On initial and terminal value problems for fractional nonclassical diffusion equations

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 149, Number 1, January 2021, Pages 143–161 https://doi.org/10.1090/proc/15131 Article electronically published on June 11, 2020 ON INITIAL AND TERMINAL VALUE PROBLEMS FOR FRACTIONAL NONCLASSICAL DIFFUSION EQUATIONS NGUYEN HUY TUAN AND TOM´ AS CARABALLO (Communicated by Wenxian Shen) Abstract. In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case 0 <α≤1 and show the ill-posedness in the sense of Hadamard in the case α>1. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in Lqnorm is first established. 1. Introduction In this paper, we consider the following nonlinear nonclassical diffusion equation (called pseudo-parabolic equation): ut−kΔut+(−Δ)αu=G(u)in(0,T]×Ω, u(t, x)=0 on(0,T]×∂Ω, (1) where k>0, and Ω ⊂Rd,(d≥1) is a bounded domain with smooth boundary ∂Ω, the operator (−Δ)αis the fractional Laplacian with α∈(0,1) ∪(1,∞). Let us divide the nonclassical diffusion equation into two different problems as follows: •Initial value problem: This problem consists of finding u(x, t)for0<t≤T from the initial state (2) u(x, 0) = u0(x),x∈Ω. •Terminal value problem: This problem is related to recovering u(x, t)for 0≤t≤Tfrom the terminal value data (or final state) (3) u(x, T)=uT(x),x∈Ω. Received by the editors November 7, 2019, and, in revised form, January 15, 2020. 2010 Mathematics Subject Classification. Primary 26A33, 35B65, 35R11. Key words and phrases. Fractional nonclassical diffusion equation, well-posedness, illposedness, regularity estimates, regularization and error estimate. This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09. The research of the second author was partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00. c 2020 American Mathematical Society 143 Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 144 N. H. TUAN AND T. CARABALLO Nonclassical diffusion equations (or pseudo-parabolic equations) are characterized by the occurrence of a time derivative appearing in the highest order term, which describes various important physical processes. It can be regarded as a Sobolev-type equation or a Sobolev-Galpern-type equation. Pseudo-parabolic equations have many applications in science and technology, especially in physical phenomena such as seepage of homogeneous fluids through a fissured rock, aggregation of populations, etc.; see, e.g., [7] and its references. If α= 1, then the fractional operator (−Δ)αbecomes the standard Laplace operator. In this special case, the nonclassical diffusion equation has been studied in [2,4–8,11,14,19–21] with various directions and motivations. Until now, the results on fractional pseudo-parabolic equations are limited and can be mentioned in just a few papers, for example, [7,22,28,29]. Since the fractional operator (−Δ)αappearing in the main equation is nonlocal and can be regarded as the infinitesimal generator of Levy stable diffusion processes, many scientists believed that it described some physical phenomena more exactly than integral differential equations. As in [7], equation (1) is of the regularity-gain-type for α>1, and of the regularity-loss-type for 0 <α<1. Our new results