Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
aa r X i v : . [ m a t h . A P ] J un Initial-Boundary Value Problems for Multi-TermTime-Fractional Diffusion Equations withPositive Constant Coefficients ∗ Zhiyuan LI † Yikan LIU † Masahiro YAMAMOTO † Abstract
In this paper, we investigate the well-posedness and the long-time asymptoticbehavior for the initial-boundary value problem for multi-term time-fractional diffusionequations, where the time differentiation consists of a finite summation of Caputo deriva-tives with decreasing orders in (0 ,
1) and positive constant coefficients. By exploitingseveral important properties of multinomial Mittag-Leffler functions, various estimates fol-low from the explicit solutions in form of these special functions. Then the uniquenessand continuous dependency upon initial value and source term are established, from whichthe continuous dependence of solution of Lipschitz type with respect to various coefficientsis also verified. Finally, by a Laplace transform argument, it turns out that the decayrate of the solution as t → ∞ is dominated by the minimum order of the time-fractionalderivatives. Keywords
Initial-boundary value problem, Time-fractional diffusion equation,Multinomial Mittag-Leffler function, Well-posedness,Long-time asymptotic behavior, Laplace transform
AMS subject classifications
Let Ω be an open bounded domain in R d with a smooth boundary (for example, of C class)and T > m , let α j and q j ( j = 1 , . . . , m ) bepositive constants such that 1 > α > · · · > α m >
0. Consider the following initial-boundaryvalue problem for the multi-term time-fractional diffusion equation m X j =1 q j ∂ α j t u ( x, t ) = Lu ( x, t ) + F ( x, t ) , x ∈ Ω , < t ≤ T,u ( x, t ) = 0 , x ∈ ∂ Ω , < t ≤ T,u ( x,
0) = a ( x ) , x ∈ Ω , (1.1)(1.2)(1.3)where L is a symmetric uniformly elliptic operator with the homogeneous Dirichlet boundarycondition, and we can assume q = 1 without lose of generality. The regularities of the initialvalue a and the source term F will be specified later. Here ∂ α j t denotes the Caputo derivative Manuscript last updated: October 2, 2018. † Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku,Tokyo 153-8914, Japan.E-mail: [email protected], [email protected], [email protected] ∗ This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
0) = a ( x ) , x ∈ Ω , (1.1)(1.2)(1.3)where L is a symmetric uniformly elliptic operator with the homogeneous Dirichlet boundarycondition, and we can assume q = 1 without lose of generality. The regularities of the initialvalue a and the source term F will be specified later. Here ∂ α j t denotes the Caputo derivative Manuscript last updated: October 2, 2018. † Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku,Tokyo 153-8914, Japan.E-mail: [email protected], [email protected], [email protected] ∗ This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations defined by ∂ α j t f ( t ) := 1Γ(1 − α j ) Z t f ′ ( s )( t − s ) α j d s, where Γ( · ) is a usual Gamma function. For various properties of the Caputo derivative, werefer to Kilbas et al. [12] and Podlubny [24]. See also [8, 28] for further contents on fractionalcalculus.In the case of m = 1, equation (1.1) is reduced to its single-term counterpart ∂ αt u = Lu + F in Ω × (0 , T ] , α ∈ (0 , . (1.4)The above formulation has been studied extensively from different aspects due to its vastcapability of modeling the anomalous diffusion phenomena in highly heterogeneous aquifer andcomplex viscoelastic material (see Adams & Gelhar [1], Ginoa et al. [6], Hatano & Hatano [9],Nigmatullin [22] and the references therein). Indeed, although the single-term time-fractionaldiffusion equation inherits certain properties from the diffusion equation with time derivativeof natural number order, it differs considerably from the traditional one especially in senseof its limited smoothing effect in space and slow decay in time. In Luchko [16], a maximumprinciple of the initial-boundary value problem for (1.4) was established, and the uniqueness ofa classical solution was proved. Luchko [17] represented the generalized solution to (1.4) with F = 0 by means of the Mittag-Leffler function and gave the unique existence result. Sakamoto& Yamamoto [26] carried out a comprehensive investigation including the well-posedness of theinitial-boundary value problem for (1.4) as well as the long-time asymptotic behavior of thesolution. It turns out that the spatial regularity of the solution is only moderately improvedfrom that of the initial value, and the solution decays with order t − α as t → ∞ . Recently, theLipschitz stability of the solution to (1.4) with respect to α and the diffusion coefficient wasproved as a byproduct of an inverse coefficient problem in Li et al. [13]. For other discussionsconcerning equation (1.4), see e.g., Gorenflo et al. [7] and Luchko [15], Pr¨uss [25]. Regardingnumerical treatments, we refer to Liu et al. [14] and Meerschaert & Tadjeran [20] for the finitedifference method and Jin et al. [11] for the finite element method.As natural extension, equation (1.1) is expected to improve the modeling accuracy in de-picting the anomalous diffusion due to its potential feasibility. However, to the authors’ bestknowledge, published works on this extension are quite limited in spite of rich literatures onits single-term version. Luchko [18] developed the maximum principle for problem (1.1)–(1.3)and constructed a generalized solution when F = 0 by means of the multinomial Mittag-Lefflerfunctions. Jiang et al. [10] considered fractional derivatives in both time and space and derivedanalytical solutions. As for the asymptotic behavior, for m = 2 it reveals in Mainardi et al. [21]that the dominated decay rate of the solution is related to the minimum order of time fractionalderivative. On the other hand, Beckers & Yamamoto [4] investigated (1.1)–(1.3) in a slightlymore general formulation and obtained a weaker regularity result than that in [26].In this paper, we are concerned with the well-posedness and the long-time asymptotic be-havior of the solution to the initial-boundary value problem (1.1)–(1.3), and we attempt toestablish results parallel to that for the single-term case. On basis of the explicit representationof the solution, we give estimates for the solution by exploiting several properties of the multi-nomial Mittag-Leffler function. Moreover, as long as the continuous dependency of the solution ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
