Well-Posedness For Degenerate Third Order Equations With Delay And Applications To Inverse Problems

In this paper, we study well-posedness for the following third-order in time equation with delay (0.1)α(Mu′)″(t)+(Nu′)′(t)=βAu(t)+γBu′(t)+Gut′+Fut+f(t),t∈[0,2π] where α, β, γ are real numbers, A and B are linear operators defined on a Banach space X with domains D(A) and D(B) such thatD(A)∩D(B)⊂D(M)∩D(N);u(t)is the state function taking values in X and u_t: (−∞, 0] → X defined as u_t(θ) = u(t+θ) for θ < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue–Bochner spaces L^p(T, X) , periodic Besov spaces Bp,qs(T,X) and periodic Triebel–Lizorkin spaces Fp,qs(T,X). A novel application to an inverse problem is given.