and main contributions in this paper are described as follows: •For the initial value problem (1)-(2),our main goal is the study of existence, uniqueness, continuous dependence, unique continuation of solutions, and a blow-up alternative under critical nonlinearity of source function G.As we know, nonlinear PDEs with critical nonlinearities are an interesting topic. This is mentioned in [3, 15] and the references therein. Studying the initial value problem for (1)-(2) in the critical case is also a challenging problem. We note that the work on global existence, blow-up criterion, and continuation of solutions for PDEs has recently attracted many authors, for example, T. Issa and W. Chen [16–18], T. Caraballo et al. [19], A. N. Carvalho et al. [15, 26], B. de Andrade et al. [23–25] and the references therein. •For the terminal value problem for pseudo-parabolic equation (1)-(3), to the best of our knowledge, there are not any results about it. Our work is the first study in this direction which is divided into various cases. Under thecase0<α≤1, we state well-posedness of the terminal value problem. However, the property of the solution in the case α>1 is very different from one in the case 0 <α≤1. The well-posedness of problems (1)-(3) with α> 1 is not guaranteed because of the ill-posedness of the backward problem in the sense of Hadamard [13]. An “ill-posed problem” (not well-posed problem) is a problem that either has no solutions in the desired class, or has many (two or more) solutions, or the solution procedure is unstable (i.e., arbitrarily small errors in the measurement data may lead to indefinitely large errors in the solutions). Our main goal in this paper is to provide some regularized solutions that are called regularized solutions for approximating u(x, t),0≤t<T.In this paper, we do not investigate the existence and uniqueness of the solution of the backward problem (1)-(3) with α>1.It is also a challenging and open problem, and should be the topic for another paper. In this paper, we assume that the backward problem (1)-(3) has a unique solution u (called a sought solution) that belongs to an appropriate space. So our main purpose in this case is to consider a regularized problem for finding an approximate solution. Furthermore, error estimates with Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINAL VALUE PROBLEM 145 the speed of convergence between the regularized solution and the sought solution under some a priori assumptions on the sought solution are also our primary purpose. •This is a first study for the backward problem for PDEs when the noisy data ϕ∈Lp(Ω) for 1 ≤p<2. Since Lp(Ω) →L2(Ω), we know that if ϕ∈Lp(Ω) for p≥2, then it belongs to L2(Ω). So, the analysis for the noisy data belongs to Lp(Ω) for p≥2 is trivial and similar to the L2(Ω) setting. However, in some physical practice, in oder to approximate uT,we only obtain the noisy data ϕwhich only belongs to class Lp(Ω) for p<2. This means that we only get the noisy level which is the upper bound of the error ϕ−uTLp(Ω). Section 3 is the first result for a regularized solution and the convergence analysis in Lpnorms. The case of L2is much easier than the other ones since one can use the Parseval equality in