Local null controllability for degenerate parabolic equations with nonlocal term

We establish a local null controllability result for following the nonlinear parabolic equation: $$u_t-{\left(b(x,{\int }_0^{1}u)u_x\right)}_x+f(t,x,u)=h{\chi }_{\omega },(t,x)\in (0,T)\times (0,1)$$ where $b(x,r)=\mathrm{\ell }(r)a(x)$ 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.