Abstract:In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modi ed quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and e ciency of this method. Keywords:Ill-posed problems, Biparabolic problem, Regularization MSC: 47A52, 65J22Formulation of the problem Throughout this paper H denotes a complex separable Hilbert space endowed with the inner product ⟨., .⟩ and the norm . , L(H) stands for the Banach algebra of bounded linear operators on H.Let A ∶ D(A) ⊂ H → H be a positive, self-adjoint operator with compact resolvent, so that A has an orthonormal basis of eigenvectors (φ n ) ⊂ H with real eigenvalues (λ n ) ⊂ R + , i.e.,In this paper, we consider the following inverse source problem of determining the unknown source term u( ) = f and the temperature distribution u(t) for ≤ t < T, of the following biparabolic problemwhere < T < ∞ and g is a given H-valued function.