A New Fourier Truncated Regularization Method for Semilinear Backward Parabolic Problems
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A New Fourier Truncated Regularization Method for Semilinear Backward Parabolic Problems Tuan Nguyen Huy1 · Mokhtar Kirane2,3 · Bessem Samet4 · Van Au Vo1
Received: 23 January 2016 / Accepted: 23 October 2016 © Springer Science+Business Media Dordrecht 2016
Abstract We study the backward problem for non-linear (semilinear) parabolic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new Fourier truncated regularization method for stabilising the ill-posed problem. In comparison with previous studies on solving the nonlinear backward problem, our method shows a significant improvement. Keywords Nonlinear parabolic problem · Backward problem · Regularization method · Ill-posed problem Mathematics Subject Classification 35K05 · 35K99 · 47J06 · 47H10
1 Introduction Let H be a Hilbert space with the inner product . and the norm . and let A be a positive self-adjoint operator defined on a dense subspace D(A) ⊂ H such that −A generates a compact contractive semi-group {S(t)}t≥0 on H . We shall consider the backward problem
B M. Kirane
[email protected]
1
Department of Mathematics, University of Science, Vietnam National University, Ho Chi Minh City, Viet Nam
2
Laboratoire de Mathématiques Pôle Sciences et Technologie, Universié de La Rochelle, Aénue M. Crépeau, 17042 La Rochelle Cedex, France
3
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4
College of Science, Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
T.N. Huy et al.
of finding a function u : [0, T ] → H such that ut + Au = f t, u(t) , u(T ) = ϕ,
t ∈ (0, T ),
(1)
where the data ϕ is given in H and the source function f will be defined later. In practice, the data ϕ ∈ H is noisy and is represented by the perturbed data ϕ ∈ H satisfying ϕ − ϕ ≤ , (2) where the constant > 0 represents an upper bound on the measurement error. The problem (1) is not well-posed because its solution may not exist and, even if it exists, it does not depend continuously on the “noisy” Cauchy data ϕ . Hence, a regularization process is required in order to obtain a stable solution. The linear homogeneous case f = 0 has been studied in many papers, we refer to some interesting ones [1–4, 7, 9, 15, 18, 24]. Although there are many papers on the linear case of the backward parabolic problem, the nonlinear cases has been less investigated [8, 17, 21–23] and it is the purpose of this study to make advances into the semi-linear problem (1). First, the study of backward problem for parabolic are very close to those that have been developed in the context of controllability of parabolic problems. The issues there are also very close to that of solving the equation backwards, although often this appears as a technical preliminary step to deal with more
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