Global Existence and Convergence Rates for the Strong Solutions in H 2 to the 3 D Chemotaxis Model

We are concerned with a 3D chemotaxis model arising from biology, which is a coupled hyperbolic-parabolic system. We prove the global existence of a strong solution when H-norm of the initial perturbation around a constant state is sufficiently small. Moreover, if additionally, L-norm of the initial perturbation is bounded; the optimal convergence rates are also obtained for such a solution. The proofs are obtained by combining spectral analysis with energy methods.


Introduction
In this paper, we investigate global existence and optimal convergence rates of strong solutions to the following 3D chemotaxis model: with initial data (V, ) (, 0) = (V 0 ,  0 ) () → (0, ) , as || → ∞, (2) where  > 0 is a positive constant.The system (1) is one of the models describing the chemotaxis phenomenon in biology and is closely related to the following system: )) , which is motivated by biological considerations and numerical computations carried out by Othmer and Stevens in [1] and Levine and Sleeman in [2].Here, (, ) denotes the particle density and (, ) is the concentration of chemicals. > 0 is the diffusion rate of particles.The function Φ is commonly referred to as the chemotactic potential and Ψ denotes the chemical kinetics.Depending on the specific modeling goals, the kinetic function Ψ(, ) has a wide variability.In this paper, we consider a class of nonlinear kinetic functions Ψ(, ): where  is a positive constant and  is a smooth function satisfying for all  under consideration.
Furthermore, by setting we can rewrite the system (6) as Finally, for positive constants , , and  1 to be determined below, let  = ,  = ,  = , and V =  1 ; then the system (8) becomes If we choose that is, then it is easy to see that  and V satisfy If we replace the variables (, ) by (, ), (12) is exactly (1).
To go directly to the theme of this paper, we now only review some former results which are closely related.For the one-dimensional version of the chemotaxis model (1), the existence and asymptotic behavior of smooth solutions have been studied by several authors.When the kinetic function is linear, that is, () =  −  with (>0) and (≥0) being given constants, the corresponding system reads as follows: The initial boundary value problem and Cauchy problem for the system (13) were considered by [3,5] and [4], respectively.In [3], they considered the initial boundary value problem for the system (13).When ‖ 0 − 1‖ 2  2 + ‖V 0 ‖ 2  2 is sufficiently small, they proved the global existence of smooth solutions to the system (13).The authors in [5] generalized the results of [3] to the arbitrarily large initial data case.In [4], the authors obtained the global existence of smooth solutions to the Cauchy problem for the system (13) with large initial data.Recently, the authors in [6][7][8] extended the results of [3][4][5] to the nonlinear kinetic function case, respectively.For high dimensions, the global well-posedness of smooth small solution to (1) with () =  was investigated in [5,9] for initial-boundary value problem and Cauchy problem, respectively.For other related results, such as nonlinear stability of waves in one dimension and so on, please refer to [6][7][8] and references therein.
However, to our knowledge, so far there is no result on the optimal convergence rates of the strong solutions to the Cauchy problem (1)- (2).The main motivation of this paper is to give a positive answer to this question.In particular, we prove the global existence of a strong solution when  2 -norm of the initial perturbation around a constant state is sufficiently small.Moreover, if in addition,  1 -norm of the initial perturbation is bounded, the optimal convergence rates are also obtained for such a solution.The proofs are based on energy methods and spectral analysis which have been developed in [32][33][34][35] and references therein.
Before stating our main results, we explain the notations and conventions used throughout this paper.We denote positive constants by .Moreover, the character "" may differ in different places.  =   (R 3 ) (1 ≤  ≤ ∞) denotes the usual Lebesgue space with the norm (R 3 ) ( ≥ 0) denotes the usual th-order Sobolev space with the norm where ‖ ⋅ ‖ = ‖ ⋅ ‖ 0 = ‖ ⋅ ‖  2 .Now, we are ready to state our main results.
Theorem 1. Assume that ∇ × V 0 = 0 and ‖(V 0 ,  0 − )‖ 2 is sufficiently small; then there exists a unique globally strong solution (V, ) of the Cauchy problem (1)-( 2) such that for any  ∈ [0, ∞), satisfying where  1 is a positive constant independent of .Moreover, if, in addition, ‖(V 0 ,  0 − )‖  1 is bounded, then there is a positive constant  2 independent of  such that for any  ≥ 0, the solution (V, ) has the following decay properties: Remark 2. As compared to the classic results in [9,17,30], where smallness conditions on  3 -norm of the initial data were proposed, we are able to prove the global existence and convergence rates for the strong solutions to the Cauchy problem under only that  2 -norm of the initial data is sufficiently small.
Remark 3. Similar ideas can be applied to study the initial boundary value problem.This will be reported in a forthcoming paper.

Remark 4.
Here and in what follows, (V, ) denotes the vector The proofs of Theorem 1 are based on energy methods and spectral analysis.The key point for the proof of global existence is to obtain some a priori estimates independent of .Then, we can combine spectral analysis with the uniform a priori estimates to prove our desired decay estimates ( 17)- (20).
The rest of this paper is organized as follows.In Section 2, we reformulate the problem.In Section 3, we give some elementary facts on the decay-in-time estimates on (V, ) for the linearized system.Sections 4 and 5 are dedicated to prove Theorem 1.

