CARLEMAN ESTIMATE AND NULL CONTROLLABILITY OF A CASCADE DEGENERATE PARABOLIC SYSTEM WITH GENERAL CONVECTION TERMS

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1 Electronic Journal of Differential Equations, Vol ), No. 195, pp ISSN: URL: or CARLEMAN ESTIMATE AND NULL CONTROLLABILITY OF A CASCADE DEGENERATE PARABOLIC SYSTEM WITH GENERAL CONVECTION TERMS JIANING XU, CHUNPENG WANG, YUANYUAN NIE Abstract. This article shows Carleman estimate and null controllability of a cascade control system governed by the semilinear degenerate parabolic equations with the general convection terms. The semilinear parabolic equations are weakly degenerate on the boundary and the convection terms cannot be controlled by the diffusion terms. We establish the Carleman estimate and the observability inequality for the linear conjugate system, and prove that the control system is null controllable. 1. Introduction In this article, we study the Carleman estimate and the null controllability of the cascade semilinear degenerate parabolic system with convection terms u t x α u x ) x P 1 x, t, u)) x F 1 x, t, u) = hx, t)χ ω, x, t) Q T, 1.1) v t x α v x ) x P 2 x, t, v)) x F 2 x, t, u, v) =, x, t) Q T, 1.2) u, t) = v, t) =, u1, t) = v1, t) =, t, T ), 1.3) ux, ) = u x), vx, ) = v x), x, 1), 1.4) where < α < 1/2, Q T =, 1), T ), h is the control function, ω, 1) is an interval, χ ω is the characteristic function of ω, u, v L 2, 1), and P 1, P 2, F 1, F 2 are measurable functions. It is noted that 1.1) and 1.2) are degenerate at the boundary x =. The cascade system 1.1) and 1.2) arises in some models from mathematical biology and physics, such as the Keller-Segel model [6] and the Lotka- Volterra model [25]. Controllability theory has been widely investigated for nondegenerate parabolic equations and systems over the last forty years and the known results are almost complete see [3, 4, 5, 16, 18, 19, 2, 21, 22] and the references therein). Recently, controllability theory for degenerate ones has been studied and there have been many results see [1, 2, 7, 8, 9, 1, 11, 12, 13, 14, 15, 17, 23, 24, 26, 27, 28, 29, 31]). But it is far from being solved. The null controllability of the following system, governed by a degenerate diffusion equation, is already investigated u t x α u x ) x cx, t)u = hx, t)χ ω, x, t) Q T, 1.5) 21 Mathematics Subject Classification. 93B5, 93C2, 35K67. Key words and phrases. Carleman estimate; null controllability; convection; degenerate parabolic system. c 218 Texas State University. Submitted April 2, 218. Published December 6,

2 2 J. XU, C. WANG, Y. NIE EJDE-218/195 u, t) = u1, t) = if < α < 1, t, T ), x α u x ), t) = u1, t) = if α 1, t, T ), 1.6) ux, ) = u x), x, 1), 1.7) where α >, c L Q T ). It was shown that the system 1.5) 1.7) is null controllable if < α < 2 [1, 1, 11, 24], while not if α 2 [9]. Although system 1.5) 1.7) is not null controllable for α 2, it was proved in [14, 26, 27, 28] and [7, 8, 9] that it is approximately controllable in L 2, 1) and regional null controllable for each α >, respectively. Flores and Teresa [17] studied the degenerate convection-diffusion equation u t x α u x ) x x α/2 bx, t)u x cx, t)u = hx, t)χ ω, x, t) Q T 1.8) with b L Q T ) and proved that the system 1.8), 1.6) and 1.7) is null controllable for < α < 2. Clearly, the convection term can be controlled by the diffusion term in 1.8). Wang and Du [29] considered the degenerate convection-diffusion equation u t x α u x ) x bx, t)u) x cx, t)u = hx, t)χ ω, x, t) Q T, 1.9) and proved that system 1.9), 1.6) and 1.7) is null controllable if < α < 1/2. Here < α < 1/2 is optimal when one establishes the Carleman estimate in such a way as in [29]. Since 1.9) is degenerate, the convection term can cause essential differences. For example, problem 1.9), 1.6) and 1.7) is well-posed in the weakly degenerate case < α < 1), while may be ill-posed in the strongly degenerate case α 1) see [3, 32]). Moreover, the system u t x α u x ) x bx, t)u x cx, t)u = hx, t)χ ω, x, t) Q T with 1.6) and 1.7) was shown to be null controllable for < α < 1 if b, b x, b xx, b t L Q T ) in [31]. As for systems, the authors in [2] studied the null controllability of system 1.1) 1.4) without convection term, i.e. the case that P 1 = P 2 =. It was shown in [2] that the system is null controllable if < α < 2 a different boundary condition at x = is prescribed if 1 α < 2). Du and Xu [15] considered the linear case of system 1.1) 1.4) when the convection terms can be controlled by the diffusion terms. Furthermore, the Carleman estimate in [15] depends on the derivatives of the coefficients of the convection terms and cannot be used to treat the semilinear case. In this article, we consider the more general system 1.1) 1.4) where the convection terms are independent of the diffusion terms. More precisely, we assume that P 1, P 2, F 1, F 2 are measurable functions satisfying F 2x, t, y, z) y P 1 x, t, ) = P 2 x, t, ) = F 1 x, t, ) = F 2 x, t,, ) =, F 2 x, t,, ) C 1 R 2 ), x, t) Q T, P 1 x, t, y 1 ) P 1 x, t, y 2 ) K y 1 y 2, P 2 x, t, z 1 ) P 2 x, t, z 2 ) K z 1 z 2, F 1 x, t, y 1 ) F 1 x, t, y 2 ) K y 1 y 2, 1.1) 1.11) x, t) Q T, y 1, y 2, z 1, z 2 R, K, x, t) QT, y, z R, 1.12) K, F 2 x, t, y, z) z

