Nonlinear Analysis 73 (2010) 3479–3488 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Sharp estimates for approximations to a nonlinear backward heat equation Nguyen Huy Tuan a,∗ , Dang Duc Trong b a Department of Mathematics, SaiGon University, 273 AnDuongVuong, HoChiMinh City, Viet Nam b Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Q.5, HoChiMinh City, Viet Nam article info Article history: Received 28 May 2009 Accepted June 2010 MSC: 35K05 35K99 47J06 47H10 abstract A nonlinear backward heat problem for an infinite strip is considered The problem is illposed in the sense that the solution (if it exists) does not depend continuously on the data In this paper, we use the Fourier regularization method to solve the problem Some sharp estimates of the error between the exact solution and its regularization approximation are given © 2010 Elsevier Ltd All rights reserved Keywords: Backward heat problem Ill-posed problem Nonlinear heat Truncation method Error estimate Introduction Let T be a positive number We consider the problem of finding the temperature u(x, t ), (x, t ) ∈ R × [0, T ], such that ut − uxx = f (x, t , u(x, t )), u(x, T ) = ϕ(x), x ∈ R, (x, t ) ∈ R × (0, T ), (1) where ϕ(x), f (x, t , z ) are given The problem is called the backward heat problem (BHP), the backward Cauchy problem or the final value problem It is known in general that the backward problem is ill-posed, i.e., a solution does not always exist, and in the case of existence, it does not depend continuously on the given datum In fact, from a small noise contaminated physical measurement, the corresponding solutions may have a large error This makes the numerical computation difficult Hence, a regularization is in order The special case where the function f is independent of u, namely f (x, t , u) = or f (x, t , u) = f (x, t ), has been studied by many authors in recent years As a few examples, we mention Lattes and Lions [1], Showalter [2], Ames and Payne [3] who approximated the BHP by a quasi-reversibility method; Tautenhahn and Schroter [4] who established an optimal error estimate for a BHP; Seidman [5] who established an optimal filtering method; and Hao [6] who studied a modification method We also refer the reader to various other works of Fu et al [7–10], Campbell et al [11], Lien et al [12], Murniz et al [13], Dokuchaev et al [14], Gilliam et al [15] and Engl et al [16] ∗ Corresponding author E-mail address: tuanhuy_bs@yahoo.com (N.H Tuan) 0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.na.2010.06.002 3480 N.H Tuan, D.D Trong / Nonlinear Analysis 73 (2010) 3479–3488 Although there have been many works on the homogeneous case and the linear inhomogeneous cases, and the backward heat problem (2), literature on the nonlinear case of the backward heat problem is quite scarce In 2005, Quan and Dung [17] offered a regularized solution by a semi-group method However, they were able to give error estimates only for the very special case where the exact solution has a finite Fourier series expansion and the Lipschitz constant k > is small enough In 2006, Trong and Quan [18] used the integral transform method to treat the nonlinear case and attained an error estimate t T of order for each t > This estimate is good at any fixed t > but useless at t = Very recently, Trong and Tuan [19] t t improved this method to give an error estimate of order T (ln(1/ )) T −1 for all t ∈ [0, T ] In the present paper, a classical Fourier method is used for solving the nonlinear backward heat problem, which will improve the results of [18,19] in two ways (although our approach is different from that of [18,19]): (1) We shall give a criterion for choosing the regularization parameter with a rigorous mathematical proof, which itself possesses an independent significance (2) We shall provide some sharp error estimates Under some suitable conditions on the exact solution u, we shall introduce t the error estimate of order 2T + This is a significant improvement in comparison with the results of [7,6,18,20,4,12,10] Some comments on the usefulness of this method are given in some remarks This paper is organized as follows: In Section 2, we give some auxiliary results In Section 3, we give the regularization solution by using the truncated method and give stable estimates of the error between the regularization solution and the exact solution Some auxiliary results Let gˆ (ξ ) denote the Fourier transform of function g ∈ L2 (R) defined formally as +∞ gˆ (ξ ) = √ 2π g (x)e−iξ x dx (2) −∞ Let H = W 1,2 (R), H = W 2,2 (R) be the Sobolev spaces which are defined by H (R) = {g ∈ L2 (R), ξ gˆ (ξ ) ∈ L2 (R)}, H (R) = {g ∈ L2 (R), ξ gˆ (ξ ) ∈ L2 (R)} We denote by , H1 , H2 the norms in L2 (R), H (R), H (R) respectively, namely g H1 = g + gx = (1 + ξ ) gˆ (ξ ) , g H2 = g + gx + gxx = (1 + ξ + ξ ) gˆ (ξ ) Let us first make clear what a weak solution to the problem (1) is Lemma Let f : R × [0, T ] × R → R be a function such that |f (x, t , u) − f (x, t , v)| ≤ K |u − v|, (3) for all (x, t ) ∈ R × [0, T ] and for some constant K > independent of x, t , u, v Let us have ϕ ∈ L (R) Assume that u ∈ C ([0, T ], H (R)) ∩ C ([0, T ], L2 (R)) is a solution of the equation T uˆ (ξ , t ) = e(T −t )ξ ϕ(ξ ˆ )− e−(t −s)ξ fˆ (ξ , s, u)ds (4) t Then ut , uxx , f (x, t , u) ∈ C ([0, T ], L2 (R)) and u is a solution to the heat equation (1) where the main equation holds in C ([0, T ], L2 (R)) Proof By letting t = T in the equation T uˆ (ξ , t ) = e(T −t )ξ ϕ(ξ ˆ )− e−(t −s)ξ fˆ (ξ , s, u)ds, 0≤t ≤T t we have immediately uˆ (ξ , T ) = ϕ(ξ ˆ ) Therefore, we get u(x, T ) = ϕ(x) in L2 (R) Multiplying the above equation with et ξ we obtain T et ξ uˆ (ξ , t ) = eT ξ ϕ(ξ ˆ )− 2 esξ fˆ (ξ , s, u)ds, t ∈ [0, T ] t Differentiating the latter equation w.r.t the time variable t we get et ξ ξ uˆ (ξ , t ) + d dt uˆ (ξ , t ) = et ξ fˆ (ξ , t , u), (5) N.H Tuan, D.D Trong / Nonlinear Analysis 73 (2010) 3479–3488 3481 namely ξ uˆ (ξ , t ) + d dt uˆ (ξ , t ) = fˆ (ξ , t , u), t ∈ [0, T ] Since u ∈ C ([0, T ], H (R))∩C ([0, T ], L2 (R)) we have ξ uˆ (ξ , t ) = uˆxx (ξ ) and ˆ (ξ , t ) belongs to C ([0, T ], L2 (R)) Therefore, fˆ (ξ , t , u) also belongs to C ([0, T ], L (R)) which is equivalent to f (x, t , u) belonging to C ([0, T ], L2 (R)) Thus the latter d u dt equation means ut − uxx = f (x, t , u) in the sense of C ([0, T ], L2 (R)) Regularization and error estimates 3.1 The approximation problem and the main results Since t < T , we know from (4) that, when ξ becomes large, the term exp{(T − t )ξ } increases rather quickly Thus for uˆ (ξ , t ) ∈ L2 (R) with respect to ξ , the exact data ϕ(ξ ˆ ) must decay rapidly as |ξ | → ∞ Small errors in high-frequency components can blow up and completely destroy the solution for ≤ t < T Define the Fourier regularization solution of problem (1) as follows: √ + A u (x, t ) = √ 2π e √ (T −t )ξ √ + A iξ x ϕ(ξ ˆ ) e dξ − √ 2π − A T √ − A e−(t −s)ξ fˆ (ξ , s, u )eiξ x dξ ds, (6) t where A is a positive constant which will be selected appropriately as a regularization parameter such that lim →0 A = ∞ We now study the properties of (6) considered as an approximation to (1), i.e., we will give some stability estimates Theorem Let f be the function defined by (3) Let ϕ ∈ L2 (R) Then the problem (6) has a unique solution u ∈ C ([0, T ]; L2 (R))∩ C ((0, T ); L2 (R)) Furthermore, this approximate solution depends continuously on the final value ϕ , i.e., let w, v be the solutions of the problem (6) corresponding to the final values ϕ and φ ; then w(., t ) − v(., t ) ≤ e(T −t )A eK (T −t )2 ϕ−φ C1 Remark (1) If A = , then the stability of the regularized solution is of order e It is of the same order as the results in [21] t (2) If A = T1 ln , then the stability of the regularized solution is of order T −1 It is of the same order as the results in [22,23] (3) If A = T T 1+ln T ln , then the stability of the regularized solution is of order T 1+ln T This is better than the results in [22,23] Theorem Let f , ϕ, u be as in Theorem Suppose that ϕ ∈ L2 (R) and let ϕ ∈ L2 (R) be measured data such that ϕ −ϕ ≤ Suppose problem (4) has a unique solution u ∈ C ([0, T ]; L2 (R)) and let w ∈ C ([0, T ]; L2 (R)) be the unique solution of problem (6) corresponding to ϕ (a) If u is such that +∞ e2t ξ |ˆu(ξ , t )|2 dξ < ∞, −∞ for all t ∈ [0, T ] then w (., t ) − u(., t ) ≤ e−tA 2 eK (T −t ) eTA + N( , t) , ∀t ∈ [0, T ] (7) where T N( , t) = K e3K T (T −t ) M ( , t )dt + M ( , t ) (8) and √ − A M( , t) = −∞ ∞ e2t ξ |ˆu(ξ , t )|2 dξ + √ A e2t ξ |ˆu(ξ , t )|2 dξ (9) 3482 N.H Tuan, D.D Trong / Nonlinear Analysis 73 (2010) 3479–3488 (b) If there exists a positive number s > such that +∞ (ξ 2s e2t ξ |ˆu(ξ , t )) |2 dξ < ∞ −∞ then 2 eK (T −t ) eTA w (., t ) − u(., t ) ≤ e−tA 6Q + e3KT (T −t )/2 (10) (A )s where +∞ Q = sup 0≤t ≤T (ξ 2s e2t ξ |ˆu(ξ , t )) |2 dξ (11) −∞ (c) If there exists a positive number p > such that +∞ Rp = sup 0≤t ≤T e2(t +p)ξ |ˆu(ξ , t )|2 dξ