POINTWISE ESTIMATES OF SOLUTIONS TO CONSERVATION LAWS WITH NONLOCAL DISSIPATION-TYPE TERMS

. This article concerns the Cauchy problem of conservation laws with nonlocal dissipation-type terms in R 3 . By using Green’s function and the time-frequency decomposition method, we study global classical solutions and their long time behavior including pointwise estimates for large initial data, for solutions near the nontrivial equilibrium state.


Introduction
In this article, we study the existence of global solutions and their decay estimates to conservation laws with nonlocal dissipation-type terms in R 3 .We consider the solution, near the equilibrium state u * = 0, of the equation subject to the initial value u(x, 0) = u 0 (x), (t, x) ∈ R + × R 3 . (1.2) Here, γ 1 , γ 2 > 0, and h(u) = (h 1 (u), h 2 (u), h 3 (u)) satisfies the condition This model is motivated by physical considerations from fluid dynamics.It describes a variety of important physical processes, such as the aggregation of populations in which u describes the population density [12].It can also be used in the analysis of a seepage of homogeneous fluids through a fissured rock and the unidirectional propagation of nonlinear dispersive long waves [1,14].There are many equations similar to problem (1.1).For instance, the equation which is (1.1) with γ 2 = 1, can be used to describe the nonstationary processes in semiconductors in the presence of sources [9,10].Here k∆u t − u t indicates the rate of change in free charge's density, ∆u indicates the electric current of linear dissipative free charge and div h(u) describes a source of free electron current [9].Ting, Showalter and Golpala Rao studied the initial-boundary value problem and the Cauchy problem for the linear equation when γ 2 = 1 and proved the existence and uniqueness of the solution [6,13,15].Afterwards, considerable attention has been paid to the study of the nonlinear equation, and the large time behavior of the solutions [7,8,21].
Cao, Yin and Wang [2] studied the equation They focused on the parameter p and sufficiently small initial data.They obtained two critical indices about p and then considered the global existence or the blow up of the solutions.
In [16], the authors considered the following equation under the condition 0 ≤ s ≤ 1/2: (1.3) They got the global classical solution and long-time behavior for arbitrary large initial data.In [17], Wang proved that in the case s = 1, the solution is global with sufficiently small initial data, but will blow up with some large initial data.
Wang and Zhang [20] investigated the problem which is what we call the Benjamin-Bona-Mahony (BBM) equation.They obtained ta global solution and the optimal decay estimates with large initial data when γ 3 = 0.In our case, i.e. γ 3 = 0, there are some new difficulties.First of all, the condition γ 3 = 0 in (1.4) gives a dissipation term ∆ 2 u, which allows using the energy method to get the L ∞ estimate, and it also gives a better decay.Without this term, we need to use some new methods such as the time-frequency decomposition.Moreover, there is no maximum principle like for the viscous Burgers equation, which increases the difficulty.
Another motivation for us to consider the solution near the nontrivial equilibrium state is that the equation in our paper actually has some hyperbolic properties which however will vanish if we just consider the trivial equilibrium state.From the main results, we will see from the solution that the phenomenon of a propagating wave appears, and the propagation direction and speed of the wave are decided by the initial data.Obviously, if we only consider the solution near zero, the speed and direction will both be zero and will not be observed.So here, to get the optimal decay estimate and the hyperbolic property of the solution, we first do the pointwise estimate of the Green's function, then combining it with the variable substitution and Green's method to obtain the L 1 and L ∞ bounded estimate.Based on this, we obtain the pointwise estimate of the solution.
Since we consider the solution near the equilibrium state u * , using u − u * as the new u, we can rewrite equation (1.1) as with initial condition The main results of this article read as follows.
Theorem 1.1.Suppose that the initial data v Theorem 1.2.Assume that the initial data u 0 satisfies the conditions in Theorem 1.1, and u is a solution of (1.1) , for any t ≥ 1, where C only depends on the initial data.
Theorem 1.3.Assume that the initial data u 0 satisfies the conditions in Theorem 1.1, and v 0 satisfies Then the solution satisfies the pointwise estimate Where B r (|x|, t) is defined by (2.1), t ≥ 1, and |α| ≤ s − 1.
From the above results, we see that the solution mainly follows the decay of heat equation and, near the equilibrium state u * = 0, the equation has some hyperbolic properties with speed b.
This article is organized as follows.In Section 2, we first recall some important lemmas that will be used in this paper.In Section 3, we introduce the properties of the Green's function, mainly focusing on the pointwise estimate and the L p estimate of the Green's function.Then in Section 4, using Green's function and the variable substitution, we obtain the L p boundedness of the solution u to equation (1.5) with initial value (1.6).In Section 5, we obtain the optimal decay estimates of the solution in Ḣs by using the time-frequency decomposition method.And, as a corollary of Theorem 1.2, the optimal L ∞ convergence rate can be obtained easily.Finally, in Section 6, based on the L p boundedness and the decay of the solution, the pointwise estimate of the solution is established.