3 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 3 F 2 y c or x,x 1),T ) R 2 F 2 y c, 1.13) x,x 1),T ) R 2 where K >, c > and < x < x 1 < 1 such that x, x 1 ) ω. Since 1.1) and 1.2) are degenerate at the boundary x =, the classical solution may not exist and weak solution should be considered. The key to prove the null controllability of system 1.1) 1.4) is the Carleman estimate for its linear conjugate problem. The degeneracy of the equations and the existence of the general convection terms cause essential difficulties for the Carleman estimate. In order to establish the needed Carleman estimate for the degenerate parabolic system, we use the Carleman estimate for a single degenerate parabolic equation in [29], where the auxiliary functions for the Carleman estimate were chosen by the method of undetermined coefficients and it was turned out that only the case < α < 1/2 can be treated in such a way. Using the Carleman estimate in [29] and the classical Carleman estimate, together with some other energy estimates, we establish the Carleman estimate for the linear conjugate problem of problem 1.1) 1.4). Some technical difficulties caused by the degeneracy of the equations and the existence of the general convection terms have to be overcome. After the Carleman estimate, we establish the observability inequality and further prove the null controllability of system 1.1) 1.4). This article is organized as follows. In 2, we introduce the well-posedness of problem 1.1) 1.4) and some a priori estimates. The Carleman estimate and the observability inequality are established in 3. Finally, the null controllability of system 1.1) 1.4) is shown in Well-posedness and some a priori estimates Consider the linear nondegenerate parabolic problem u η t x η) α u η x) x c 1 x, t)u η ) x c 3 x, t)u η c 4 x, t)v η = f 1 x, t), x, t) Q T, v η t x η) α v η x) x c 2 x, t)v η ) x c 5 x, t)u η c 6 x, t)v η = f 2 x, t), x, t) Q T, 2.1) 2.2) u η, t) = v η, t) =, u η 1, t) = v η 1, t) =, t, T ), 2.3) u η x, ) = u x), v η x, ) = v x), x, 1), 2.4) where < α < 1, < η < 1, c i L Q T ) 1 i 6), f 1, f 2 L 2 Q T ), and u, v L 2, 1). The problem 2.1) 2.4) admits a unique solution u η, v η ) with u η, v η L, T ; L 2, 1)) L 2, T ; H 1, 1)). Moreover, u η, v η ) satisfies the following a priori estimates. Lemma 2.1. Assume that < α < 1, < η < 1, c i L Q T ) with c i L Q T ) K 1 i 6), f 1, f 2 L 2 Q T ), and u, v L 2, 1). Then the solution u η, v η ) to problem 2.1) 2.4) satisfies u η L,T ;L 2,1)) x η) α/2 u η x L 2 Q T ) v η L,T ;L 2,1)) x η) α/2 v η x L 2 Q T ) N f 1 L2 Q T ) f 2 L2 Q T ) u L2,1) v L2,1)), 2.5)

4 4 J. XU, C. WANG, Y. NIE EJDE-218/195 u η x, t 2 ) u η x, t 1 ))ςx) dx Nt 2 t 1 f 1 L2 Q T ) f 2 L2 Q T ) u L2,1) v η x, t 2 ) v η x, t 1 ))ςx) dx v L 2,1)) ς H 1,1), t 1 < t 2 T, ς H 1, 1), δ δ u η x, τ δ) u η x, τ)) 2 dx dτ v η x, τ δ) v η x, τ)) 2 dx dτ Nδ 1/2 f 1 2 L 2 Q T ) f 2 2 L 2 Q T ) u 2 L 2,1) v 2 L 2,1) ), < δ < T, 2.6) 2.7) where N > depends only on K, T, and α. Proof. Without loss of generality, it is assumed that u η, v η ) is a smooth solution. Otherwise, one can mollify c i 1 i 6), f 1, f 2, u, v, and prove the lemma by a standard limit process. For convenience, u η and v η are abbreviated as u and v, respectively, in this proof. For < s < T, multiplying 2.1) and 2.2) by u and v, respectively, then integrating over, 1), s) by parts and summing up, we obtain 1 2 = s s K 1 2 Therefore, s s s s u 2 ) t v 2 ) t ) dx dt c 1 uu x c 2 vv x ) dx dt f 1 u f 2 v) dx dt s s uu x vv x ) dx dt 2K f 1 u f 2 v ) dx dt x η) α u 2 x v 2 x) dx dt c 3 u 2 c 4 c 5 )uv c 6 v 2 ) dx dt s x η) α u 2 x v 2 x) dx dt 1 2 K2 s 2K 1 ) s 2 u 2 v 2 ) dx dt 1 2 u 2 v 2 ) dx dt s x η) α u 2 v 2 ) dx dt f 2 1 f 2 2 ) dx dt. u 2 x, s) v 2 x, s)) dx s u 2 x) v 2 x)) dx K 2 s κ α K 2 4K 1) x η) α u 2 x v 2 x) dx dt κ s x η) α u 2 v 2 ) dx dt u 2 v 2 ) dx dt s f 2 1 f 2 2 ) dx dt, 2.8)

