Higher gradient bounds for a class of higher-order nonlinear systems

In this article, fundamental L p -estimates and the large-time behavior for higher derivatives of solutions to a generalized class of higher-order nonlinear parabolic systems are studied.


Statements of the main results
This paper is devoted to the study of solutions to the nonlinear parabolic system where t > 0, κ + σ ≤ 2m − 1 and ∇ k denotes the vector D γ |γ|=k with γ = (γ 1 , · · · , γ n ) ∈ N n . The constant vector a is in R n and the vectorial function b : t → b(t) satisfies the inequality for a small enough > 0. The nonlinear function f is such that f ∈ C θ (R) and f (0) = 0. The class of homogeneous operators L m = m χ m is of order 4m and the function χ is bounded, measurable, positive, and independent of time.
The particular case, when L m = , a = b = 1, θ = κ = 1, σ = 0 and f is the identity function, corresponds to the parabolic system, modeling the well-known chemotaxis biological phenomenon dealing with the movement of an organism in response to a chemical stimulus. The function (x, t) −→ u(x, t) stands for the population density of the organism at position x and time t, and the function (x, t) −→ v(x, t) represents the concentration of the chemical [9]. Also, we cite the Keller-Segel model describing the movement of amoebae with density u, in the presence of a chemoattractant with the concentration v [13].
The literature concerning the existence, uniqueness, large-time behavior of solutions and the blow-up problem is abundant, see [4, 6, 8-10, 13, 15]. For example, Nagai et al. in [9] studied the existence and the uniqueness of bounded solutions to the parabolic system in R n when n ≥ 2. Also, the large time behavior was discussed under the condition sup t>0 u(t) p + v(t) p < ∞ for p = 1, ∞. In fact, it was shown that for every 1 < p ≤ ∞, Under the same condition as mentioned before and n ≥ 2, Nagai and Yamada established in [10] the following decay: Also, it is important to mention the reference [15], where sharp asymptotic expansions are obtained under the conditions (|x| n u 0 ) ∈ L 1 (R n ) and sup x∈R n , t>0 , 0≤µ≤n Now, let us turn to the core of our work, which is an extension of the work reported in the article [12]. The class of higherorder operators L m = m χ m , used in the system (1), is homogeneous and of order 4m. It was shown in [1] that a more general class of operators with non-smooth coefficients of type T 0 = L * 0 aL 0 , where L 0 = |α|=|β|=m a αβ D α+β with constant coefficients a αβ and a ∈ L ∞ (R n , C) with Re(a) ≥ δ > 0, verifies an elliptic condition of De Giorgi type which is equivalent to Gaussian estimates of the kernel and its derivatives. More precisely, the distributional kernel K : (x, t) −→ K(x, t) of the semi-group e −tT0 (i.e. the heat kernel associated with T 0 ) satisfies the following Gaussian estimates: (i) There are constants c 0 > 0, c 1 > 0 such that for all x = (x 1 , · · · , x n ) ∈ R n , all t > 0 and all multi-index γ = (γ 1 , · · · , γ n ) ∈ N n such that |γ| ≤ 2m − 1, one has where |γ| = γ 1 + · · · + γ n and D γ (ii) The Gaussian estimates (2) hold for the kernel of t d dt e −tT0 and as a consequence, the following L p -estimates are obtained on the heat kernel K and its time derivatives: for all p ∈ [1, ∞] with 1/p + 1/q = 1 and all γ ∈ N n such that |γ| ≤ 2m − 1.
On the other hand, by using the estimates (3), among others, it was shown in [7] that the solution w of the heat equation where while p and γ are the same as defined before.
The main objective of this article is to establish L p -estimates for the higher-order gradient of solutions to the system (1) and then to derive an asymptotic behavior for these solutions under some conditions on the nonlinearity f . The solutions of our interest are mild solutions. More precisely, by a solution (u, v) to the system (1), we mean functions u, v ∈ C [0, ∞) , L 1 (R n ) satisfying the Duhamel integral formula or by using the heat kernel K : (x, t) −→ K(x, t) associated with L m , where the symbol * stands for the Laplace convolution operator.
We prove that the solutions and their derivatives decay to 0 when t → ∞ and behave like those of the heat kernel. More precisely, we show that for all multi-index λ ∈ N n such that |λ| + θ ≤ 2m − 1, we have when the function f is dominated by power functions depending on m, n, θ and κ. Alternatively, To obtain these estimates, we rely, in particular, on the following useful result giving L p -estimates of φ: which holds for all p ∈ [1, ∞] and all t > 0. The constant C depends on n, m and p, and p + q = pq. Also, we mention that by using the arguments employed in the forthcoming analysis and the L p -contraction property, valid for all p ∈ [1, ∞], it is possible to handle a more general system of the form: with |g(t)| ≤ C t N . The adaptation and details will be left to the reader. The present work is organized as follows. Section 2 is dedicated to fundamental L p -estimates of higher derivatives of solutions to (1). For the reader's comfort, the computations are divided into three steps. We start by establishing the classical conservation of mass and L p -contraction properties, which are very useful for the sequel. The second step focuses on the estimates of the concentration function v, which are essential for the derivation of estimates on the density function u in the last step. The first part of Section 3 is concerned with the asymptotic behavior of higher-order derivatives using the heat kernel as a regulator factor. The second part of of Section 3 deals with the rapid decay when the initial data φ 0 belongs to L 1 (R n ) ∩ L ∞ (R n ). More precisely, we prove that for λ ∈ N n such that |λ| + θ < 2m − 1, we have depending on the power of the dominant function associated with f while µ depends on n, m, θ and κ.