obtaining stability estimates. The technique in the Lpestimate here is more complex than the L2estimate, since we do not have the Parseval equality. Our new technique in this section is based on applying some Sobolev embedding. This is a new and strong point of this paper. We also emphasize that our method in the current paper can be applied for various PDEs such as parabolic equations, elliptic equations, etc. The paper is organized as follows. In Section 1, we introduce our problem and the motivation for our study. In Section 2, we state and prove a local well-posedness and the regularity result for (1)-(2). The existence of a unique continuation and a blow-up alternative for the mild solution of the abstract problems (1)-(2) are also mentioned in Section 2. In Section 3, we establish a regularized solution using a truncation method as in [9, 10, 12] and show the well-posedness of our problem. We also mention our results concerning the error estimate in the Lpnorm of the regularized solution and the sought solution. Let us consider the operator A=−ΔonV:=H1 0(Ω) ∩H2(Ω), and assume that the operator Ahas the eigenvalues λjsuch that 0 <λ 1≤λ2≤ ··· ≤ λj≤... which approach ∞as jgoes to ∞. The corresponding eigenfunctions are denoted by ej∈V. Now, let us define fractional powers of Aand its domain. For all s≥0, we define by Asthe following operator Asv:=∞ j=1v,ejλs jej,v∈ D(As)={v∈L2(Ω) : ∞ j=1 |v,ej|2λ2s j<∞}.The domain D(As)isaBanach space equipped with the norm vD(As):=∞ j=1 |v,ej|2λ2s j1 2,v∈D(As). The definition of the negative fractional power A−swith s>0 can be found in H. Brezis [1]. Its domain D(A−s) is a Hilbert space endowed with the dual inner product ., .−s,s taken between D(A−s)andD(As). This generates the norm vD(A−s)=∞ j=1 |v,ej−s,s |2λ−2s j1 2.Let L∞(0,T;Gα,k(Ω)) (see [27]) be the following space: (4) L∞(0,T;Gα,k(Ω)) :=v∈L∞(0,T;L2(Ω)),sup 0≤t≤T exp 2tk−1λα−1 jv,ej2<∞ for any α>1. Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 146 N. H. TUAN AND T. CARABALLO 2. Well-posedness of initial value problems (1)-(2) In this section, we study the well-posedness of problems (1)-(2). Our main purposes are to ensure sufficient conditions for existence and uniqueness of mild solutions to (1)-(2), analyze the possible continuation of this solution to a maximal interval of existence, and consider the problem of continuous dependence with respect to initial data. Let us set Sα,k(t)w= ∞  j=1 exp −tλα j(1 + kλj)−1w, ejej,Bα,k(t)w = ∞  j=1 exp −tλα j(1 + kλj)−1 1+kλj w, ejej. Then from [7], we deduce the mild solution of the following initial value problem: u(t)=Sα,k(t)u0+t 0 Bα,k(t−s)G(u(s))ds.(5) In this section, for simplicity, we only study problems (1)-(2) with the case 0 < α≤1. The case α>1 may be more complex but we omit it here since it can be considered by a similar argument. 2.1. Local well-posedness and regularity. In this subsection, we consider the following function with the critical nonlinearity form G:D(Aη)−→ D(Aν),ν< η<ν+αand satisfy the following: ⎧ ⎨ ⎩ G(v)−G(w)D(Aν)≤K1+vp−1 D(Aη)+wp−1 D(Aη)v−wD(Aη), G(v)D(Aν)≤Kvp D(Aη)+1  (6) for p>1, K>0, and v, w ∈D(Aη). Lemma 2.1. Let any w0∈D(Aη)and u∈C[0,T0],D(Aη)such that sup 0≤t≤T0 u(t)−w0D(Aη)≤M. If ν<η<ν+α,then     t 0 Bα,k(t−s)G(u(s))ds    D(Aη) ≤Kw0D(Aη)+Mp+1 tα+ν−η α.