5 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 5 where < κ < 1 will be determined. It follows from 2.3) and < α < 1 that s κ s κ = s κ 1 1 α x η) α u 2 x, t) dx dt x η) α x 2 u x x, t) d x) dx dt κ x η) α x ) x ) x η) α d x x η) α u 2 x x, t) d x dx dt x η) 1 2α dx κ η)2 2α η 2 2α 21 α) 2 Similarly, it holds that s κ s κ s x η) α v 2 x, t) dx dt κ η)2 2α η 2 2α 21 α) 2 Take < κ < 1 so small that K 2 Substituting 2.9) 2.11) into 2.8) yields u 2 x, s) v 2 x, s)) dx 1 2 s x η) α u 2 xx, t) dx dt x η) α u 2 xx, t) dx dt. s x η) α v 2 xx, t) dx dt. 2.9) 2.1) max κ γ 1 γ)2 2α γ 2 2α ) 1 α) ) s u 2 x) v 2 x)) dx κ α K 2 4K 1) f 2 1 f 2 2 ) dx dt. x η) α u 2 x v 2 x) dx dt s u 2 v 2 ) dx dt 2.12) Then, 2.5) follows from 2.12) and the Gronwall inequality. For t 1 < t 2 T and ς H 1, 1), multiplying 2.1) and 2.2) by ς, and then integrating over, 1) t 1, t 2 ) by parts, we obtain ux, t 2 ) ux, t 1 ))ςx) dx t2 = t 1 t2 = x η) α u x ς x c 1 uς x c 3 uς c 4 vς f 1 ς) dx dt, vx, t 2 ) vx, t 1 ))ςx) dx t 1 x η) α v x ς x c 2 vς x c 5 uς c 6 vς f 2 ς) dx dt, which, together with the Hölder inequality and 2.5), lead to 2.6). It remains to prove 2.7). For < δ < T and < τ < T δ, multiplying 2.1) and 2.2) by ux, τ δ) ux, τ) and vx, τ δ) vx, τ), respectively, then integrating over

6 6 J. XU, C. WANG, Y. NIE EJDE-218/195 τ, τ δ) with respect to t and summing up, we obtain ux, τ δ) ux, τ)) 2 vx, τ δ) vx, τ)) 2 = τδ τ τδ τ τδ τ τδ τ τδ τ x η) α u x ) x ux, τ δ) ux, τ)) dt x η) α v x ) x vx, τ δ) vx, τ)) dt c 1 u) x ux, τ δ) ux, τ)) dt τδ f 1 c 3 u c 4 v)ux, τ δ) ux, τ)) dt f 2 c 5 u c 6 v)vx, τ δ) vx, τ)) dt. Integrating this equality by parts, over, 1), T δ), yields δ = ux, τ δ) ux, τ)) 2 dx dτ δ τδ τ δ τδ τ δ τδ τ δ τδ τ δ τδ τ ux, τ δ) ux, τ)) dt dx dτ δ τδ τ vx, τ δ) vx, τ)) dt dx dτ τδ T x η) α u 2 x dx dt δ T δ K T τ δ τδ τ δ τδ τ τ δ c 2 v) x vx, τ δ) vx, τ)) dt vx, τ δ) vx, τ)) 2 dx dτ x η) α u x x, t)u x x, τ δ) u x x, τ)) dt dx dτ x η) α v x x, t)v x x, τ δ) v x x, τ)) dt dx dτ c 1 x, t)ux, t)u x x, τ δ) u x x, τ)) dt dx dτ c 2 x, t)vx, t)v x x, τ δ) v x x, τ)) dt dx dτ f 1 x, t) c 3 x, t)ux, t) c 4 x, t)vx, t)) f 2 x, t) c 5 x, t)ux, t) c 6 x, t)vx, t)) x η) α u x x, τ δ) u x x, τ)) 2 dx dτ x η) α v 2 x dx dt x η) α v x x, τ δ) v x x, τ)) 2 dx dτ x η) α u 2 dx dt

7 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 7 δ K T δ T δ T δ δ τδ τ δ τδ τ δ τδ τ δ 2T δ x η) α u x x, τ δ) u x x, τ)) 2 dx dτ x η) α v 2 dx dt x η) α v x x, τ δ) v x x, τ)) 2 dx dτ f 1 K u K v ) 2 dx dt ux, τ δ) ux, τ)) 2 dx dτ f 2 K u K v ) 2 dx dt vx, τ δ) vx, τ)) 2 dx dτ 2KT δ 2KT δ 23T δ 23T δ x η) α u 2 x dx dt 2T δ x η) α u 2 dx dt x η) α v 2 dx dt f1 2 K 2 u 2 K 2 v 2 ) dx dt f2 2 K 2 u 2 K 2 v 2 ) dx dt x η) α v 2 x dx dt x η) α u 2 x dx dt x η) α v 2 x dx dt For < κ < 1 satisfying 2.11), it follows from 2.9) and 2.1) that 1 2K 2 1 2K 2 x η) α u 2 dx dt x η) α u 2 x dx dt κ α x η) α v 2 dx dt x η) α v 2 x dx dt κ α Then, 2.7) follows from 2.13) 2.15) and 2.5). u 2 dx dt, v 2 dx dt. u 2 dx dt v 2 dx dt. 2.13) 2.14) 2.15) Using the a priori estimates in Lemma 2.1, one can prove the well-posedness of problem 1.1) 1.4). Proposition 2.2. Assume that < α < 1, and P 1, P 2, F 1, F 2 satisfy 1.1) 1.13). For h L 2 Q T ) and u, v L 2, 1), problem 1.1) 1.4) admits a unique solution u, v) H α Q T ) H α Q T ). Furthermore, u, v L, T ; L 2, 1))