0) = a ( x ) , x ∈ Ω , (1.1)(1.2)(1.3)where L is a symmetric uniformly elliptic operator with the homogeneous Dirichlet boundarycondition, and we can assume q = 1 without lose of generality. The regularities of the initialvalue a and the source term F will be specified later. Here ∂ α j t denotes the Caputo derivative Manuscript last updated: October 2, 2018. † Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku,Tokyo 153-8914, Japan.E-mail: [email protected], [email protected], [email protected] ∗ This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations defined by ∂ α j t f ( t ) := 1Γ(1 − α j ) Z t f ′ ( s )( t − s ) α j d s, where Γ( · ) is a usual Gamma function. For various properties of the Caputo derivative, werefer to Kilbas et al. [12] and Podlubny [24]. See also [8, 28] for further contents on fractionalcalculus.In the case of m = 1, equation (1.1) is reduced to its single-term counterpart ∂ αt u = Lu + F in Ω × (0 , T ] , α ∈ (0 , . (1.4)The above formulation has been studied extensively from different aspects due to its vastcapability of modeling the anomalous diffusion phenomena in highly heterogeneous aquifer andcomplex viscoelastic material (see Adams & Gelhar [1], Ginoa et al. [6], Hatano & Hatano [9],Nigmatullin [22] and the references therein). Indeed, although the single-term time-fractionaldiffusion equation inherits certain properties from the diffusion equation with time derivativeof natural number order, it differs considerably from the traditional one especially in senseof its limited smoothing effect in space and slow decay in time. In Luchko [16], a maximumprinciple of the initial-boundary value problem for (1.4) was established, and the uniqueness ofa classical solution was proved. Luchko [17] represented the generalized solution to (1.4) with F = 0 by means of the Mittag-Leffler function and gave the unique existence result. Sakamoto& Yamamoto [26] carried out a comprehensive investigation including the well-posedness of theinitial-boundary value problem for (1.4) as well as the long-time asymptotic behavior of thesolution. It turns out that the spatial regularity of the solution is only moderately improvedfrom that of the initial value, and the solution decays with order t − α as t → ∞ . Recently, theLipschitz stability of the solution to (1.4) with respect to α and the diffusion coefficient wasproved as a byproduct of an inverse coefficient problem in Li et al. [13]. For other discussionsconcerning equation (1.4), see e.g., Gorenflo et al. [7] and Luchko [15], Pr¨uss [25]. Regardingnumerical treatments, we refer to Liu et al. [14] and Meerschaert & Tadjeran [20] for the finitedifference method and Jin et al. [11] for the finite element method.As natural extension, equation (1.1) is expected to improve the modeling accuracy in de-picting the anomalous diffusion due to its potential feasibility. However, to the authors’ bestknowledge, published works on this extension are quite limited in spite of rich literatures onits single-term version. Luchko [18] developed the maximum principle for problem (1.1)–(1.3)and constructed a generalized solution when F = 0 by means of the multinomial Mittag-Lefflerfunctions. Jiang et al. [10] considered fractional derivatives in both time and space and derivedanalytical solutions. As for the asymptotic behavior, for m = 2 it reveals in Mainardi et al. [21]that the dominated decay rate of the solution is related to the minimum order of time fractionalderivative. On the other hand, Beckers & Yamamoto [4] investigated (1.1)–(1.3) in a slightlymore general formulation and obtained a weaker regularity result than that in [26].In this paper, we are concerned with the well-posedness and the long-time asymptotic be-havior of the solution to the initial-boundary value problem (1.1)–(1.3), and we attempt toestablish results parallel to that for the single-term case. On basis of the explicit representationof the solution, we give estimates for the solution by exploiting several properties of the multi-nomial Mittag-Leffler function. Moreover, as long as the continuous dependency of the solution ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations on the initial value and the source term is verified, we can also deduce the Lipschitz stabilityof the solution to (1.1)–(1.3) with respect to α j , q j ( j = 1 , . . . , m ) and the diffusion coefficientimmediately. For the long-time asymptotic behavior, we employ the Laplace transform in timeto show that the decay rate as t → ∞ is indeed t − α m , where α m is the minimum order ofCaputo derivatives in time.The rest of this paper is organized as follows. The main results concerning problem (1.1)–(1.3) are collected in Section 2, which includes theorems on well-posedness and long-time asymp-totic behavior of the solution. The proofs of the main theorems are postponed to Section 3,which is further divided into three subsections. Subsection 3.1 is devoted to a close scrutinyof the multinomial Mittag-Leffler functions, which serves as essential keys in the proofs ofwell-posedness results in Subsection 3.2. Due to the difference of techniques, the asymptoticbehavior is proved independently in Subsection 3.3. Finally, concluding remarks are given inSection 4. In this section, we state the main results obtained in this paper. More precisely, we givea priori estimates for the solution u to (1.1)–(1.3) with respect to the initial value (Theorem2.1), the source term (Theorem 2.2), and Lipschitz continuous dependence of the solutions oncoefficients and orders (Theorem 2.3) so that stability and uniqueness follow, and we describethe asymptotic behavior of the solution in Theorem 2.4.To this end, we first fix some general settings and notations. Let L (Ω) be a usual L -space with the inner product ( · , · ) and H (Ω), H (Ω) denote the Sobolev spaces (see, e.g.,Adams [2]). The elliptic operator L is defined for f ∈ D ( − L ) := H (Ω) ∩ H (Ω) as Lf ( x ) = d X i,j =1 ∂ j ( a ij ( x ) ∂ i f ( x )) + c ( x ) f ( x ) , x ∈ Ω , where a ij = a ji (1 ≤ i, j ≤ d ) and c ≤ a ij ∈ C (Ω), c ∈ C (Ω) and there exists a constant δ > δ d X i =1 ξ i ≤ d X i,j =1 a ij ( x ) ξ i ξ j , ∀ x ∈ Ω , ∀ ( ξ , . . . , ξ d ) ∈ R d . On the other hand, let { λ n , ϕ n } ∞ n =1 be the eigensystem of the elliptic operator − L such that0 < λ < λ ≤ · · · , λ n → ∞ as n → ∞ and { ϕ n } ⊂ H (Ω) ∩ H (Ω) forms an orthonormalbasis of L (Ω). Then the fractional power ( − L ) γ is well-defined for γ ∈ R (see Pazy [23]) with D (( − L ) γ ) = ( f ∈ L (Ω); ∞ X n =1 | λ γn ( f, ϕ n ) | < ∞ ) , and D (( − L ) γ ) is a Hilbert space with the norm k f k D (( − L ) γ ) = ∞ X n =1 | λ γn ( f, ϕ n ) | ! / . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
0) = a ( x ) , x ∈ Ω , (1.1)(1.2)(1.3)where L is a symmetric uniformly elliptic operator with the homogeneous Dirichlet boundarycondition, and we can assume q = 1 without lose of generality. The regularities of the initialvalue a and the source term F will be specified later. Here ∂ α j t denotes the Caputo derivative Manuscript last updated: October 2, 2018. † Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku,Tokyo 153-8914, Japan.E-mail: [email protected], [email protected], [email protected] ∗ This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations defined by ∂ α j t f ( t ) := 1Γ(1 − α j ) Z t f ′ ( s )( t − s ) α j d s, where Γ( · ) is a usual Gamma function. For various properties of the Caputo derivative, werefer to Kilbas et al. [12] and Podlubny [24]. See also [8, 28] for further contents on fractionalcalculus.In the case of m = 1, equation (1.1) is reduced to its single-term counterpart ∂ αt u = Lu + F in Ω × (0 , T ] , α ∈ (0 , . (1.4)The above formulation has been studied extensively from different aspects due to its vastcapability of modeling the anomalous diffusion phenomena in highly heterogeneous aquifer andcomplex viscoelastic material (see Adams & Gelhar [1], Ginoa et al. [6], Hatano & Hatano [9],Nigmatullin [22] and the references therein). Indeed, although the single-term time-fractionaldiffusion equation inherits certain properties from the diffusion equation with time derivativeof natural number order, it differs considerably from the traditional one especially in senseof its limited smoothing effect in space and slow decay in time. In Luchko [16], a maximumprinciple of the initial-boundary value problem for (1.4) was established, and the uniqueness ofa classical solution was proved. Luchko [17] represented the generalized solution to (1.4) with F = 0 by means of the Mittag-Leffler function and gave the unique existence result. Sakamoto& Yamamoto [26] carried out a comprehensive investigation including the well-posedness of theinitial-boundary value problem for (1.4) as well as the long-time asymptotic behavior of thesolution. It turns out that the spatial regularity of the solution is only moderately improvedfrom that of the initial value, and the solution decays with order t − α as t → ∞ . Recently, theLipschitz stability of the solution to (1.4) with respect to α and the diffusion coefficient wasproved as a byproduct of an inverse coefficient problem in Li et al. [13]. For other discussionsconcerning equation (1.4), see e.g., Gorenflo et al. [7] and Luchko [15], Pr¨uss [25]. Regardingnumerical treatments, we refer to Liu et al. [14] and Meerschaert & Tadjeran [20] for the finitedifference method and Jin et al. [11] for the finite element method.As natural extension, equation (1.1) is expected to improve the modeling accuracy in de-picting the anomalous diffusion due to its potential feasibility. However, to the authors’ bestknowledge, published works on this extension are quite limited in spite of rich literatures onits single-term version. Luchko [18] developed the maximum principle for problem (1.1)–(1.3)and constructed a generalized solution when F = 0 by means of the multinomial Mittag-Lefflerfunctions. Jiang et al. [10] considered fractional derivatives in both time and space and derivedanalytical solutions. As for the asymptotic behavior, for m = 2 it reveals in Mainardi et al. [21]that the dominated decay rate of the solution is related to the minimum order of time fractionalderivative. On the other hand, Beckers & Yamamoto [4] investigated (1.1)–(1.3) in a slightlymore general formulation and obtained a weaker regularity result than that in [26].In this paper, we are concerned with the well-posedness and the long-time asymptotic be-havior of the solution to the initial-boundary value problem (1.1)–(1.3), and we attempt toestablish results parallel to that for the single-term case. On basis of the explicit representationof the solution, we give estimates for the solution by exploiting several properties of the multi-nomial Mittag-Leffler function. Moreover, as long as the continuous dependency of the solution ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations on the initial value and the source term is verified, we can also deduce the Lipschitz stabilityof the solution to (1.1)–(1.3) with respect to α j , q j ( j = 1 , . . . , m ) and the diffusion coefficientimmediately. For the long-time asymptotic behavior, we employ the Laplace transform in timeto show that the decay rate as t → ∞ is indeed t − α m , where α m is the minimum order ofCaputo derivatives in time.The rest of this paper is organized as follows. The main results concerning problem (1.1)–(1.3) are collected in Section 2, which includes theorems on well-posedness and long-time asymp-totic behavior of the solution. The proofs of the main theorems are postponed to