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A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. (English) Zbl 1140.34397

Summary: We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

MSC:

34G10 Linear differential equations in abstract spaces
35K90 Abstract parabolic equations
35R25 Ill-posed problems for PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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