8 8 J. XU, C. WANG, Y. NIE EJDE-218/195 C w [, T ]; L 2, 1)). Here, H α Q T ) = { w L 2 Q T ) : x α/2 w x L 2 Q T ) } and a function ξ C w [, T ]; L 2, 1)) means that ξx, t)γx) dx C[, T ]) for each γ L 2, 1). The proof of Proposition 2.2 is standard and is omitted here. We refer to [29, Theorem 2.1] for the single equation case. 3. Uniform Carleman estimate and observability inequality In this section, we prove the uniform Carleman estimate and the observability inequality for the linear nondegenerate conjugate system U η t x η) α U η x ) x c 1 x, t)u η x c 3 x, t)u η c 4 x, t)v η =, x, t) Q T, 3.1) V η t x η) α V η x ) x c 2 x, t)v η x c 5 x, t)v η =, x, t) Q T, 3.2) U η, t) = V η, t) =, U η 1, t) = V η 1, t) =, t, T ), 3.3) U η x, T ) = U T x), V η x, T ) = V T x), x, 1), 3.4) where < α < 1/2, < η < 1, c i L Q T ) 1 i 5), and U T, V T L 2, 1). Let ψ C [, 1]) satisfy = 1, x [, 3x 2x 1 )/5], ψ [, 1], x ω, =, x [2x 3x 1 )/5, 1], where ω = 3x 2x 1 )/5, 2x 3x 1 )/5) and ˆω = 4x x 1 )/5, x 4x 1 )/5). Set where ϕx, t) = θt)gx), Ψx, t) = θt) e 2ζ) e ζx)), x, t) Q T, Φx, t) = ψx)ϕx, t) 1 ψx))ψx, t), x, t) Q T, θt) = 1, t, T ), tt t)) 4 gx) = 8x η) 4 2α)/3 8), ζx) = 1 η) 1 α/2 x η) 1 α/2, x, 1). The Carleman estimate for the solution to problem 3.1) 3.4) is as follows. Theorem 3.1 Uniform Carleman estimate). Assume that < α < 1/2, < η < 1, c i L Q T ) with c i L Q T ) K 1 i 5), and c 4 x,x 1),T ) c or c 4 x,x 1),T ) c. There exist two constants s > and M > depending only on x, x 1, K, c, T, and α, such that for each U T, V T L 2, 1), the solution U η, V η ) to problem 3.1) 3.4) satisfies sθu η x ) 2 s 3 θ 3 U η ) 2 sθv η x ) 2 s 3 θ 3 V η ) 2 )e 2sΦ dx dt M U η ) 2 dx dt, s s. ω Proof. For convenience, U η and V η are abbreviated by U and V, respectively, in the proof. Without loss of generality, it is assumed that U, V C 2 Q T ). Set wx, t) = ψx)ux, t), W x, t) = ψx)v x, t), x, t) Q T. 3.5)

9 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 9 Then, w, W ) solves where Set w t x η) α w x ) x = ρ 1, W t x η) α W x ) x = ρ 2, x, t) Q T, 3.6) ρ 1 = ϱ 1 c 1 w x c 3 w, ρ 2 = ϱ 2 c 2 W x c 5 W, x, t) Q T, 3.7) ϱ 1 = x η) α ψ U) x ψ x η) α U x c 1 ψ U c 4 ψv, x, t) Q T, 3.8) ϱ 2 = x η) α ψ V ) x ψ x η) α V x c 2 ψ V, x, t) Q T. 3.9) Y x, t) = e sϕx,t) wx, t), Zx, t) = e sϕx,t) W x, t), x, t) Q T. 3.1) From 3.6) it follows that e sϕ e sϕ Y ) t x η) α e sϕ Y ) x ) x ) = e sϕ ρ 1, x, t) Q T, e sϕ e sϕ Z) t x η) α e sϕ Z) x ) x ) = e sϕ ρ 2, x, t) Q T. From [29, Proposition 3.1], there exist three positive constants M 1, M 2, and s 1 depending only on T and α, such that for each s s 1, M 1 s x η) 4α 2)/3 θy 2 x Z 2 x) dx dt M 2 s 3 ρ 2 1 ρ 2 2)e 2sϕ dx dt. θ 3 Y 2 Z 2 ) dx dt This formula, together with 3.7) and 3.1), leads to that for each s s 1, M 1 s 2 M 3 x η) 4α 2)/3 θy 2 x Z 2 x) dx dt M 2 s 3 ϱ 2 1 ϱ 2 2)e 2sϕ dx dt θ 3 Y 2 Z 2 ) dx dt Y 2 x Z 2 x Y 2 Z 2 s 2 x η) 21 2α)/3 θ 2 Y 2 Z 2 )) dx dt, where M 3 > depends only on K, T, and α. Therefore, there exist s 2 s 1 and M 4 > depend only on K, T, and α, such that for each s s 2, s s 3 M 4 x η) 4α 2)/3 θe sϕ Y ) 2 x e sϕ Z) 2 x)e 2sϕ dx dt θ 3 Y 2 Z 2 ) dx dt ϱ 2 1 ϱ 2 2)e 2sϕ dx dt. From 3.11), < α < 1/2, 3.8) 3.1) and 3.5), we obtain sθw 2 x s 3 θ 3 w 2 sθw 2 x s 3 θ 3 W 2 )e 2sϕ dx dt M 5 U 2 Ux 2 V 2 Vx 2 )e 2sϕ dx dt ω 3.11) ) W 2 e 2sϕ dx dt, s s 2,