Preliminaries
In this section, we first introduce some notation, and then give some lemmas which will be used later.
For a multi-index α As usual, the Fourier transform of f (t, x) with respect to the variable and the inverse Fourier transform is Here, n denotes the space dimension.The space W s,p (R n ), with s a positive integer and p ∈ [1, +∞], is the usual Sobolev space with the norm In particular, when We denote by Ẇ s,p (R n ) the Sobolev space equipped with the norm where Λ s is defined by ) is the homogeneous Sobolev space.Next, we recall some useful lemmas.The first lemma gives some estimate of f (u), its proof can be found in [11].
, where C is a constant and only depends on s and ν 0 , and 1 r = 1 p + 1 q , with 1 ≤ p, q, r ≤ ∞.
The following two lemmas are very useful for obtaining the pointwise estimate of the solutions.Their proofs can be found in [19].
Lemma 2.2.If f (ξ, t) has a compact support about ξ, and there exists b > 0 such that for all |β| ≤ 2N , with N a positive integer, then where k and m are positive integers, and where δ(x) is the Dirac function.Furthermore, for positive integers N with 2N > 3 + |α|, we have 18]).For |y| ≤ M , t ≥ 4M 2 and N > 0, there exists C N > 0 such that Finally, note that for any g(u) ∈ L p (R n ) and p ≥ 1, we have where Thus, we have (2.2)

Existence of a local solution
In this section, we construct a convergent Cauchy sequence to obtain the local solution.Let v m (x, t) be the solution of the Cauchy problem where m ≥ 1, v 0 (x, t) = 0 and v 0 (x) ∈ H s (R 3 ).For any given integer s ≥ 1 + [3/2], define the function space One readily checks that X is a complete metric space.Next we prove that the function sequence v m (x, t) converges in the space X.Lemma 3.1.For any m ≥ 1, there exists a sufficiently small constant T 0 , such that v m (x, t) ∈ X.
Proof.We use induction.For m = 1, we know that ). Taking derivatives with respect to x, α times, on both sides of the above equation, where |α| ≤ s − 1, we have x v 1 on both sides, and integrating with respect to x and Thus for T 0 > 0, we have v 1 (x, t) ∈ X.
Assume that there is a sufficiently small T 0 such that for each j ≤ m, v j (x, t) ∈ X.We shall show that v m+1 (x, t) ∈ X.In fact, from (3.1) we have 1 2 Here, we have used the inequalities ]. Integrating with respect to t and taking the summation of |α| from 0 to s − 1, we have Thus for T 0 small enough, we obtain v m+1 (x, t) ∈ X.

Lemma 3.2.
There is a sufficiently small constant T 0 , such that v m (x, t) is a Cauchy sequence in the complete metric space X.
Proof.We are going to show that for some 0 < κ < 1, Choosing sufficiently small T 0 such that Since X is a complete metric space, combining Lemmas 3.1 and 3.2, the limit function v(x, t) ∈ X satisfies the Cauchy problem (3.1).Hence we obtain the existence of a local solution.