(7) Proof. We have     t 0 Bα,k(t−s)G(u(s))ds    D(Aη) ≤t 0  Bα,k(t−s)G(u(s))  D(Aη)ds ≤t 0∞  j=1 λ2η j (1+kλj)2exp −2(t−s)λα j(1 + kλj)−1G(u(s)),e j21 2 ds. Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINAL VALUE PROBLEM 147 Using the inequality e−y≤Mβy−βfor any β>0andMβ>0 is a positive constant, we derive the estimate exp −2(t−s)λα j(1 + kλj)−1≤|Mβ|21+kλ12β λ−2αβ j(t−s)−2β, which implies that     t 0 Bα,k(t−s)G(u(s))ds    D(Aη) ≤Mβ1+kλ1β−1t 0 (t−s)−β  G(u(s))  D(Aη−αβ)ds ≤Mβ1+kλ1β−1t 0 (t−s)−β  G(u(s))  D(Aν)ds ≤KMβ1+kλ1β−1t 0 (t−s)−β u(s) p D(Aη)+1 ds ≤Kw0D(Aη)+Mp+1 t 0 (t−s)−βds =Kw0D(Aη)+Mp+1 tα+ν−η α.(8) wherewehaveusedβ=η−ν α<1, and set K= KαMβ1+kλ1β−1 α+ν−η. Theorem 2.2. Let Gsatisfy (6). Then there exists T0>0such that problem (1) has a unique mild solution u∈C(0,T0); Hη(Ω). Proof. Let any 0 <M<1andR=(M+w0D(Aη))p−1.Let us choose T0such that ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ T01−β≤1−β 2KMβ1+kλ1β−11+2R, T0kλα−1 1 w0 D(Aη)+Kw0D(Aη)+Mp+1 T1−β 0 1−β≤M. (9) Using the inequality 1 −e−y≤yand noting that α≤1, we have   Sα,k(t)w0−w0   2 D(Aη)= ∞  j=1 λ2η j1−exp −tλα j(1 + kλj)−12w0,e j2 L2(Ω) ≤ ∞  j=1 λ2η jtλα j 1+kλj2w0,e j2 L2(Ω) ≤t2k2λ2α−2 1 w0 2 D(Aη).(10) Let us define the following set: F=u∈C[0,T0]; D(Aη),sup 0≤t≤T0 u(t)−w0D(Aη)≤M . It is not difficult to see that Fis a complete space. Let us define the operator J:F→Fas follows: (11) Ju(t)=Sα,k(t)u0+t 0 Bα,k(t−s)G(u(s))ds. Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 148 N. H. TUAN AND T. CARABALLO Let us consider u∈Fand 0 <t≤t+h≤T 0. Then,   Ju(t+h)−Ju(t)  D(Aη) ≤  Sα,k(t+h)−S α,k(t)u0  D(Aη) +    t+h t Bα,k(t+h−s)G(u(s))ds    D(Aη) +    t 0Bα,k(t+h−s)−S α,k(t−s)G(u(s))ds    D(Aη) =N1+N2+N3.(12) Now, we estimate the term N1. Indeed, using the inequality e−a−e−b≤|a−b|, we have |N1|2= ∞  j=1 λ2η jexp −(t+h)λα j(1 + kλj)−1−exp −tλα j(1 + kλj)−1u0 j2 ≤k2λ2α−2 1h2 ∞  j=1 λ2η j|u0 j|2=k2λ2α−2 1h2u02 D(Aη).(13) Notice that, if u∈F, then for any 0 ≤t≤T 0 (14) G(u(t))D(Aν)≤Ku(t)p D(Aη)+1 ≤Kw0D(Aη)+Mp+1 . Using (14), the term N2can be estimated as follows: N2≤t+h t  G(u(s))  D(Aη)ds ≤Kw0D(Aη)+Mp+1 t+h t (t+h−s)−βds =Kw0D(Aη)+Mp+1 h1−β 1−β,(15) wherewehaveusedthefactthatBα,k(t)vD(Aη)≤vD(Aη). Using again the inequality e−a−e−b≤|a−b|, we deduce that, for any 0 ≤s≤t,h>0, (16) exp −(t+h−s)λα j(1 + kλj)−1−exp −(t−s)λα j(1 + kλj)−1 1+kλj ≤hk−2λα−2 1. This inequality leads to the following bound for the term N3:     t 0Bα,k(t+h−s)−B α,k(t−s)G(u(s))ds    D(Aη) ≤k−2λα−2 1ht 0  G(u(s))  D(Aη)ds ≤Kk−2λα−2 1w0D(Aη)+Mp+1 ht.(17) Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINAL VALUE PROBLEM 149 Combining some of the preceding estimates, we deduce that if u∈F,thenJu∈ C[0,T0]; D(Aη). It follows from (10) that   Ju(t)−w0  D(Aη) ≤  Sα,k(t)w0−w0  D(Aη)+  t 0 Bα,k(t−s)G(u(s))ds  D(Aη) ≤T 0kλα−1 1 w0 D(Aη)+Kw0D(Aη)+Mp+1 T1−β 0 1−β≤M.(18) Now let w, v ∈F. For any t∈[0,T0], using (8), we deduce the following estimate:   Jw(t)−Jv(t)  D(Aη) =    t 0 Bα,k(t−s)G(w(s)) −G(v(s))ds    D(Aη) ≤Mβ1+kλ1β−1t 0 (t−s)−β  G(w(s)) −G(v(s))  D(Aν)ds ≤KMβ1+kλ1β−1t 0 (t−s)−β1+wp−1 D(Aη)+vp−1 D(Aη) w−v D(Aη)ds ≤KMβ1+kλ1β−1 1+2M+w0D(Aη)p−1t 0 (t−s)−β w−v D(Aη)ds. In light of (9) and noting that t 0(t−s)−βds =t1−β 1−β, we find that sup 0≤t≤T0  Jw(t)−Jv(t)  D(Aη) ≤ KMβ1+kλ1β−1T01−β 1−β1+2Rsup 0≤t≤T0 w(t)−v(t)D(Aη) ≤1 2sup 0≤t≤T0 w(t)−v(t)D(Aη).