10 1 J. XU, C. WANG, Y. NIE EJDE-218/195 where M 5 > depends only on K, T, x, x 1, and α. Therefore, there exist s 3 s 2 and M 6 > depend only on K, T, x, x 1, and α, such that for each s s 3, sθw 2 x s 3 θ 3 w 2 sθw 2 x s 3 θ 3 W 2 )e 2sϕ dx dt M 6 U 2 Ux 2 V 2 Vx 2 )e 2sϕ dx dt. ω This formula, together with 3.5) and the definition of ψ, leads to that for each s s 3, Set 3x2x 1)/5 sθu 2 x s 3 θ 3 U 2 sθv 2 x s 3 θ 3 V 2 )e 2sΦ dx dt M 6 U 2 Ux 2 V 2 Vx 2 )e 2sϕ dx dt. ω qx, t) = 1 ψx))ux, t), Qx, t) = 1 ψx))v x, t), x, t) Q T. 3.12) From the classical Carleman estimate [1, Proposition 4.2], there exist s 4 > and M 7 > depend only on K, T, x, x 1, and α, such that for each s s 4, sθe ζ q 2 x s 3 θ 3 e 3ζ q 2 sθe ζ Q 2 x s 3 θ 3 e 3ζ Q 2 )e 2sΨ dx dt M 7 U 2 Ux 2 V 2 Vx 2 )e 2sΨ dx dt. ω This formula, and the definition of ψ, leads to 2x 3x 1)/5 sθe ζ U 2 x s 3 θ 3 e 3ζ U 2 sθe ζ V 2 x s 3 θ 3 e 3ζ V 2 )e 2sΦ dx dt M 7 U 2 Ux 2 V 2 Vx 2 )e 2sΨ dx dt, s s 4. ω It follows from 3.12), 3.13) and the definition of Φ, ϕ, Ψ, ψ that sθu 2 x s 3 θ 3 U 2 sθv 2 x s 3 θ 3 V 2 )e 2sΦ dx dt 3.13) M 8 U 2 Ux 2 V 2 Vx 2 )e 2sϕ e 2sΨ e 2sΦ ) dx dt 3.14) ω 3M 8 U 2 Ux 2 V 2 Vx 2 )e 2sΦ dx dt, s max{s 3, s 4 }, ω where M 8 > depends only on K, T, x, x 1, and α. Let ξ 1 C [, 1]) such that supp ξ 1 ˆω, ξ 1 1 in, 1) and ξ 1 1 in ω. For s >, 3.1) 3.3) show = = 2s d dt 2 ξ 2 1U 2 V 2 )e 2sΦ dx dt ξ 2 1Φ t U 2 V 2 )e 2sΦ dx dt ξ 2 1x η) α U 2 x x η) α V 2 x )e 2sΦ dx dt

11 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY s 2 ξ 1 ξ 1Ux η) α U x V x η) α V x )e 2sΦ dx dt ξ 2 1c 1 UU x c 2 V V x )e 2sΦ dx dt ξ 2 1Φ x Ux η) α U x V x η) α V x )e 2sΦ dx dt ξ 2 1c 3 U 2 c 5 V 2 c 4 UV )e 2sΦ dx dt, which leads to ξ 2 1x η) α U 2 x x η) α V 2 x )e 2sΦ dx dt 1 ξ 2 1x 2 η) α Ux 2 x η) α Vx 2 )e 2sΦ dx dt M 9 1 s 2 ) θ 2 U 2 V 2 )e 2sΦ dx dt, ˆω with M 9 > depending only on K, T, x, x 1, and α. Hence, for s >, Ux 2 Vx 2 )e 2sΦ dx dt M 1 1 s 2 ) ω θ 2 U 2 V 2 )e 2sΦ dx dt, 3.15) ˆω where M 1 > depends only on K, T, x, x 1, and α. From 3.14) and 3.15), there exist s 5 max{s 3, s 4 } and M 11 > depend only on K, T, x, x 1, and α, such that for each s s 5, sθux 2 s 3 θ 3 U 2 sθvx 2 s 3 θ 3 V 2 )e 2sΦ dx dt 3.16) M 11 U 2 V 2 )e 2sΦ dx dt. ˆω Let ξ 2 C [, 1]) such that supp ξ 2 x, x 1 ), ξ 2 1 in, 1) and ξ 2 1 in ˆω. Multiplying 3.1) by ξ 2 V e 2sΦ, then integrating by parts and using 3.2), one gets = 2 c 4 ξ 2 V 2 e 2sΦ dx dt x η) α ξ 2 U x V x e 2sΦ dx dt c 2 ξ 2 UV x e 2sΦ dx dt x η) α ξ 2 e 2sΦ ) x ) x e 2sΦ) UV e 2sΦ dx dt. c 1 ξ 2 U x V e 2sΦ dx dt c 3 c 5 2sΦ t )ξ ) The Hölder inequality gives 1 2 x η) α ξ 2 U x V x e 2sΦ dx dt θu 2 xe 2sΦ dx dt 1 2 θ 1 V 2 x e 2sΦ dx dt, 3.18)

12 12 J. XU, C. WANG, Y. NIE EJDE-218/195 and s2 ) c 1 ξ 2 U x V e 2sΦ dx dt θu 2 xe 2sΦ dx dt 1 2 K2 c 2 ξ 2 UV x e 2sΦ dx dt θvx 2 e 2sΦ dx dt 1 2 K2 θ 1 V 2 e 2sΦ dx dt, ω θ 1 U 2 e 2sΦ dx dt, c 3 c 5 2sΦ t )ξ 2 x η) α ξ 2 e 2sΦ ) x ) x e 2sΦ )UV e 2sΦ dx dt M 12 1 s 2 ) θ 3 V 2 e 2sΦ dx dt ω θ 3 U 2 e 2sΦ dx dt, 3.19) 3.2) 3.21) where M 12 > depends only on K, T, x, x 1, and α. Since c 4 x,x 1),T ) c or c 4 x,x 1),T ) c, it follows from 3.17) 3.21) and the definition of ξ 2 that c 3 2 V 2 e 2sΦ dx dt ˆω s2 ) M 12 1 s 2 ) θu 2 xe 2sΦ dx dt 1 2 θ 1 V 2 x e 2sΦ dx dt 1 2 K2 θ 3 V 2 e 2sΦ dx dt ω θv 2 x e 2sΦ dx dt θ 3 U 2 e 2sΦ dx dt. Then the theorem follows from 3.16) and 3.22). θ 1 U 2 V 2 )e 2sΦ dx dt 3.22) Below we prove the observability inequality for the solution to problem 3.1) 3.4). Theorem 3.2 Uniform observability inequality). Assume that < α < 1/2, < η < 1, c i L Q T ) with c i L Q T ) K 1 i 5), and c 4 x,x 1),T ) c or c 4 x,x 1),T ) c. There exists M > depending only on x, x 1, K, c, T, and α, but independent of η, such that for each U T, V T L 2, 1), the solution U η, V η ) to problem 3.1) 3.4) satisfies U η ) 2 x, ) V η ) 2 x, ))dx M U η ) 2 dx dt. Proof. As in Theorem 3.1, U η and V η are abbreviated by U and V, respectively, and it is assumed that U, V C 2 Q T ). Multiplying 3.1) and 3.2) by U and V, ω