L p decay estimate of Green's function
In this section, we consider the behavior of the solution to the linear form of (1.5).Consider the following Cauchy problem Let G(t, x) be the inverse Fourier transform corresponding to Ĝ(t, ξ).Taking the Fourier transform with respect to x and solving the ordinary differential equation directly, we obtain Ĝ(t, ξ) = e µ(ξ)t+η(ξ)t , with Note that the solution of the Cauchy problem (1.5) with initial data (1.6) has the integral representation Next, we use the frequency decomposition to get an estimate for the Green's function G.We divide the solution into the low frequency part and the high frequency part and discuss them separately to obtain the decay property of the solution.Let , where ε and R are positive constants satisfying 2ε < R−1.We define three pseudo-differential operators χ 1 (D), χ 2 (D), χ 3 (D) with the symbols χ 1 (ξ), χ 2 (ξ), χ 3 (ξ) respectively and set Ĝi (t, ξ) Now we estimate the three parts separately.Using Lemma 2.2, we have the following estimate on G 1 (t, x).
Proposition 4.1.For sufficiently small , there exists a constant Proof.For ξ sufficiently small, the Taylor expansion gives Now we take the Fourier transform and set By properties of the Fourier transform, we have Since Ĥ(t, ξ) is smooth in the variable ξ near |ξ| = 0, for |β| ≤ 2N , we obtain Thus, by Lemma 2.2, we have This completes the proof.
Proposition 4.2.For fixed ε and R, there exist positive numbers m 0 and C such that Thus, Next, we give an estimate for x β G 2 (t, x) using induction on β.Assume that for any |β| ≤ l − 1, it holds for |β| = 0 by (4.2).
Taking the Fourier transform of (4.1) with respect to x and then multiplying by χ 2 (ξ) we have Applying the operator D β ξ on (4.5), we obtain where a 0 is a polynomial of |ξ|, and Then for |β| = l, by the ODE theory, we have Using the induction hypothesis, we obtain which implies that (4.4) is also valid for |β| = l.Then for 1 ≤ |β| ≤ l, we have Using (4.3) when |x| 2 ≤ t and using (4.6) when |x| 2 > t, we obtain (|β| = 2N ) Finally, we consider G 3 (t, x) by using Lemma 2.3.Proposition 4.3.For sufficiently large R, there exist positive numbers C 0 and C such that Here Gf α is the distribution of the form where p j (t)s are polynomials of degree j.
Proof.Since R is large enough, Then, it is easy to see that Thus, Using Lemma 2.3, we obtain ]+1 In conclusion, we obtain the following estimate for the regular part of G. Proof.

Noting that
we can use the Hausdorff-Young's inequality to obtain the following theorems about the Green's function, which are essential for the proof of the existence and the decay estimate of the Cauchy problem (1.5)-(1.6).
Theorem 4.5.For any multi-index α and p ∈ [1, ∞], we have Theorem 4.6.For all ϕ and N ≥ 0 integer, we have Here C p depends only on p.

L p bound estimate of solutions
Now we prove an L p (p ∈ [0, ∞]) estimate of the solution u of (1.5)-(1.6).First, by the fundamental energy estimate, we have Integrating with respect to t, we obtain Therefore, we have the following basic bounds for u: be a smooth cut-off function in C ∞ (R 3 ).Define the time-frequency cut-off operator χ(t, D) with symbol χ(t, ξ) = χ 0 1+t µ |ξ| 2 , where µ is a constant and µ > max{ 3 2γ1 , 3 2γ2 }.A solution u of (1.5) can be decomposed it into two parts: the low frequency part u L and the high frequency part u H , where u L (t, x) = χ(t, D)u(t, x) and u H (t, x) = (1 − χ(t, D))u(t, x).Lemma 5.1.Assume that u 0 ∈ L 1 (R 3 ).Then where C is a positive constant depending only on u 0 .
From the decay estimate of u L L 2 , we have the following interesting relation between the decay rates of u L and u.
, where C is a positive constant depending only on u 0 .
Proof.We already have the basic energy estimate (5.1), that is From the above inequality, we deduce that d dt ( (5.5) By the assumption, (5.5) becomes Multiplying the above inequality by e t 0 C 0 µ 1+τ dτ = (1 + t) C0µ and integrating from 0 to t, we obtain Thus Since C 0 µ > 2σ, we have Remark 5.3.Combining Lemmas 5.2 with 5.1, we obtain where C is a positive constant depending only on u 0 .
5.2.L ∞ bounds.We now estimate u L ∞ .Unlike the method in [20], the energy method is not enough.However, using the variable substitution and Green's function method, we can first obtain the bounded estimate of u H 2 and then apply the embedding theorem to get the L ∞ bounded estimate.Set Then (1.5) can be rewritten as By Duhamel Principle, we know that the solution will be of the form Now we use Green's function to estimate w.
Proposition 5.5.Suppose that where C is a positive constant depending only on u 0 .
Proof.Applying Theorem 4.7 to (5.11) and using the Hölder inequality, we obtain (5.12) Note that we have used the Sobolev inequality u L 6 (R 3 ) ≤ C ∇u L 2 (R 3 ) above.Combining (5.12) with (2.2), we obtain Combining these with the Green's function again, we obtain (5.13) For I 1 , we have For I 2 , we have (5.15) In the last inequality we used that ∇u L 3 (R 3 ) ≤ C ∆u L 3/2 (R 3 ) .From (5.13), (5.14) and (5.15), we obtain w Remark 5.6.By Proposition 5.5, we can easily obtain Then by the Sobolev embedding theorem 5.3.L p bounds.Since we already have L 1 and L ∞ bounds, by the interpolation inequality we easily get Here C is a positive constant depending only on the initial data u 0 .
5.4.H s bounds.With the L ∞ boundedness, we can improve the regularity of u as follows.
where C is a positive constant depending only on u 0 .
Proof.Multiplying Λ 2j u to both sides of (1.5) and integrating with respect to x, we obtain By Hölder's inequality, we have 2 Thus, for any j ≥ 1, we obtain By summation, we have One readily checks that this implies Integrating the above inequality with respect to t, we obtain Then, it follows that u H s ≤ C.
From the H s boundedness and the existence of local solutions, we obtain the existence of global solutions to (1.5), by the continuity method.Moreover,