Reformulated System
In this section, we will first reformulate the problem.Set Taking change of variables (V, ) → (V,  + ) and linearizing the system around (0, ), we can reformulate the Cauchy problem (1)-( 2) as where Here and in the sequel, for the notational simplicity, we still denote the reformulated variables by (V, ).
To prove the global existence of a solution to ( 22), we will combine the local existence result together with a priori estimates.To begin with, we state the following local existence, the proof of which can be found in [4,[36][37][38].
Proposition 5 (local existence).Assume that (V 0 ,  0 ) ∈  2 (R 3 ).Then, there exists a sufficiently small positive constant  0 depending only on ‖(V 0 ,  0 )‖ 2 such that the Cauchy problem (22) Proposition 6 (a priori estimate).Let ∇×V 0 = 0 and (V 0 ,  0 ) ∈  2 (R 3 ).Assume that the Cauchy problem (22) has a solution (V, )(, ) on R 3 × [0, ] for some  > 0 in the same function class as in Proposition 5.Then, there exist a small constant  > 0 and a constant  3 , which are independent of T, such that if then for any  ∈ [0, ], it holds that Moreover, if, in addition, ‖(V 0 ,  0 )‖  1 is bounded, then there is a positive constant  4 independent of  such that for any  ∈ [0, ], the solution (V, ) has the following decay properties: Theorem 1 follows from Propositions 5 and 6 and standard continuity arguments.The proof of Proposition 6 will be given in Sections 4 and 5.

Linear Decay Estimates
In this section, we consider the Cauchy problem for the linearized equations corresponding to (22) 1 -( 22) 2 : The solutions (V, ) of the linear system (31) can be expressed as Here, () := (, ) is the Green's matrix for the system (31).
To derive the large time behavior of the solutions, we first give an explicit expression for the Fourier transform Ĝ(, ) of Green's matrix (, ).Lemma 7. The Fourier transform Ĝ of Green's matrix for the linear system (31) where The representation above holds for || ̸ = ± 2 1 /.
Proof.The proof is in spirit of Hoff and Zumbrun [39,40].
To derive the long-time decay rate of solutions in  2framework, we need to verify the approximation of Green's function Ĝ(, ) for both lower frequency and high frequency.
In terms of the definition of the eigenvalues (34), we are able to obtain that it holds for || ≪ 1 that where This approximation gives the leading order terms of the elements of Green's function as follows: where  and  are some positive constants.With the help of the formula (33) for the Fourier transform Ĝ of Green's matrix and the asymptotical analysis on its elements, we are able to establish the  2 time decay rate.Indeed, we have the  2 time decay rate of the global solution to the Cauchy problem for the linear problem (31) as follows.
Proof.From ( 33), (42), and (45), we have with  > 0 being a constant here and below, and where here and below  > 0 denotes a small but fixed constant.Therefore, to prove (46), we only need to deal with the case for V since the case for  can be proved similarly.By virtue of the pointwise estimate (47), the Parseval theorem, and Hausdorff-Young's inequality, for 0 ≤  ≤ 3, we have Thus, (49) gives (46) immediately, and the proof of Lemma 8 is completed.

A Priori Estimates
Throughout this section and the next section, we assume that all conditions of Proposition 6 are satisfied.Moreover, we make a priori assumption: where  is a sufficiently small positive constant.
In the following, a series of lemmas on the energy estimates are given.Firstly, we will obtain the lower-order energy estimate for (V, ) in the following lemma.Lemma 9.There exists a positive constant  1 , which is sufficiently large and independent of , such that for any 0 ≤  ≤ .
Proof.Multiplying (22) 1 -( 22) 2 by V, , respectively, then summing up and integrating the resultant equation, we obtain Applying mean value theorem, Hölder's inequality, and Sobolev's inequality, it is clear that the two terms on the righthand side of (52) can be estimated as follows: Combining ( 52) with (53) and using the fact that  is sufficiently small, we have Next, we will estimate ‖∇V‖ 2 .Multiplying (22) 2 by ∇ ⋅ V and integrating the resulting equation over R 3 , we have where from (22) 1 , the first term on the right-hand side can be written as Then, it follows from (55)-( 56), ( 23), Hölder's inequality, and Young's inequality that This together with the fact that  is sufficiently small implies that Taking the curl for (22) 1 and noting that ∇ × V 0 = 0, we have Combining ( 58) with (60) yields Finally, multiplying (54) by  1 which is suitably large and adding it to (61), one has (51) since  > 0 is sufficiently small.This completes the proof of Lemma 9.
The following lemma is concerned with the higher-order energy estimate on (V, ).Lemma 10.There exists a positive constant  2 , which is sufficiently large and independent of , such that for any 0 ≤  ≤ .
Proof.Applying ∇ to (22) 1 -( 22) 2 and multiplying them by ∇V, ∇, respectively, and then integrating them over R 3 , we obtain Next, we estimate the terms  1 and  2 one by one.To begin with, by using ( 23 Combining ( 66) and (67) and using the fact that  is sufficiently small, we have Next, we estimate ‖∇ 2 V‖ 2 .To begin with, applying ∇ to (22) 2 and then multiplying by ∇(∇⋅V), we have from Cauchy's inequality that By integrating by parts several times, we estimate the first term on the right-hand side of (69) as follows: Finally, since  is sufficiently small, multiplying (68) by a suitably large positive constant  2 and adding it to (74) give (62), and this completes the proof of Lemma 10.

Global Existence and Convergence Rate
In this section, we devote ourselves to prove Proposition 6.To begin with, we give the following lemma which is concerned with the a priori decay-in-time estimates on (V, ).
5.1.Proof of Proposition 6.We will do it in two steps.