13 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 13 respectively, and then integrating over, 1) with respect to x, we obtain 1 d 2 dt =, 1 d 2 dt U 2 dx V 2 dx x η) α U 2 xdx x η) α V 2 x dx c 1 UU x dx c 2 V V x dx c 3 U 2 dx c 5 V 2 dx =, for t, T ). Using the Hölder inequality and the Hardy inequality yields d dt U 2 V 2 ) dx M where M > depends only on K and α. Hence U 2 x, ) V 2 x, )) dx e Mt Integrating 3.23) over [T/4, 3T/4] leads to T 2 U 2 x, ) V 2 x, )) dx e 3 MT/4 U 2 V 2 ) dx, t, T ), c 4 UV dx U 2 x, t) V 2 x, t)) dx, t, T ). 3.23) 3T/4 T/4 U 2 V 2 ) dx dt. 3.24) The theorem follows from 3.24), the Hardy inequality and Theorem 3.1. By a standard limit process, one can get the Carleman estimate and the observability inequality for the degenerate parabolic system U t x α U x ) x c 1 x, t)u x c 3 x, t)u c 4 x, t)v =, x, t) Q T, 3.25) V t x α V x ) x c 2 x, t)v x c 5 x, t)v =, x, t) Q T, 3.26) U, t) = V, t) =, U1, t) = V 1, t) =, t, T ), 3.27) Ux, T ) = U T x), V x, T ) = V T x), x, 1), 3.28) where < α < 1/2, c i L Q T ) 1 i 5), and U T, V T L 2, 1). Theorem 3.3. Assume that < α < 1/2, < η < 1, c i L Q T ) with c i L Q T ) K 1 i 5), and c x,x 4 c 1),T ) or c x,x 4 c 1),T ). There exist three positive constants s, M and M depending only on x, x 1, K, c, T, and α, such that for each U T, V T L 2, 1), the solution U, V ) to problem 3.25) 3.28) satisfies where M sθu 2 x s 3 θ 3 U 2 sθv 2 x s 3 θ 3 V 2 )e 2s Φ dx dt ω U 2 dx dt, s s, U 2 x, ) V 2 x, ))dx M ω U 2 dx dt, Φx, t) = 8θt)ψx)x 4 2α)/3 8) θt)1 ψx)) e 2 e 1 x1 α/2), for x, t) Q T.

14 14 J. XU, C. WANG, Y. NIE EJDE-218/ Null controllability In this section, we study the null controllability of system 1.1) 1.4). consider the nondegenerate parabolic system First u η t x η) α u η x) x P 1 x, t, u η )) x F 1 x, t, u η ) = h η χ ω, x, t) Q T, 4.1) v η t x η) α v η x) x P 2 x, t, v η )) x F 2 x, t, u η, v η ) =, x, t) Q T, 4.2) u η, t) = v η, t) =, u η 1, t) = v η 1, t) =, t, T ), 4.3) u η x, ) = u x), v η x, ) = v x), x, 1), 4.4) where < α < 1/2, < η < 1, and u, v L 2, 1). Lemma 4.1. Assume that < α < 1/2, < η < 1, and P 1, P 2, F 1, F 2 satisfy 1.1) 1.13). For each u, v L 2, 1), there exists h η L 2 Q T ), such that the solution u η, v η ) to problem 4.1) 4.4) satisfies u η x, T ) = v η x, T ) =, x, 1). 4.5) Furthermore, there exists M > depending only on x, x 1, K, c, T, and α, such that h η L2 Q T ) M u L2,1) v L2,1)). 4.6) Proof. For x, t, y, z) Q T R 2, set { P1x,t,y) P 1x,t,) y, y, c 1 x, t, y) =, y =, { P2x,t,z) P 2x,t,) c 2 x, t, z) = z, z,, z =, { F1x,t,y) F 1x,t,) y, y, c 3 x, t, y) =, y =, c 4 x, t, y, z) = F 2 y x, t, λy, λz) dλ, c 5x, t, y, z) = F 2 x, t, λy, λz) dλ. z Then, 1.11) 1.13) show that c 1, c 2, c 3 L Q T R) and c 4, c 5 L Q T R 2 ) satisfy c i L Q T R) K i = 1, 2, 3) c j L Q T R 2 ) K j = 4, 5), c 4 x,x 1),T ) R 2 c or c 4 x,x 1),T ) R 2 c. Let y, z L 2 Q T ). For ε >, consider the problem { min h 2 dx dt 1 u 2 x, T ) dx 1 } v 2 x, T ) dx : h L 2 Q T ), 4.7) ε ε where u, v) is the solution to the problem u t x η) α u x ) x c 1 x, t, yx, t))u) x c 3 x, t, yx, t))u = hχ ω, x, t) Q T, v t x η) α v x ) x c 2 x, t, zx, t))v) x c 4 x, t, yx, t), zx, t))u c 5 x, t, yx, t), zx, t))v =, x, t) Q T, 4.8) 4.9)