Section 3,which is further divided into three subsections. Subsection 3.1 is devoted to a close scrutinyof the multinomial Mittag-Leffler functions, which serves as essential keys in the proofs ofwell-posedness results in Subsection 3.2. Due to the difference of techniques, the asymptoticbehavior is proved independently in Subsection 3.3. Finally, concluding remarks are given inSection 4. In this section, we state the main results obtained in this paper. More precisely, we givea priori estimates for the solution u to (1.1)–(1.3) with respect to the initial value (Theorem2.1), the source term (Theorem 2.2), and Lipschitz continuous dependence of the solutions oncoefficients and orders (Theorem 2.3) so that stability and uniqueness follow, and we describethe asymptotic behavior of the solution in Theorem 2.4.To this end, we first fix some general settings and notations. Let L (Ω) be a usual L -space with the inner product ( · , · ) and H (Ω), H (Ω) denote the Sobolev spaces (see, e.g.,Adams [2]). The elliptic operator L is defined for f ∈ D ( − L ) := H (Ω) ∩ H (Ω) as Lf ( x ) = d X i,j =1 ∂ j ( a ij ( x ) ∂ i f ( x )) + c ( x ) f ( x ) , x ∈ Ω , where a ij = a ji (1 ≤ i, j ≤ d ) and c ≤ a ij ∈ C (Ω), c ∈ C (Ω) and there exists a constant δ > δ d X i =1 ξ i ≤ d X i,j =1 a ij ( x ) ξ i ξ j , ∀ x ∈ Ω , ∀ ( ξ , . . . , ξ d ) ∈ R d . On the other hand, let { λ n , ϕ n } ∞ n =1 be the eigensystem of the elliptic operator − L such that0 < λ < λ ≤ · · · , λ n → ∞ as n → ∞ and { ϕ n } ⊂ H (Ω) ∩ H (Ω) forms an orthonormalbasis of L (Ω). Then the fractional power ( − L ) γ is well-defined for γ ∈ R (see Pazy [23]) with D (( − L ) γ ) = ( f ∈ L (Ω); ∞ X n =1 | λ γn ( f, ϕ n ) | < ∞ ) , and D (( − L ) γ ) is a Hilbert space with the norm k f k D (( − L ) γ ) = ∞ X n =1 | λ γn ( f, ϕ n ) | ! / . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Also we note that D (( − L ) γ ) ⊂ H γ (Ω) for γ > D (( − L ) / ) = H (Ω).Now we are well-prepared to consider the dependency of the solution u to the initial-boundary value problem (1.1)–(1.3) upon the initial value a and the source term F . In view ofthe superposition principle, it suffices to deal with the cases F = 0, a = 0 and a = 0, F = 0separately. Theorem 2.1
Let F = 0 , ≤ γ ≤ and a ∈ D (( − L ) γ ) , where we interpret − γ = ∞ if γ = 1 . Concerning the solution u to the initial-boundary value problem (1.1) – (1.3) , thefollowings hold true. (a) There exists a unique solution u ∈ C ([0 , T ]; L (Ω)) ∩ C ((0 , T ]; H (Ω) ∩ H (Ω)) to (1.1) – (1.3) . Actually, u ∈ L − γ (0 , T ; H (Ω) ∩ H (Ω)) and there exists a constant C > such that k u k C ([0 ,T ]; L (Ω)) ≤ C k a k L (Ω) , (2.1) k u ( · , t ) k H (Ω) ≤ C k a k D (( − L ) γ ) t α ( γ − , < t ≤ T. (2.2)(b) We have lim t → k u ( · , t ) − a k D (( − L ) γ ) = 0 . (2.3)(c) There holds ∂ t u ∈ C ((0 , T ]; L (Ω)) . Moreover, there exists a constant C > such that k ∂ t u ( · , t ) k L (Ω) ≤ C k a k D (( − L ) γ ) t α γ − , < t ≤ T. (2.4)(d) If γ > , then ∂ βt u ∈ L − γ (0 , T ; L (Ω)) for < β ≤ α . Moreover, for < β < , thereexists a constant C > such that k ∂ βt u ( · , t ) k L (Ω) ≤ C k a k D (( − L ) γ ) t α γ − β , < t ≤ T. (2.5) Theorem 2.2
Let a = 0 , ≤ p ≤ ∞ , ≤ γ ≤ and F ∈ L p (0 , T ; D (( − L ) γ )) , where weinterpret /p = 0 if p = ∞ . Concerning the solution u to the initial-boundary value problem (1.1) – (1.3) , the followings hold true. (a) If p = 2 , then there exists a unique solution u ∈ L (0 , T ; D (( − L ) γ +1 )) to (1.1) – (1.3) .Moreover, there exists a constant C > such that k u k L (0 ,T ; D (( − L ) γ +1 )) ≤ C k F k L (0 ,T ; D (( − L ) γ )) . (2.6)(b) If p = 2 , then there exists a unique solution u ∈ L p (0 , T ; D (( − L ) γ +1 − τ )) to (1.1) – (1.3) for any τ ∈ (0 , . Moreover, there exists a constant C > such that k u k L p (0 ,T ; D (( − L ) γ +1 − τ )) ≤ Cτ k F k L p (0 ,T ; D (( − L ) γ )) . (2.7)(c) If α p > , then for any τ ∈ ( α p , , there holds lim t → k u ( · , t ) k D (( − L ) γ +1 − τ ) = 0 . (2.8) Remark 2.1
We compare the conclusions in Theorems 2.1–2.2 with those of single-termcases obtained in [26]. In case of the homogeneous source term, i.e. F = 0 in (1.1), it turns outthat Theorem 2.1 is a parallel extension of its single-term counterpart. For instance, in Theorem ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
We compare the conclusions in Theorems 2.1–2.2 with those of single-termcases obtained in [26]. In case of the homogeneous source term, i.e. F = 0 in (1.1), it turns outthat Theorem 2.1 is a parallel extension of its single-term counterpart. For instance, in Theorem ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations a ∈ L (Ω), a ∈ H (Ω) and a ∈ H (Ω) ∩ H (Ω) agreewith those in [26, Theorem 2.1]. Especially, it will be readily seen from the proof of Theorem2.1 that the regularity of the solution u at any positive time can be improved from the initialregularity by 2 orders in space, namely, u ( · , t ) ∈ D (( − L ) γ +1 ) if a ∈ D (( − L ) γ ) for 0 < t ≤ T .On the other hand, if the source term F does not vanish, the improvement of regularity inspace is strictly less than 2 orders except for the special case that F is L in time. For example,if F ∈ L (Ω × (0 , T )), then it follows from Theorem 2.2(a) that u ∈ L (0 , T ; H (Ω) ∩ H (Ω)),which coincides with [26, Theorem 2.2]. However, if F ∈ L p (0 , T ; L (Ω)) with p = 2, thenTheorem 2.2(b) asserts u ∈ L p (0 , T ; D (( − L ) − τ )), where τ ∈ (0 ,
1] can be arbitrarily small butis never zero. The technical reason is that only in case of p = 2 one can take advantage of anewly established property in Bazhlekova [3] (see Lemma 3.4).On basis of these established results, we can consider the dependency of the solution uponsome specified coefficients, especially the orders of Caputo derivatives. More precisely, weevaluate the difference between the solutions u and e u to m X j =1 q j ∂ α j t u = L D u in Ω × (0 , T ] ,u = 0 on ∂ Ω × (0 , T ] ,u | t =0 = a in Ω (2.9)and m X j =1 e q j ∂ e α j t e u = L e D e u in Ω × (0 , T ] , e u = 0 on ∂ Ω × (0 , T ] , e u | t =0 = a in Ω (2.10)respectively, where L D u ( x, t ) := div( D ( x ) ∇ u ( x, t )) and D denotes the diffusion coefficient. Tothis end, we fix 1 > α > α > q > q > δ > M > A := { ( α , . . . , α m ) ∈ R m ; α ≥ α > α > · · · > α m ≥ α } , Q := { ( q , . . . , q m ) ∈ R m ; q = 1 , q j ∈ [ q, q ] ( j = 2 , . . . , m ) } , U := { D ∈ C (Ω); D ≥ δ in Ω , k D k C (Ω) ≤ M } . (2.11)Under these settings, we can show the following result on the Lipschitz stability. Theorem 2.3