(19) From the Banach fixed point theorem, it turns out that Jhas a unique fixed point u∈F. 2.2. Continuation and blow-up alternative. In this subsection, we give continuity to our study of problems (1)-(2) proving that the mild solution provided by Theorem 2.2 has a unique continuation to a larger interval of existence. Definition 2.3. Given a mild solution u∈C([0,T0],D(Aη)) to problem (1), it is said that uis a continuation of uin [0,T1]ifu∈C([0,T1],D(Aη)) is a mild solution for T1>T0and u(t)=u(t) for any t∈[0,T0]. Theorem 2.4. Let ubeamildsolutionofproblems(1)-(2) on [0,T0].IfGsatisfies (6), then there exist T1>T0and a unique continuation uof uin [0,T1]. Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 150 N. H. TUAN AND T. CARABALLO Proof. Fix 0 <M≤1andtakeT1>T0such that for t∈[T0,T1], we denote (20) UM=v∈C[0,T1],D(Aη):v(t)−u(T0)D(Aη)≤M,v(t)=u(t),t∈[0,T0]. It is easy to see that UMis a complete metric space with the norm of supremum in D(Aη). Let us define G:UM→UMby (21) Gv(t)=Sα,k(t)u0+t 0 Bα,k(t−s)G(v(s))ds. If v∈UM, then it is obvious to obtain that Gv(t)=u(t) for any t∈[0,T0]. Let any t∈[T0,T1]andletanyv∈UM. By some simple computations, we have Gv(t)−u(T0) =Sα,k(t)−S α,k(T0)u0 +T0 0Sα,k(t−s)−Sα,k(T0−s)G(u(s))ds +t T0 Bα,k(t−s)G(u(s))ds =Sα,k(t−T 0)u(T0)−u(T0)+t T0 Bα,k(t−s)G(v(s))ds,(22) wherewehaveusedthefactthatu(T0)=Sα,k(T0)u0+T0 0Sα,k(T0−s)G(u(s))ds. In view of (8) and (6), Gv(t)−u(T0)D(Aη) ≤  Sα,k(t−T 0)u(T0)−u(T0)  D(Aη) +Mβ1+kλ1β−1t T0 (t−s)−β  G(v(s))  D(Aν)ds ≤(T1−T 0)kλα−1 1 u(T0) D(Aη) +KMβ1+kλ1β−1t T0 (t−s)−βv(s)p D(Aη)+1 ds.(23) Since v∈UM, we deduce that v(t)D(Aη)≤u(T0)D(Aη)+Mfor any t∈[T0,T1]. Therefore, we have the following estimate: t T0 (t−s)−βv(s)p D(Aη)+1 ds ≤u(T0)D(Aη)+Mp+1 t T0 (t−s)−βds ≤u(T0)D(Aη)+Mp+1 (T1−T 0)1−β 1−β.(24) Let us choose T1such that ⎧ ⎪ ⎨ ⎪ ⎩ (T1−T 0)kλα−1 1 u(T0) D(Aη)≤M 2, KMβ1+kλ1β−1u(T0)D(Aη)+Mp+1 (T1−T0)1−β 1−β≤M 2. (25) Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINAL VALUE PROBLEM 151 Take any v, w ∈UM.Wehave Gv(t)−Gw(t)D(Aη) =  t T0 Bα,k(t−s)G(v(s)) −G(w(s))ds  D(Aη) ≤Mβ1+kλ1β−1t T0 (t−s)−β  G(v(s)) −G(w(s))  D(Aν)ds ≤KMβ1+kλ1β−1t T0 (t−s)−β1+vp−1 D(Aη)+wp−1 D(Aη)v−wD(Aη)ds. Since v,w ∈UM, we deduce 1 + v(t)p−1 Hη(Ω) +w(t)p−1 Hη(Ω) ≤1+2(u(T0)Hη(Ω) + M)p−1for any t∈[T0,T1]. From the preceding estimates,  Gv(t)−Gw(t) D(Aη) ≤KMβ1+kλ1β−11+2 u(T0)Hη(Ω) +Mp−1 ×t T0 (t−s)−βv−wD(Aη)ds ≤KMβ1+kλ1β−11+2 u(T0)Hη(Ω) +Mp−1 ×(T1−T 0)1−β 1−βsup 0≤t≤T1 w(t)−v(t) D(Aη).(26)  The next theorem is our result on global existence or noncontinuation by blowup. Theorem 2.5. Assume that Gsatisfies (6).Letube the mild solution of problem (1) defined on [0,Tmax),whereTmax is the maximal time of existence of u.Then we have Tmax =+∞or lim supt→T − max u(t)D(Aη)=∞. Proof. Suppose that Tmax <∞and there exists a constant Bsuch that max u0D(Aη),sup 0≤t≤T u(t)Hη(Ω)≤B for any t∈[0,Tmax). Let us pick a sequence of positive numbers tn→T− max;we consider the sequence {u(tn)}in D(Aη). We will prove that this sequence is a Cauchy sequence in the space D(Aη). For tm,t n∈[0,Tmax) such that tm<t n,we obtain after some simple caculations u(tn)−u(tm)=Sα,k(tn)−S α,k(tm)u0+tn tm Bα,k(tn−s)G(u(s))ds.