15 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 15 u, t) = v, t) =, u1, t) = v1, t) =, t, T ), 4.1) ux, ) = u x), vx, ) = v x), x, 1). 4.11) As shown in [5], one can prove that problem 4.7) 4.11) admits a unique solution h η,ε,y,z = U η,ε,y,z χ ω, where U η,ε,y,z, V η,ε,y,z ) is the solution to the problem U η,ε,y,z t x η) α Ux η,ε,y,z ) x c 1 x, t, yx, t))ux η,ε,y,z c 3 x, t, yx, t))u η,ε,y,z c 4 x, t, yx, t), zx, t))v η,ε,y,z =, x, t) Q T, 4.12) V η,ε,y,z t x η) α Vx η,ε,y,z ) x c 2 x, t, zx, t))vx η,ε,y,z c 5 x, t, yx, t), zx, t))v η,ε,y,z =, x, t) Q T, 4.13) U η,ε,y,z, t) = V η,ε,y,z, t) =, U η,ε,y,z 1, t) = V η,ε,y,z 1, t) =, t, T ), 4.14) U η,ε,y,z x, T ) = 1 ε uη,ε,y,z x, T ), V η,ε,y,z x, T ) = 1 ε vη,ε,y,z x, T ), x, 1), 4.15) with u η,ε,y,z, v η,ε,y,z ) solving problem 4.8) 4.11) for h = h η,ε,y,z. Multiplying 4.8) with h = h η,ε,y,z, 4.9), 4.12) and 4.13) by U η,ε,y,z, V η,ε,y,z, u η,ε,y,z and v η,ε,y,z, respectively, and then integrating over Q T by parts, we obtain 1 ε = h η,ε,y,z χ ω U η,ε,y,z dx dt 1 ε v η,ε,y,z x, T )) 2 dx U η,ε,y,z x, )u x) dx u η,ε,y,z x, T )) 2 dx V η,ε,y,z x, )v x) dx. 4.16) It follows from 4.16), h η,ε,y,z = U η,ε,y,z χ ω, the Hölder inequality and Theorem 3.2 that h η,ε,y,z ) 2 dx dt 1 u η,ε,y,z x, T )) 2 dx 1 v η,ε,y,z x, T )) 2 dx ε ε M 2 M 2 U η,ε,y,z x, )u x) dx u 2 x) vx)) 2 dx 1 2M u 2 x) v 2 x)) dx 1 2 V η,ε,y,z x, )v x) dx U η,ε,y,z x, )) 2 V η,ε,y,z x, )) 2 ) dx h η,ε,y,z ) 2 dx dt, where M > depending only on x, x 1, K, c, T and α, is given in Theorem 3.2. Hence h η,ε,y,z ) 2 dx dt 2 u η,ε,y,z x, T )) 2 dx 2 v η,ε,y,z x, T )) 2 dx ε ε M u 2 x) v 2 x)) dx, 4.17)

16 16 J. XU, C. WANG, Y. NIE EJDE-218/195 which, together with Lemma 2.1, leads to u η,ε,y,z L,T ;L 2,1)) x η) α/2 u η,ε,y,z x L 2 Q T ) v η,ε,y,z L,T ;L 2,1)) x η) α/2 v η,ε,y,z x L2 Q T ) N h η,ε,y,z L 2 Q T ) u L 2,1) v L 2,1)) NM 1/2 1) u L 2,1) v L 2,1)), δ δ u η,ε,y,z x, τ δ) u η,ε,y,z x, τ)) 2 dx dτ v η,ε,y,z x, τ δ) v η,ε,y,z x, τ)) 2 dx dτ Nδ 1/2 h η,ε,y,z 2 L 2 Q T ) u 2 L 2,1) v 2 L 2,1) ) Nδ 1/2 M 1) u 2 L 2,1) v 2 L 2,1)), < δ < T, where N > depending only on K, T and α, is given in Lemma 2.1. Define with Λ ε : y, z) u η,ε,y,z, v η,ε,y,z ), y, z B R = {w L 2 Q T ) : w L 2 Q T ) R} R = NT 1/2 M 1/2 1) u L 2,1) v L 2,1)). 4.18) 4.19) It follows from 4.18) and 4.19) that Λ ε is a mapping from B R to B R, and Λ ε is compact and continuous. Therefore, the Schauder fixed point theorem yields that Λ ε admits a fixed point u η,ε, v η,ε ) B R B R, which solves problem 4.1) 4.4) with h = h η,ε = h η,ε,uη,ε,v η,ε. Moreover, 4.17) 4.19) yield M h η,ε ) 2 dx dt 2 ε u 2 x) v 2 x)) dx, u η,ε x, T )) 2 dx 2 ε v η,ε x, T )) 2 dx u η,ε L,T ;L 2,1)) x η) α/2 u η,ε x L 2 Q T ) v η,ε L,T ;L 2,1)) x η) α/2 v η,ε x L2 Q T ) NM 1/2 1) u L 2,1) v L 2,1)), δ δ u η,ε x, τ δ) u η,ε x, τ)) 2 dx dτ v η,ε x, τ δ) v η,ε x, τ)) 2 dx dτ Nδ 1/2 M 1) u 2 L 2,1) v 2 L 2,1) ). Then there is ε n, 1) with lim n ε n =, E m Q T with lim m meas E m =, h η L 2 Q T ) and u η, v η L, T ; L 2, 1)) L 2, T ; H 1, 1)), such that h η,εn h η, u η,εn u η, v η,εn v η, u η,εn x u η x, vx η,εn vx η in L 2 Q T ) as n, u η,εn u η, v η,εn v η uniformly in Q T \E m as n for each positive integer m, 4.2) 4.21) u η x, T ) = v η x, T ) =, x, 1), 4.22)

