Local null controllability for degenerate parabolic equations with nonlocal term

TL;DR

This paper proves local null controllability for a nonlinear degenerate parabolic equation with a nonlocal term, using Carleman estimates and inverse mapping theorem, addressing challenges posed by degeneracy and nonlocality.

Contribution

It introduces a novel controllability result for degenerate parabolic equations with nonlocal terms, employing Carleman estimates and inverse mapping techniques.

Findings

Established local null controllability for the specified nonlinear degenerate parabolic equation.

Developed a Carleman estimate suitable for equations with degeneracy and nonlocal terms.

Applied inverse mapping theorem to prove controllability result.

Abstract

We establish a local null controllability result for following the nonlinear parabolic equation: $$u_t-\left(b\left(x,\int_0^1u \ \right)u_x \right)_x+f(t,x,u)=h\chi_\omega,\ (t,x)\in (0,T)\times (0,1) $$ where $b(x,r)=\ell(r)a(x)$ 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.