Fundamental L p -estimates
The objective of this section is to prove that, for 1 ≤ p ≤ ∞ and under conditions on f and the parameters of the system (1), there exists a constant C > 0 such that the coupled solution (u, v) verifies for all t > 0 and all λ ∈ N n such that max |λ| + θ, |λ| + σ, κ + σ ≤ 2m − 1. The first step to prove the estimates (9) concerns essentially some well-known conservation and L 1 -contraction properties.

Conservation and contraction properties
Proof. To obtain the conservation property, we integrate equations in (1) with respect to the space variable x. Firstly, we have and by using the Fourier transform where φ stands for u or v. This leads to d dt R n φ(x, t)dx = 0 and the conservation property (i).
For the proof of the contraction property, a straightforward adaptation of the classical arguments used (for example) in [3] (Theorem 3.1) and [5] (Proposition 1), allows to derive (ii).
It is worth mentioning that we can use the arguments mentioned in Proposition 2.1 to derive the L p -contraction property (8) valid for all p ∈ [1, ∞]. Now, let us state the main results of this section concerning higher-order derivatives estimates for the couple (u, v) solution to the system (1). We derive the estimates of D λ x v, u, and D λ x u.

Higher Gradient estimates of the concentration function
We start with the L p -estimates of D λ x v(t).

Higher Gradient estimates related to the population density function
Now, we move towards the L p -estimates related to the density function u. Proposition 2.3. Let 1 ≤ p , q ≤ ∞ be such that p + q = pq, M = 4m−θ n+κ and suppose that for all t ∈ {s ∈ R/|s| ≤ 1}, Then there is a constant C > 0 such that the function u of (1) verifies the following estimates for all t > 0 and all λ ∈ N n such that max(θ , κ + σ) ≤ 2m − 1.
Proof. The arguments used in more detail above and the inequality (10) lead to the following estimates As before, it follows by using the contraction property and the interpolation inequality that which corresponds to the estimates (12).
Finally, in order to complete the proof of the estimates (9), we extend the above estimates to the higher derivatives for all t > 0 and all λ ∈ N n , provided max |λ| + θ , κ + σ ≤ 2m − 1.
By applying Young's inequality and the estimates (3), (10), (12) to I 1 and I 2 , we derive the following estimates.
(i) Behavior of higher derivatives of u. According to (14), write and According to (4) , we have and then To complete our estimates, we need to show that Observe that || · · · || p ds := I 1 + I 2 .
Since lim t→∞ ||h(|∇ κ v(t)|)|| ∞ = 0 then for all ε > 0, there exists δ > 0 such that for all s ≥ δ and then where C is a constant depending on m and θ. Also, as then we obtain lim t→∞ τ (t) = 0. Similar computations on I 2 provide we have lim t→∞ t n+q|λ| 4mq and this ends the proof of (i) of (15).
(ii) Behavior of higher derivatives of v. According to (11), the same decomposition and the estimates used above for D λ u(t), it is sufficient to show that Observe that the decomposition J ≤ J 1 + J 2 as above, gives ||b(s + 1).∇ σ (D λ K(t − s))|| 1 ||u(s + 1)|| p + ||v(s + 1)|| p ds This implies which completes the proof of (ii) of (15) and hence the proof of Theorem 3.1 is completed.

Sharp large time behavior
Suppose that for all p ∈ [1, ∞], there is a constant C > 0 such that the solution (u, v) of (1) satisfies for all t > 0, where 1/p + 1/q = 1. This condition seems to be valid when u 0 and v 0 are in L 1 (R n ) ∩ L ∞ (R n ) (see for example, Corollary 3.2 and its extension in [3]).