17 EJDE-218/195 CARLEMAN ESTIMATE AND NULL CONTROLLABILITY 17 h η ) 2 dx dt M It follows from 1.11), 1.12) and 4.21) that u 2 x) v 2 x)) dx. 4.23) P 1 x, t, u η,εn ) P 1 x, t, u η ), P 2 x, t, v η,εn ) P 2 x, t, v η ), F 1 x, t, u η,εn ) F 1 x, t, u η ), F 2 x, t, u η,εn, v η,εn ) F 2 x, t, u η, v η ) in L 2 Q T ) as n. 4.24) From 4.2) and 4.24), one can show that u η, v η ) is the solution to problem 4.1) 4.4). Finally, 4.5) and 4.6) follow from 4.22) and 4.23). Now we are ready to prove the null controllability of the degenerate parabolic system 1.1) 1.4). Theorem 4.2. Assume that < α < 1/2, and P 1, P 2, F 1, F 2 satisfy 1.1) 1.13). The system 1.1) 1.4) is null controllable. More precisely, for each u, v L 2, 1), there exists h L 2 Q T ), such that the solution u, v) to problem 1.1) 1.4) satisfies ux, T ) = vx, T ) =, x, 1). 4.25) Furthermore, there exists M > depending only on x, x 1, K, c, T, and α, such that h L2 Q T ) M u L2,1) v L2,1)). 4.26) Proof. For each < η < 1, Lemma 4.1 shows that there exists h η L 2 Q T ) with h η L2 Q T ) M u L2,1) v L2,1)), 4.27) such that the solution u η, v η ) to problem 4.1) 4.4) satisfies u η x, T ) = v η x, T ) =, x, 1), 4.28) where M > depends only on x, x 1, K, c, T, and α. Rewrite 4.1) and 4.2) into u η t x η) α u η x) x c 1 x, t)u η ) x c 3 x, t)u η = h η χ ω, x, t) Q T, v η t x η) α v η x) x c 2 x, t)v η ) x c 4 x, t)u η c 5 x, t)v η =, x, t) Q T, where for x, t) Q T, c 1 x, t) = c 2 x, t) = c 3 x, t) = c 4 x, t) = c 5 x, t) = { P1x,t,u η x,t)) P 1x,t,) u η x,t), u η x, t),, u η x, t) =, { P2x,t,v η x,t)) P 2x,t,) v η x,t), v η x, t),, v η x, t) =, { F1x,t,u η x,t)) F 1x,t,) u η x,t), u η x, t),, u η x, t) =, F 2 y x, t, λuη x, t), λv η x, t)) dλ, F 2 z x, t, λuη x, t), λv η x, t)) dλ.

18 18 J. XU, C. WANG, Y. NIE EJDE-218/195 Then, c i L Q T ) with c i L Q T ) K 1 i 5). Lemma 2.1 yields u η L,T ;L 2,1)) x η) α/2 u η x L 2 Q T ) v η L,T ;L 2,1)) x η) α/2 v η x L 2 Q T ) N h η L2 Q T ) u L2,1) v L2,1)), δ δ u η x, τ δ) u η x, τ)) 2 dx dτ v η x, τ δ) v η x, τ)) 2 dx dτ Nδ 1/2 h η 2 L 2 Q T ) u 2 L 2,1) v 2 L 2,1)), < δ < T, 4.29) 4.3) where N > depending only on K, T and α, is given in Lemma 2.1. By 4.27) 4.3), there exist η n, 1) with lim n η n =, E m Q T with lim m meas E m =, h L 2 Q T ) and u, v L, T ; L 2, 1)) H α, such that h ηn h, u ηn u, v ηn v, x η) α/2 u ηn x x α/2 u x, x η) α/2 vx ηn x α/2 v x in L 2 Q T ) as n, u ηn u, v ηn v uniformly in Q T \E m as n for each positive integer m, 4.31) 4.32) ux, T ) = vx, T ) =, x, 1), 4.33) h 2 dx dt M It follows from 1.11), 1.12) and 4.32) that P 1 x, t, u ηn ) P 1 x, t, u), u 2 x) v 2 x)) dx. 4.34) P 2 x, t, v ηn ) P 2 x, t, v), F 1 x, t, u ηn ) F 1 x, t, u), F 2 x, t, u ηn, v ηn ) F 2 x, t, u, v) 4.35) in L 2 Q T ) as n. From 4.31) and 4.35), one can show that u, v) solves problem 1.1) 1.4). Finally, 4.25) and 4.26) follow from 4.33) and 4.34). Acknowledgments. This research was supported by the National Natural Science Foundation of China No and ), the Natural Science Foundation for Young Scientists of Jilin Province No JH), and by the Scientific and Technological project of Jilin Provinces Education Department in Thirteenth Five-Year No. JJKH218114KJ). The authors would like to express their sincerely thanks to the referees and to the editor for their helpful comments on the original version of the paper. References [1] F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli; Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 62) 26), [2] E. M. Ait, B. Hassi, F. Ammar Khodja, A. Hajjaj, L. Maniar; Null controllability of degenerate parabolic cascade systems, Portugal. Math., 683) 211), [3] F. Ammar Khodja, A. Benabdallah, C. Dupaix, I. Kostin; Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 425) 23), [4] F. Ammar Khodja, A. Benabdallah, C. Dupaix, I. Kostin; Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., 113) 25),

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20 2 J. XU, C. WANG, Y. NIE EJDE-218/195 Jianing Xu School of Mathematics, Jilin University, Changchun 1312, China address: Chunpeng Wang corresponding author) School of Mathematics, Jilin University, Changchun 1312, China address: Yuanyuan Nie School of Mathematics, Jilin University, Changchun 1312, China address: