Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations

Abstract

Abstract Let X be a Banach space with norm ∥ ⋅ ∥ {\|\cdot\|} . Let A : D ⁢ ( A ) ⊂ X → X {A:D(A)\subset X\rightarrow X} be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 {\varepsilon>0} and T > 0 {T>0} are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X {u:[0,T]\rightarrow X} satisfying u t + A ⁢ u = 0 , 0 < t < T , ∥ u ⁢ ( T ) - φ ∥ ⩽ ε , u_{t}+Au=0,\quad 0<t<T,\;\|u(T)-\varphi\|\leqslant\varepsilon, for φ in X, is regularized by the generalized Sobolev equation u α ⁢ t + A α ⁢ u α = 0 , 0 < t < T , u α ⁢ ( T ) = φ , u_{\alpha t}+A_{\alpha}u_{\alpha}=0,\quad 0<t<T,\;u_{\alpha}(T)=\varphi, where 0 < α < 1 {0<\alpha<1} and A α = A ⁢ ( I + α ⁢ A b ) - 1 {A_{\alpha}=A(I+\alpha A^{b})^{-1}} with b ⩾ 1 {b\geqslant 1} . Error estimates of the method with respect to the noise level are proved.