Regularization of backward parabolic
equations in Banach spaces by generalized
Sobolev equations
Tác giả: Nguyen Van Duc, Dinh Nho Hào and Maxim Shishlenin
J. Inverse Ill-Posed Probl.
Quyển: 32(1) Trang: 9-20
Năm xuất bản: 2/2024
Tóm tắt
Let X be a Banach space with norm ‖ ⋅ ‖. Let A : D(A) ⊂ X → X be an (possibly unbounded) operator
that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [0, T] → X satisfying
ut + Au = 0, 0 < t < T, ‖u(T) − φ‖ ⩽ ε,
for φ in X, is regularized by the generalized Sobolev equation
uαt + Aαuα = 0, 0 < t < T, uα(T) = φ,
where 0 < α < 1 and Aα = A(I + αAb)−1 with b ⩾ 1. Error estimates of the method with respect to the noise level
are proved.
Từ khóa
Backward parabolic equations, ill-posed problems, regularization, Sobolev equation