Fix γ, τ ∈ (0 , . Let u and e u be the solutions to (2.9) and (2.10) respectively,where a ∈ D (( − L ) γ ) , ( α , . . . , α m ) , ( e α , . . . , e α m ) ∈ A , ( q , . . . , q m ) , ( e q , . . . , e q m ) ∈ Q , D, e D ∈ U and A , Q , U are defined in (2.11) . Then there exists a constant C > depending only on a, A , Q and U such that k u − e u k L − γ (0 ,T ; D (( − L ) − τ )) ≤ Cτ m X j =1 | α j − e α j | + m X j =2 | q j − e q j | + k D − e D k C (Ω) (2.12) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
Fix γ, τ ∈ (0 , . Let u and e u be the solutions to (2.9) and (2.10) respectively,where a ∈ D (( − L ) γ ) , ( α , . . . , α m ) , ( e α , . . . , e α m ) ∈ A , ( q , . . . , q m ) , ( e q , . . . , e q m ) ∈ Q , D, e D ∈ U and A , Q , U are defined in (2.11) . Then there exists a constant C > depending only on a, A , Q and U such that k u − e u k L − γ (0 ,T ; D (( − L ) − τ )) ≤ Cτ m X j =1 | α j − e α j | + m X j =2 | q j − e q j | + k D − e D k C (Ω) (2.12) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations for < γ < and k u − e u k L (0 ,T ; H (Ω)) ≤ C m X j =1 | α j − e α j | + m X j =2 | q j − e q j | + k D − e D k C (Ω) (2.13) for γ ≥ . The above theorem extends a similar result in [13] for the single-term case. It is alsofundamental for the optimization method for an inverse problem of determining α j , q j , D ( x ) byextra data of the solution.In Sakamoto & Yamamoto [26], the decay rate of the solution to the single-term time-fractional diffusion equation (1.4) was shown to be t − α as t → ∞ . Here we give generalizationfor the multi-term case where we specify the principal term of the solution as t → ∞ . Theorem 2.4
Let F = 0 and a ∈ L (Ω) . Then there exists a unique solution u ∈ C ([0 , ∞ ); L (Ω)) ∩ C ((0 , ∞ ); H (Ω) ∩ H (Ω)) to (1.1) – (1.3) . Moreover, there exists a constant C > such that (cid:13)(cid:13)(cid:13)(cid:13) u ( · , t ) − ( − L ) − ( q m a )Γ(1 − α m ) t α m (cid:13)(cid:13)(cid:13)(cid:13) H (Ω) ≤ C k a k L (Ω) t α m − as t → ∞ . (2.14) Remark 2.2
We explain the significance of the Theorem 2.4. It reveals that the decay rateof u ( · , t ) in sense of H (Ω) is exactly t − α m as t → ∞ . In fact, inequality (2.14) implies thatthere exist constants C > C > C k a k L (Ω) t − α m ≤ k u ( · , t ) k H (Ω) ≤ C k a k L (Ω) t − α m as t → ∞ . (2.15)Consequently, it turns out that the decay rate t − α m is the best possible. In other words, if k u ( · , t ) k H (Ω) ≤ C t − β as t → ∞ for any order β > α m and some constant C >
0, then u ( x, t ) = 0 for x ∈ Ω and t >
0. Actually,in this case it is easily inferred from the lower bound in (2.15) that there should be a = 0 in Ω.Therefore, Theorem 2.1 and the upper bound in (2.15) immediately imply u ≡ × (0 , ∞ ).Furthermore, (2.14) also gives the convergence rate of the approximation u ( · , t ) − ( − L ) − ( q m a )Γ(1 − α m ) t − α m → H (Ω) as t → ∞ , that is, t − α m − . In this section, we give proofs for the theorems stated in Section 2.In the discussion of single-term time-fractional diffusion equations, it turns out that thesolutions can be explicitly represented by the usual Mittag-Leffler function E α,β ( z ) := ∞ X k =0 z k Γ( αk + β ) , z ∈ C , α > , β ∈ R , (3.1) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations
0. Actually,in this case it is easily inferred from the lower bound in (2.15) that there should be a = 0 in Ω.Therefore, Theorem 2.1 and the upper bound in (2.15) immediately imply u ≡ × (0 , ∞ ).Furthermore, (2.14) also gives the convergence rate of the approximation u ( · , t ) − ( − L ) − ( q m a )Γ(1 − α m ) t − α m → H (Ω) as t → ∞ , that is, t − α m − . In this section, we give proofs for the theorems stated in Section 2.In the discussion of single-term time-fractional diffusion equations, it turns out that thesolutions can be explicitly represented by the usual Mittag-Leffler function E α,β ( z ) := ∞ X k =0 z k Γ( αk + β ) , z ∈ C , α > , β ∈ R , (3.1) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations and several basic properties play remarkable roles especially for obtaining estimates for the sta-bility. Since explicit solutions to the multi-term case are also available by using a generalizedform of (3.1) called the multinomial Mittag-Leffler function, we shall first investigate this gen-eralization so that similar arguments are still feasible for multi-term time-fractional diffusionequations. The multinomial Mittag-Leffler function is defined as (see Luchko & Gorenflo [19]) E ( β ,...,β m ) ,β ( z , . . . , z m ) := ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 z k j j Γ( β + P mj =1 β j k j ) , (3.2)where we assume 0 < β <
2, 0 < β j < z j ∈ C ( j = 1 , . . . , m ), and ( k ; k , . . . , k m ) denotesthe multinomial coefficient( k ; k , . . . , k m ) := k ! k ! · · · k m ! with k = m X j =1 k j , where k j , 1 ≤ j ≤ m , are non-negative integers. We recall the following formula for multinomialcoefficients (see Berge [5]) m X j =1 ( k − k , . . . , k j − , k j − , k j +1 , . . . , k m ) = ( k ; k , . . . , k m ) . (3.3)If some k j vanishes, we understand ( k − k , . . . , k j − , k j − , k j +1 , . . . , k m ) = 0 and (3.3)degenerates to its lower dimensional version.Concerning the relation between multinomial Mittag-Leffler functions with different param-eters, we have the following lemma. Lemma 3.1
Let < β < , < β j < j = 1 , . . . , m ) and z j ∈ C ( j = 1 , . . . , m ) be fixed.Then β ) + m X j =1 z j E ( β ,...,β m ) ,β + β j ( z , . . . , z m ) = E ( β ,...,β m ) ,β ( z , . . . , z m ) . Proof.
