Local null controllability for degenerate parabolic equations with nonlocal term

We establish a local null controllability result for following nonlinear parabolic equation: ut−bx,∫01uuxx+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.