(27) Therefore,   u(tn)−u(tm)  D(Aη) ≤  Sα,k(tn)−S α,k(tm)u0  D(Aη)  ! A1 +  tn tm Bα,k(tn−s)G(u(s))ds  D(Aη)  ! A2 . (28) Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 158 N. H. TUAN AND T. CARABALLO of u. Using (51), we obtain u(t)L2(Ω) ≤C1,σ,p,d(N)−σexp (T−t)k−1(N)α−1ϕLp(Ω) +K(1+kλ1)−1T t exp (s−t)Nα−1 (1+kN)−1u(s)L2(Ω)ds.(56) By applying Gronwall’s inequality, exp tk−1(N)α−1u(t)L2(Ω) ≤H1H2()C1,σ,p,dϕLp(Ω) (57) which implies that (58) u(t)L2(Ω) ≤H1C1,σ,p,dH2()exp−tk−1(N)α−1ϕLp(Ω). Since the Sobolev embedding D(Aγ)→L2d d−4γ(Ω), we deduce for any 0 <γ<d 4 u(t)L 2d d−4γ(Ω) ≤C2,γ,du(t)D(Aγ)=C2,γ,d λj≤N λ2γ ju(t),e j21 2 ≤C2,γ,d(N)γu(t)L2(Ω) ≤BϕLp(Ω),(59) where Bdepends on . Theorem 3.4. Let us assume problem (1) has a unique solution u∈ L∞(0,T;Gα,k(Ω)) such that uL∞(0,T ;Gα,k (Ω)) ≤M. Let us choose Nsuch that for any t>0 (60) lim →0N=∞,lim →0(N)−σexp Tk−1Nα−1 = lim →0(N)γexp −tk−1Nα−1 =0. Then u(t)−u(t)L 2d d−4γ(Ω) is of order max (N)γexp −tk−1(N)α−1,(N)−p. Remark 3.2.Let us choose N=1 rTk−1log(1 )1 α−1for any 0 <r<1. Then it is easy to check that (60) holds. The error u(t)−u(t)L 2d d−4γ(Ω) is of order max 1 Tk−1rlog(1 )γ α−1r T,1 Tk−1rlog(1 )−p α−1. Proof. Note that u(t)−PNu(t)L2(Ω) = λj>N exp −2tk−1λα−1 jexp 2tk−1λα−1 ju(t),e j21 2 ≤exp −tk−1Nα−1 M. Since ϕ−uTLp(Ω) ≤, and observing that −d 4<σ≤(p−2)d 4we have the Sobolev embedding Lp(Ω) →D(Aσ), and we then find that ϕ−uTD(Aσ)≤ C1,σ,p,dϕ−uTLp(Ω) ≤C1,σ,p,d. The latter inequality leads to   S−1 α,k(T−t)PN(ϕ−uT)  L2(Ω) ≤(N)−σexp (T−t)k−1(N)α−1ϕ−uTAσ). Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINAL VALUE PROBLEM 159 Using the triangle inequality and combining some previous estimates, we find that u(t)−u(t)L2(Ω) ≤  S−1 α,k(T−t)PN(ϕ−uT)  L2(Ω) +u(t)−PNu(t)L2(Ω) +T t  B−1 α,k(s−t)PNG(u(s)) −G(u(s))  L2(Ω) ≤exp −tk−1(N)α−1M+C1,σ,p,d(N)−σexp Tk−1(N)α−1ds +K(1+kλ1)−1T t exp (s−t)k−1Nα−1 u(s)−u(s)L2(Ω)ds. (61) Multiplying both sides by exp(tk−1(N)α−1) and then applying Gronwall’s inequality, exp tk−1(N)α−1u(t)−u(t)L2(Ω) ≤M+C1,σ,p,d(N)−σexp Tk−1(N)α−1exp K(1+kλ1)−1T. (62) Since the Sobolev embedding D(Aγ)→L2d d−4γ(Ω) holds true for any 0 <γ<d 4, then u(t)−u(t)L 2d d−4γ(Ω) ≤C2,γ,du(t)−u(t)D(Aγ).(63) Now, we continue to estimate the term u(t)−u(t)D(Aγ). Indeed, u(t)−PNu(t)2 D(Aγ)= λj≤N λ2γ ju(t)−u(t),e j2≤(N)2γu(t)−u(t)2 L2(Ω) (64) and u(t)−PNu(t)2 D(Aγ)= λj>N λ−2p jλ2γ+2p ju(t),e j2 ≤(N)−2pu(t)2 D(Ap+γ)≤(N)−2pu2 L∞(0,T ;D(Ap+γ)).(65) From the two preceding estimates, we find that  u(t)−u(t) D(Aγ) ≤u(t)−PNu(t)D(Aγ)+u(t)−PNu(t)D(Aγ) ≤(N)γu(t)−u(t)L2(Ω) +(N)−puL∞(0,T ;D(Ap+γ)) ≤M+C1,σ,p,dH2()H1(N)γexp −tk−1(N)α−1 +(N)−puL∞(0,T ;D(Ap+γ)). (66) wherewesetH1=exp K(1 + kλ1)−1T,H 2()=(N)−σexp Tk−1Nα−1 . From (63) and (66), we obtain the desired result.  Licensed to Universidad de Sevilla. Prepared on Mon Feb 1 08:25:37 EST 2021 for download from IP 150.214.182.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 160 N. H. TUAN AND T. CARABALLO References [1] Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations,Universitext, Springer, New York, 2011. 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License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use