According to definition (3.2), direct calculations yield m X j =1 z j E ( β ,...,β m ) ,β + β j ( z , . . . , z m )= m X j =1 ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) z j Q mℓ =1 z k ℓ ℓ Γ( β + β j + P mℓ =1 β ℓ k ℓ )= ∞ X k =0 m X j =1 z k +1 j Γ( β + β j ( k + 1)) + X k + ··· + k m = kk j According to definition (3.2), direct calculations yield m X j =1 z j E ( β ,...,β m ) ,β + β j ( z , . . . , z m )= m X j =1 ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) z j Q mℓ =1 z k ℓ ℓ Γ( β + β j + P mℓ =1 β ℓ k ℓ )= ∞ X k =0 m X j =1 z k +1 j Γ( β + β j ( k + 1)) + X k + ··· + k m = kk j Let < β < and > α > · · · > α m > be given. Assume that α π/ <µ < α π, µ ≤ | arg( z ) | ≤ π and there exists K > such that − K ≤ z j < j = 2 , . . . , m ) .Then there exists a constant C > depending only on µ, K, α j ( j = 1 , . . . , m ) and β such that | E ( α ,α − α ,...,α − α m ) ,β ( z , . . . , z m ) | ≤ C | z | . Proof. Let α j , z j ( j = 1 , . . . , m ) and β be assumed as above and introduce the notation E α ′ ,β ( z , . . . , z m ) := E ( α ,α − α ,...,α − α m ) ,β ( z , . . . , z m ) . In the sequel, we denote by C a general positive constant depending at most on µ , K , α j ( j = 1 , . . . , m ) and β . First we rewrite the multinomial Mittag-Leffler function (3.2) in analternative form with the aid of the contour integral representation of 1 / Γ( z ) (see [24, § z ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − z − α ) /α d ζ, where R > α π/ < θ < µ . Here γ ( R, θ ) denotes thecontour γ ( R, θ ) := { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } ∪ { ζ ∈ C ; | ζ | > R, | arg( ζ ) | = ± θ } . Then it follows from the multinomial formula that E α ′ ,β ( z , . . . , z m )= 12 α π i ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) m Y j =1 z k j j ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Let α j , z j ( j = 1 , . . . , m ) and β be assumed as above and introduce the notation E α ′ ,β ( z , . . . , z m ) := E ( α ,α − α ,...,α − α m ) ,β ( z , . . . , z m ) . In the sequel, we denote by C a general positive constant depending at most on µ , K , α j ( j = 1 , . . . , m ) and β . First we rewrite the multinomial Mittag-Leffler function (3.2) in analternative form with the aid of the contour integral representation of 1 / Γ( z ) (see [24, § z ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − z − α ) /α d ζ, where R > α π/ < θ < µ . Here γ ( R, θ ) denotes thecontour γ ( R, θ ) := { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } ∪ { ζ ∈ C ; | ζ | > R, | arg( ζ ) | = ± θ } . Then it follows from the multinomial formula that E α ′ ,β ( z , . . . , z m )= 12 α π i ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) m Y j =1 z k j j ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations × (Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β − α ( k +1) − α k −···− α m k m ) /α d ζ ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − × ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) (cid:18) z ζ (cid:19) k m Y j =2 (cid:18) z j ζ − α j /α (cid:19) k j d ζ = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − ∞ X k =0 z ζ + m X j =2 z k ζ − α j /α k d ζ. In order to guarantee the convergence of the summation with respect to k , it is required that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ζ + m X j =2 z j ζ − α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < , ∀ ζ ∈ γ ( R, θ ) . Since | z j | ≤ K for j = 2 , . . . , m , the above inequality is achieved by taking R such that R > | z | + K m X j =2 R α j /α . Moreover, if we restrict, for example, | z | ≤ K , then R can be fixed as a constant dependingonly on K and α j ( j = 1 , . . . , m ). Now we deduce for | z j | ≤ K ( j = 1 , . . . , m ) that E α ′ ,β ( z , . . . , z m ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ. (3.6)Next we fix z , . . . , z m as negative parameters and regard both sides of (3.6) as functionsof the single complex variable z , which allows the application of the principle of analyticcontinuation to extend equality (3.6) to a domain including { z ∈ C ; µ ≤ | arg( z ) | ≤ π } (seeFigure 1).For | z | > R , we investigate the denominator of the integrand in (3.6). Since z j < α j < α for j = 2 , . . . , m , it turns out that the curve ζ − P mj =2 z j ζ α j /α ( ζ ∈ γ ( R, θ )) locateson the right-hand side of γ ( R, θ ); that is, γ ( R, θ ) is shifted by the term − P mj =2 z j ζ α j /α to thepositive direction. This observation immediately impliesmin ζ ∈ γ ( R,θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ − z − m X j =2 z j ζ α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ min ζ ∈ γ ( R,θ ) | ζ − z | ≥ | z | sin( µ − θ ) . Therefore, we come up with the estimate | E α ′ ,β ( z , . . . , z m ) | = 12 α π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α π sin( µ − θ ) Z γ ( R,θ ) | exp( ζ /α ) || ζ (1 − β ) /α | d ζ ! | z | . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Let α j , z j ( j = 1 , . . . , m ) and β be assumed as above and introduce the notation E α ′ ,β ( z , . . . , z m ) := E ( α ,α − α ,...,α − α m ) ,β ( z , . . . , z m ) . In the sequel, we denote by C a general positive constant depending at most on µ , K , α j ( j = 1 , . . . , m ) and β . First we rewrite the multinomial Mittag-Leffler function (3.2) in analternative form with the aid of the contour integral representation of 1 / Γ( z ) (see [24, § z ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − z − α ) /α d ζ, where R > α π/ < θ < µ . Here γ ( R, θ ) denotes thecontour γ ( R, θ ) := { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } ∪ { ζ ∈ C ; | ζ | > R, | arg( ζ ) | = ± θ } . Then it follows from the multinomial formula that E α ′ ,β ( z , . . . , z m )= 12 α π i ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) m Y j =1 z k j j ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations × (Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β − α ( k +1) − α k −···− α m k m ) /α d ζ ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − × ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) (cid:18) z ζ (cid:19) k m Y j =2 (cid:18) z j ζ − α j /α (cid:19) k j d ζ = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − ∞ X k =0 z ζ + m X j =2 z k ζ − α j /α k d ζ. In order to guarantee the convergence of the summation with respect to k , it is required that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ζ + m X j =2 z j ζ − α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < , ∀ ζ ∈ γ ( R, θ ) . Since | z j | ≤ K for j = 2 , . . . , m , the above inequality is achieved by taking R such that R > | z | + K m X j =2 R α j /α . Moreover, if we restrict, for example, | z | ≤ K , then R can be fixed as a constant dependingonly on K and α j ( j = 1 , . . . , m ). Now we deduce for | z j | ≤ K ( j = 1 , . . . , m ) that E α ′ ,β ( z , . . . , z m ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ. (3.6)Next we fix z , . . . , z m as negative parameters and regard both sides of (3.6) as functionsof the single complex variable z , which allows the application of the principle of analyticcontinuation to extend equality (3.6) to a domain including { z ∈ C ; µ ≤ | arg( z ) | ≤ π } (seeFigure 1).For | z | > R , we investigate the denominator of the integrand in (3.6). Since z j < α j < α for j = 2 , . . . , m , it turns out that the curve ζ − P mj =2 z j ζ α j /α ( ζ ∈ γ ( R, θ )) locateson the right-hand side of γ ( R, θ ); that is, γ ( R, θ ) is shifted by the term − P mj =2 z j ζ α j /α to thepositive direction. This observation immediately impliesmin ζ ∈ γ ( R,θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ − z − m X j =2 z j ζ α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ min ζ ∈ γ ( R,θ ) | ζ − z | ≥ | z | sin( µ − θ ) . Therefore, we come up with the estimate | E α ′ ,β ( z , . . . , z m ) | = 12 α π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α π sin( µ − θ ) Z γ ( R,θ ) | exp( ζ /α ) || ζ (1 − β ) /α | d ζ ! | z | . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations xy C Rθµ γ ( R, θ ) AA BB − K − K Figure 1. Settings of Lemma 3.2 and the contour γ ( R, θ ). If z is located in the shaded domain A , we employ the principle of analytic continuation and the contour integral representation(3.6). When z is in the shaded domain B , it suffices to argue by definition (3.2).The integral along γ ( R, θ ) converges, because for ζ such that arg( ζ ) = ± θ and | ζ | > R , thereholds | exp( ζ /α ) | = exp( | ζ | /α cos( θ/α )) with cos( θ/α ) < , while the integral on the arc { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } is a constant. Consequently | E α ′ ,β ( z , . . . , z m ) | ≤ C | z | , µ ≤ | arg( z ) | ≤ π, | z | > R. (3.7)For µ ≤ | arg( z ) | ≤ π such that | z | ≤ R , it is directly verified that | E α ′ ,β ( z , . . . , z m ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 z k j j Γ( β + α k − P mj =2 α j k j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 | z j | k j Γ( β + α k − P mj =2 α j k j ) ≤ C ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 | z j | k j Γ( β + ( α − α ) k )= C ∞ X k =0 β + ( α − α ) k ) m X j =1 | z j | k ≤ C ∞ X k =0 ( R + ( m − K ) k Γ( β + ( α − α ) k ) ≤ C, which, together with (3.7), finishes the proof. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Let α j , z j ( j = 1 , . . . , m ) and β be assumed as above and introduce the notation E α ′ ,β ( z , . . . , z m ) := E ( α ,α − α ,...,α − α m ) ,β ( z , . . . , z m ) . In the sequel, we denote by C a general positive constant depending at most on µ , K , α j ( j = 1 , . . . , m ) and β . First we rewrite the multinomial Mittag-Leffler function (3.2) in analternative form with the aid of the contour integral representation of 1 / Γ( z ) (see [24, § z ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − z − α ) /α d ζ, where R > α π/ < θ < µ . Here γ ( R, θ ) denotes thecontour γ ( R, θ ) := { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } ∪ { ζ ∈ C ; | ζ | > R, | arg( ζ ) | = ± θ } . Then it follows from the multinomial formula that E α ′ ,β ( z , . . . , z m )= 12 α π i ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) m Y j =1 z k j j ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations × (Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β − α ( k +1) − α k −···− α m k m ) /α d ζ ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − × ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) (cid:18) z ζ (cid:19) k m Y j =2 (cid:18) z j ζ − α j /α (cid:19) k j d ζ = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α − ∞ X k =0 z ζ + m X j =2 z k ζ − α j /α k d ζ. In order to guarantee the convergence of the summation with respect to k , it is required that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ζ + m X j =2 z j ζ − α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < , ∀ ζ ∈ γ ( R, θ ) . Since | z j | ≤ K for j = 2 , . . . , m , the above inequality is achieved by taking R such that R > | z | + K m X j =2 R α j /α . Moreover, if we restrict, for example, | z | ≤ K , then R can be fixed as a constant dependingonly on K and α j ( j = 1 , . . . , m ). Now