LOCAL NULL CONTROLLABILITY FOR DEGENERATE PARABOLIC EQUATIONS WITH NONLOCAL TERM R. DEMARQUE * , J. L ´ IMACO, AND L. VIANA Abstract. We establish a local null controllability result for following the nonlinear parabolic equation: ut -  b  x,  1 0 u  ux  x + f (t, x, u)= hχω, (t, x) ∈ (0,T ) × (0, 1) where b(x, r)= ℓ(r)a(x) is a function with separated variables that defines an operator which degenerates at x = 0 and has a nonlocal term. Our approach relies on an application of Liusternik’s inverse mapping theorem that demands the proof of a suitable Carleman estimate. 1. Introduction In this paper we study the null controllability for the degenerate parabolic problem      u t −  b  x,  1 0 u  u x  x + f (t,x,u)= hχ ω , u(t, 1) = u(t, 0) = 0, u(0,x)= u 0 (x), (1.1) where T> 0 is given, (t,x) ∈ (0,T ) × (0, 1), u 0 ∈ L 2 (0, 1) and h ∈ L 2 ((0,T ) × (0, 1)) is a control that acts on the system through ω =(α,β) ⊂⊂ (0, 1). We also specify some properties of b and f : A.1. Let ℓ : R → R be a C 1 function with bounded derivative and suppose that ℓ(0) = 1. We also consider a ∈ C ([0, 1]) ∩ C 1 ((0, 1]) satisfying a(0) = 0, a> 0 on (0, 1], a ′ ≥ 0 and xa ′ (x) ≤ Ka(x), ∀x ∈ [0, 1] and some K ∈ [0, 1). (1.2) The function b : [0, 1] × R → R is defined by b(x,r)= ℓ(r)a(x). Remark 1.1. Let α ∈ (0, 1), then a typical example of function satisfying (A.1) is a(x)= x α . If we define β = arctan(α), then an other example is a(x)= x α cos(βx). A.2. Let f : [0,T ] × [0, 1] × R → R be a C 1 function with bounded derivatives such that f (t,x, 0) = 0. We suppose that c = c(t,x) := D 3 f (t,x, 0) ∈ L ∞ ((0,T ) × (0, 1)) The main goal of this work is to prove that there exists h ∈ L 2 ((0,T ) × (0, 1)) such that the associated state u = u(t,x) of (1.1) satisfies u(T,x) ≡ 0 for any x ∈ [0, 1], at least if ‖u 0 ‖ H 1 a is sufficiently small, where H 1 a is a suitable weighted Hilbert space which will be defined later. 2010 Mathematics Subject Classification. Primary 35K65, 93B05; Secondary 35K55. Key words and phrases. Degenerate parabolic equations, controllability, nonlinear parabolic equations, nonlocal term. * Partially supported by FAPERJ E-26/111.039/2013 and Proppi/PDI/UFF. 1 arXiv:1612.08774v2 [math.AP] 19 Apr 2018