrating over t, letting p → ∞, and recalling v = u x , we have for sufficiently small Step 3: Control of ∂ xx u(t) p for 2 ≤ p ≤ +∞. First take 2 ≤ p < +∞ and compute We further estimate by using integration by parts, Using Cauchy Schwartz inequality: we bound the second term in (2.11) as Hence, Also it is obvious that Similarly we have Collecting all the estimates and we have Integrating over t we have To estimate u xx ∞ we need to estimate u xxx 2 . We have Finally we get Gronwall then implies that

ON A NONLOCAL AGGREGATION MODEL WITH NONLINEAR DIFFUSION13
Step 4: Control of ∂ 2k+1 x u 2 . We compute For I, we integrate by parts and obtain Discarding the negative term and using Hölder's inequality, we bound (2.14) as (2.14) ( u To estimate the summand in (2.14), we discuss several cases. Case 1. n = 2k + 1. In view of the constraint, we have (l, m) = (0, 2) or (1, 1). Hölder's inequality then gives that Case 2. n ≤ 2k, l = 0. Since l + m + n = 2k + 3 we must have m ≥ 3. We choose 2 < p, q < ∞ such that 1 p + 1 q = 1 2 and use Hölder to get Using the interpolation inequality Case 3. n ≤ 2k, l = 1. In this case m, n ≥ 2, we can choose 2 < p, q < ∞ such that 1 p + 1 q = 1 2 and use the interpolation inequality Case 4. n ≤ 2k, l, m, n ≥ 2. Choosing 2 < p, q, r < ∞ such that 1 p + 1 q + 1 r = 1 2 and using the interpolation inequality (2.17), we get Collecting all the estimates together, we conclude For term II, we simply have For III, we compute: Using the interpolation inequality we finally get III ( u ∞ + u x 2 ) ∂ 2k+1 x u 2 2 . Summarizing the estimates of I, II, III we obtain . Using Gronwall and (2.7), (2.12), (2.13) we get This concludes the estimate of the H 2k+1 x -norm of u ǫ . Finally if u 0 ≥ 0, then by the weak maximum principle we have u ǫ (t) ≥ 0 for any 0 ≤ t ≤ T 0 . The lemma is proved.
We are now ready to complete the Proof of Proposition 2.1. This follows directly from Lemma 2.5 and 2.6. In particular note that in Lemma 2.6 the time of existence of the local solution does not depend on ǫ.
Proof of Corollary 2.2. Assume u 0 ∈ ∞ m=0 H m x (R) L p x (R) for some 1 ≤ p < 2. Then using (2.1) and Duhamel's formula, we can write u ǫ (t) as Since e ǫt∂xx f p f p for any ǫ > 0, we can estimate the L p x norm of u ǫ as where the last inequality follows from the Sobolev embedding. This estimate shows that u ǫ (t) ∈ L p x for any t. The continuity (including right continuity at t = 0) follows from similar estimates. We omit the details. Finally since for 1 ≤ p < 2, u 0 2 + ∂ x u 0 ∞ u 0 p + ∂ x u 0 ∞ , one can choose the time interval sufficiently small depending only on u 0 p + ∂ x u 0 ∞ .

2.2.
Proof of Proposition 2.3. In the case u 0 ≥ 0, we shall show the regularized equation has a global solution. The key point here is that by using the positivity, we can obtain the apriori boundedness of the L p x -norm (2 ≤ p ≤ ∞) of the solution on any finite time interval. To control the L ∞ x -norm of the gradient, we need the following lemma from [39].
(2) There is a constant M > 0 such that (3) There exists L 0 > 0 such that Under all the above assumptions, we have the following bound: Proof. See Lemma 11.16 of [39].
Incorporating Lemma 2.7 with the idea of the proof of Lemma 2.6, we get the following global analogue of Lemma 2.6.
Proof. The positivity of u ǫ follows easily from the weak maximum principle. By Lemma 2.5 we only need to get an apriori control of the H 2k+1 x -norm of u on an arbitrary time interval [0, T ].
Step 1: L p x -norm control for 2 ≤ p ≤ +∞. By the same estimates as in Step 1 of the proof of Lemma 2.6 we have the following a priori L 2 x -norm estimate: To obtain the L ∞ x -norm estimate, we take 2 < p < +∞ and compute 1 p Since u ≥ 0 we can drop the first term and continue to estimate where C is a generic constant independent of p. Gronwall then implies that Letting p → ∞, we obtain This is the L ∞ x -norm estimate we needed.
Step 2: Control of ∂ x u(t) ∞ . We shall apply Lemma 2.7 with b(t, x, z, p) = z 2 + ǫ, By step 1 we have Also Collecting all these estimates, we see that This concludes the gradient estimate.
Step 3: Control of the higher derivatives. This part of the estimates is exactly the same as the corresponding estimates in the proof of Lemma 2.6. Note in particular that we do not need T to be small once we obtain the a priori control of u p and ∂ x u ∞ norms.

Now we are ready to complete
Proof of Proposition 2.3. This follows directly from the local existence Lemma 2.5 and the a priori estimate Lemma 2.8. If u 0 ∈ L p x (R) for some 1 ≤ p < 2, then by repeating the proof of Corollary 2.2 we obtain u ∈ C([0, T ], L p x ) for any T > 0. The proof is finished.
3. Proof of Theorem 1.1, 1.3 and 1.4 In this section we prove our main theorems. Our solution is going to be the limit of the sequence of regularized solutions u ε which we constructed in the previous section. To obtain uniform control of the Sobolev norms of the regularized solutions, we have the following T 0 be the corresponding unique solution to (2.1) (see Lemma 2.6). Then the set of functions (u ǫ ) 0<ǫ<1 satisfies the following uniform estimates: x can be recovered from step 4 of the proof of Lemma 2.6. To bound ∂ t u ǫ L 2 t H 2k , we use (2.1) to obtain (here we again drop the superscript ε for simplicity): Since T 0 = T 0 ( u 0 2 + ∂ x u 0 ∞ ), we obtain the desired bounds. The last estimate is a simple application of Hölder and Young's inequality using the given estimates.

Now we define the set
We shall prove the existence and the uniqueness of the solution of (1.1) in this set. This is x ). For any two ǫ 1 , ǫ 2 > 0, let u = u ǫ 1 , v = u ǫ 2 solve (2.1) with the same initial data u 0 . Denote w = u − v, then for w we have the equation .
Denote F (u, v) = u 2 + v 2 + uv. Clearly we always have F (u, v) ≥ 0. We compute . For II we have the estimate The term III can be bounded as: If in addition u 0 ≥ 0, then u(t) ≥ 0 and we also have the p-independent estimate for all 2 ≤ p ≤ +∞: In particular if p = +∞, then Proof. This follows directly from Theorem 3.2 and Lemma 2.6 (see (2.5) for the growth estimate of L p x -norm for any T > 0). Since u ǫ → u uniformly in C([0, T ]; H 2k x ), u(t) satisfies the same estimate (2.5). Similarly if u 0 ≥ 0, then the estimate (3.1) follows from the corresponding estimate for u ǫ (see (2.18) and (2.19) in the proof of Lemma 2.8). Now that the L 2 norm of the constructed local solution remains finite for any T > 0, it can be continued as long as ∂ x u(t) ∞ does not blow up.
We are now ready to complete the . By using (1.1), a standard bootstrap argument implies that u ∈ C ∞ ([0, T 0 ] × R). If u 0 ≥ 0, then the positivity of u follows from Lemma 2.8 and the uniform convergence of u ǫ to u. The theorem is proved.
Proof of Theorem 1.3. This follows immediately from Theorem 1.1 and Corollary 3.3.
Proof of Theorem 1.4. This is only slightly different from the proof of Corollary 2.2. Assume u 0 ∈ ∞ m=0 H m x (R) L p x (R) for some 1 ≤ p < 2. Then using (1.1), we can bound This shows that u(t) ∈ L p x for any t. The continuity (including right continuity at t = 0) follows from similar estimates. We omit the standard details. Finally in the case u 0 ≥ 0, u 0 ∈ L 1 x (R), the L 1 x preservation follows from direct integration.

Proof of Theorem 1.5
We argue by contradiction. Let u 0 ∈ C ∞ c (R), u 0 ≥ 0 and assume that the corresponding solution u is global. By Theorem 1.1, 1.3, 1.4, u ∈ C([0, T ), H m x ) for any m ≥ 0 and the L 1 x -norm of u is preserved. Our intuition of proving the blowup is based on the observation that as time goes on, the boundary of the solution(which in 1D case consists of two points) will move face to face at a certain speed which has a lower bound independent of time. This clearly will lead to the collapse of the solution. To realize this intuition, we will carry out a detailed analysis on the characterstic lines of the solution which satisfy    d dt X(t, α) = (K * ∂ x u)(X(t, α), t), (4.1) By standard ODE theory and our assumption that u is a smooth global solution, X(t) is well defined and smooth for all time. Moreover, we have the following lemma.
Similarly once can show that if y > X(t, L) then u(t, y) = 0. This shows that supp(u(t, ·)) ⊂ [X(t, −L), X(t, L)]. Now using this fact and (4.1), we compute where in the last step we used the fact that the L 1 norm of u is preserved. This shows that X(t, −L) grows linearly with time which is contradiction to (4.3). Thus we have shown that the solution u with u 0 as initial data cannot exist globally. Finally (1.4)