Decay estimate in Ḣs and L ∞
In this section, we establish the optimal decay rates of the solution u in the homogeneous Sobolev spaces.
6.1.Decay rate of the low frequency part in Ḣs .Based on the boundedness of u L 1 , Plancherel's theorem, and Hausdorff-Young's inequality, we can directly obtain the following estimate for the low frequency part, Here, the constant C depends on the choice of µ, which is the same as in the last section.
6.2.Decay rate of the high frequency part in Ḣs .According to the definition of u H , u H satisfies the equation Lemma 6.1.If u 0 ∈ H 1 (R 3 ), then we have the estimate Proof.As in Lemma 5.2, using (5.1), we obtain The last inequality uses (6.1).Therefore, Multiplying the inequality above by e t 0 C 0 µ 1+τ dτ = (1 + t) C0µ and integrating from 0 to t, we obtain This completes the proof.
The next result gives the optimal decay estimate for u H Ḣs .
] a given integer, and u is a solution of (1.5) in L ∞ (0, ∞; H s (R 3 )) with initial data u 0 , then for any t ≥ 1, where C only depends on the initial data u 0 .
Proof.Multiplying (6.2) by Λ 2s u H and integrating it in x, we obtain For R 1 , by Hölder's inequality and Lemma 2.1, we have By the Galgliardo-Nirenberg-Sobolev inequality, we have Ḣs+1 u 1/s Ḣ1 .Then, using the Young's inequality, we obtain The same calculation can be done to For R 2 , by Proposition 5.4, we have For R 3 , similar to R 2 , we obtain We used (6.1) in the last inequality above.Also, according to the definition of the cut-off operator, we find that Using Lemma 6.1 and multiplying the inequality above by e t 0 C 0 µ 1+τ dτ = (1 + t) C0µ and integrating from 0 to t, we have Since C 0 µ > 3/2, we have Combining the above with (6.1) and setting s = 3, we obtain . Therefore, by the Gagliardo-Nirenberg-Sobolev inequality, we have Hence, we can use the estimate of u L ∞ to get a better decay as follows.
Similar to the proof above and note that C 0 µ > 3 2 , we have Ḣs )dτ Ḣs )dτ.By Gronwall's inequality [3], we have which is the optimal decay estimates of u H .This completes the proof.
Corollary 6.3.Suppose that u 0 satisfies the same assumptions as in Theorem 6.1 and u is a solution of for t ≥ 1, where C only depends on u 0 L 1 .

Pointwise estimate of the solution
Finally, we come to the pointwise estimate of the solution u.Our goal is to prove the following theorem.
then the solution has the pointwise estimate Proof.From (5.11), we have Obviously Π 1 means the initial term while Π 2 represents the nonlinear term.
7.1.Estimate of the initial term.For Π 1 = D α x G * u 0 , since G has a distribution Gf α, we need to separate it into two parts: There are two cases:   we conclude that M (t) ≤ 2C, t > T , which implies that Then, by the structure of the equation, we have This completes the proof.