Null controllability of a semilinear degenerate parabolic equation with a gradient term

This paper concerns the null controllability of a semilinear control system governed by degenerate parabolic equation with a gradient term, where the nonlinearity of the problem is involved with the first derivative. We first establish the well-posedness and prove the approximate null controllability of the linearized system, then we can get the approximate null controllability of the semilinear control system by a fixed point argument. Finally, the semilinear control system with a gradient term is shown to be null controllable.


Introduction
In this paper, we investigate the null controllability of the following semilinear degenerate system: u tx α u x x + g(x, t, u, u x ) = h(x, t)χ ω , (x, t) ∈ Q T , (1.1) u(0, t) = u(1, t) = 0 if 0 < α < 1, t ∈ (0, T), (1.2) x α u x (0, t) = u(1, t) = 0 if 1 ≤ α < 2, t ∈ (0, T), (1.3) u(x, 0) = u 0 (x), x ∈ (0, 1), (1.4) where Q T = (0, 1) × (0, T), ω is a nonempty open subset of (0,1), u 0 ∈ L 2 (0, 1), h(x, t) ∈ L 2 (Q T ) is a control function, g(x, t, s, p) is Lebesgue measurable in Q T × R × R and C 1 continuous with respect to s, p uniformly for (x, t) ∈ Q T . Furthermore, we assume that g satisfies g(·, ·, 0, 0) = 0 and g s (x, t, s, p) + x -α/2 g p (x, t, s, p) ≤ K, ∀(x, t, s, p) ∈ Q T × R × R, (1.5) example of a Crocco-type equation coming from the study on the velocity field of a laminar flow on a flat plate. In the last forty years, many authors have been devoted to studying control systems, the interested readers can refer to  and the references therein. For instance, Wang in [27][28][29] studied the approximate controllability of a class of systems governed by degenerate parabolic equations. In 2013, Du and Wang in [11] investigated the null controllability of a class of coupled degenerate systems. Later, Du and Xu in [13] studied the boundary controllability of a semilinear degenerate system with convection term. Recently, Xu, Wang and Nie in [30] considered the Carleman estimate and null controllability of a cascade control system with convection terms. For degenerate equations, one must overcome some technical difficulties to get some necessary estimates for controllability theory. In particular, the following system governed by a single degenerate parabolic equation has been widely studied: where k ∈ L ∞ (Q T ). The system is null controllable if 0 < α < 2 [8,9,26], while not if α ≥ 2 [7]. It is noted that the degeneracy of (1.6) is weak if 0 < α < 1 and strong if α ≥ 1. The null controllability of system (1.6)-(1.9) for 0 < α < 2 is based on the Carleman estimate for solutions to its conjugate problem Since the problem may be not null controllable, the authors introduced some new concepts on controllability, the regional null controllability and the persistent regional null controllability, which is weaker than the null controllability [7]. They proved that the problem is regional null controllable and persistent regional null controllable for all α > 0. For semilinear problem (1.1)-(1.4), the authors also showed the regional and persistent regional null controllability in [3,5]. Moreover, the approximate controllability of degenerate equation (1.1) with suitable boundary and initial conditions has been proved in [12,[27][28][29] for all α > 0. In [1,19], the authors proved the null controllability of problem (1.1)-(1.4) with respectively.
In this paper, we investigate the null controllability of semilinear problem (1.1)-(1.4). First, we prove the approximate null controllability of linear problem (1.1)-(1.4) with (1.14). Next, we prove the approximate null controllability of semilinear problem (1.1)-(1.4) by using the Schauder fixed point theorem. At last, we state the null controllability of semilinear problem (1.1)-(1.4) with the method inspired by [3]. The paper is organized as follows: In Sect. 2, we introduce function spaces that are needed for the well-posedness and prove the well-posedness of system (1.1)-(1.4). In Sect. 3, we prove that the semilinear system is null controllable.
Due to the degeneracy of the coefficient x α , problem (2.1)-(2.4) may not have classical solutions, so we need to give the definition of weak solutions.
then the function u is called the weak solution of the system (2.1)-(2.4).
where C > 0 is a constant depending on T, C 1 , and C 2 . (ii) If u 0 ∈ H 1 α (0, 1), then u ∈ N and it holds that where C > 0 is a constant depending on T, C 1 , and C 2 .
Similar to the linear problem (2.1)-(2.4), one can give the definition of weak solution to the following semilinear problem: Definition 2.2 A function u is called the weak solution of the problem (2.5)-(2.8) if u ∈ M, and for any function ϕ ∈ M with ϕ t ∈ L 2 (Q T ) and ϕ(·, T)| (0,1) = 0, the following integral equality holds: The semilinear problem (2.5)-(2.8) is well posed, which is proved in Theorem 3.1 [12] and Theorem 3.7 [3].
For any w ∈ L 2 (0, T; H 1 α (0, 1)), define the functions Then (1.5) yields that Moreover, we can obtain that Furthermore, c(x, t, w) and b(x, t, w) satisfy the following property.
Proof For convenience, we denote First, we will prove (2.14) For each δ > 0, let Combined lim δ→0 meas E δ = 0 with (2.9), we only need to prove For any integers m, j > 0, denote  Fix ϕ(x, t) ∈ L 1 (Q T ), for any integer n > 0, we can deduce from (2.9) that Let k → ∞ and then n → ∞ in (2.21) and (2.22), it follows from ϕ ∈ L 1 (Q T ) and (2.14) that The convergence for b[w](x, t) can be proved similarly, the proof is complete.
Proof We divide the proof into two steps.
Step 1. Let us prove the existence of the weak solution to the problem by using the Schauder fixed point theorem. It follows from Proposition 2.1 that the problem
Step 2. Let us prove the uniqueness of the weak solution. Assume that u and v are two weak solutions to the problem (2.5)-(2.8) and set Then w(x, t) is the solution to the following problem: It follows from Proposition 2.1 that The proof is complete.

Null controllability
In this section, we first consider the approximate null controllability of the linear problem 2) where b, c ∈ L ∞ (Q T ), h ∈ L 2 (Q T ), and u 0 ∈ H 1 α (0, 1).

Theorem 3.1 The problem (3.1)-(3.4)
is approximately null controllable, which means that, for any ε > 0, there exists a function h ε ∈ L 2 (Q T ) such that

5)
where C > 0 is a constant independent of ε and u ε is the solution of (3.1)-(3.4) with h = h ε .
Proof Define a functional where u is the solution to problem (3.1)- (3.4). It is not hard to prove that the functional has a unique minimum point where ϕ ε is the solution to the conjugate problem Multiplying (3.7) by u ε and then integrating by parts, one can get that A combination of (3.6), (3.10) and (3.11) implies that As shown in Lemma 3.1 [19], there exists a constant C such that Using Hölder's inequality with (3.12) and (3.13), one has , thus (3.5) holds and the proof is complete.  4) is approximately null controllable, it means that, for any ε > 0, there exists a control function h ε ∈ L 2 (Q T ) such that where C > 0 is a constant independent of ε and u ε is the solution of (1.
Proof For any w ∈ M, we first consider the following problem: