Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ 0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.