we deduce for | z j | ≤ K ( j = 1 , . . . , m ) that E α ′ ,β ( z , . . . , z m ) = 12 α π i Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ. (3.6)Next we fix z , . . . , z m as negative parameters and regard both sides of (3.6) as functionsof the single complex variable z , which allows the application of the principle of analyticcontinuation to extend equality (3.6) to a domain including { z ∈ C ; µ ≤ | arg( z ) | ≤ π } (seeFigure 1).For | z | > R , we investigate the denominator of the integrand in (3.6). Since z j < α j < α for j = 2 , . . . , m , it turns out that the curve ζ − P mj =2 z j ζ α j /α ( ζ ∈ γ ( R, θ )) locateson the right-hand side of γ ( R, θ ); that is, γ ( R, θ ) is shifted by the term − P mj =2 z j ζ α j /α to thepositive direction. This observation immediately impliesmin ζ ∈ γ ( R,θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ − z − m X j =2 z j ζ α j /α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ min ζ ∈ γ ( R,θ ) | ζ − z | ≥ | z | sin( µ − θ ) . Therefore, we come up with the estimate | E α ′ ,β ( z , . . . , z m ) | = 12 α π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z γ ( R,θ ) exp( ζ /α ) ζ (1 − β ) /α ζ − z − P mj =2 z j ζ α j /α d ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α π sin( µ − θ ) Z γ ( R,θ ) | exp( ζ /α ) || ζ (1 − β ) /α | d ζ ! | z | . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations xy C Rθµ γ ( R, θ ) AA BB − K − K Figure 1. Settings of Lemma 3.2 and the contour γ ( R, θ ). If z is located in the shaded domain A , we employ the principle of analytic continuation and the contour integral representation(3.6). When z is in the shaded domain B , it suffices to argue by definition (3.2).The integral along γ ( R, θ ) converges, because for ζ such that arg( ζ ) = ± θ and | ζ | > R , thereholds | exp( ζ /α ) | = exp( | ζ | /α cos( θ/α )) with cos( θ/α ) < , while the integral on the arc { ζ ∈ C ; | ζ | = R, | arg( ζ ) | ≤ θ } is a constant. Consequently | E α ′ ,β ( z , . . . , z m ) | ≤ C | z | , µ ≤ | arg( z ) | ≤ π, | z | > R. (3.7)For µ ≤ | arg( z ) | ≤ π such that | z | ≤ R , it is directly verified that | E α ′ ,β ( z , . . . , z m ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 z k j j Γ( β + α k − P mj =2 α j k j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 | z j | k j Γ( β + α k − P mj =2 α j k j ) ≤ C ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m ) Q mj =1 | z j | k j Γ( β + ( α − α ) k )= C ∞ X k =0 β + ( α − α ) k ) m X j =1 | z j | k ≤ C ∞ X k =0 ( R + ( m − K ) k Γ( β + ( α − α ) k ) ≤ C, which, together with (3.7), finishes the proof. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations For later use, we adopt the abbreviation E ( n ) α ′ ,β ( t ) := E ( α ,α − α ,...,α − α m ) ,β ( − λ n t α , − q t α − α , . . . , − q m t α − α m ) , t > , (3.8)where λ n is the n -th eigenvalue of − L , 0 < β < 2, and α j , q j are those positive constants in(1.1). Especially, regarding the derivative of t α E ( n ) α ′ , α ( t ) with respect to t > 0, we state thefollowing technical lemma. Lemma 3.3 Let > α > · · · > α m > . Then dd t n t α E ( n ) α ′ , α ( t ) o = t α − E ( n ) α ′ ,α ( t ) , t > . Proof. By definition, we carry out a direct differentiation and utilize the formula Γ( s ) = Γ( s +1) /s to derivedd t n t α E ( n ) α ′ , α ( t ) o = dd t ( ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m )( − λ n ) k Q mj =2 ( − q j ) k j t α ( k +1) − α k −···− α m k m Γ(1 + α ( k + 1) − P mj =2 α j k j ) ) = ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m )( − λ n ) k Q mj =2 ( − q j ) k j t α ( k +1) − α k −···− α m k m − Γ( α ( k + 1) − P mj =2 α j k j )= t α − ∞ X k =0 X k + ··· + k m = k ( k ; k , . . . , k m )( − λ n t α ) k Q mj =2 ( − q j t α − α j ) k j Γ( α + α k + P mj =2 ( α − α j ) k j )= t α − E ( n ) α ′ ,α ( t ) . Here we use the fact that t α E ( n ) α ′ , α ( t ) is real analytic for t > Now we are ready to employ the multinomial Mittag-Leffler functions to show results onthe well-posedness. For later use we recall the eigensystem { λ n , ϕ n } of the elliptic operator − L and the abbreviation E ( n ) α ′ ,β ( t ) (0 < β < 2) in (3.8).First we prove Theorem 2.1, that is, the case of vanishing source term F . It was shownin [18] that the explicit solution to (1.1)–(1.3) is given by u ( · , t ) = ∞ X n =1 (cid:16) − λ n t α E ( n ) α ′ , α ( t ) (cid:17) ( a, ϕ n ) ϕ n . (3.9)With the aid of Lemmata 3.1–3.3, it is straightforward to demonstrate the well-posedness bydominating the solution by the initial value. Proof of Theorem 2.1. Let a ∈ D (( − L ) γ ) with 0 ≤ γ ≤ 1. In the sequel, by C we refer topositive constants independent of the initial value a which may vary from line by line.(a) First, a direct application of Lemma 3.2 yields (cid:12)(cid:12)(cid:12) − λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) ≤ λ n t α C λ n t α ≤ C. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 1. In the sequel, by C we refer topositive constants independent of the initial value a which may vary from line by line.(a) First, a direct application of Lemma 3.2 yields (cid:12)(cid:12)(cid:12) − λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) ≤ λ n t α C λ n t α ≤ C. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Thus, we take advantage of (3.9) to derive k u ( · , t ) k L (Ω) = ( ∞ X n =1 (cid:12)(cid:12)(cid:12) − λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) | ( a, ϕ n ) | ) / ≤ C k a k L (Ω) (3.10)for 0 < t ≤ T , where we use the fact that { ϕ n } forms an orthonormal basis of L (Ω). Since thesummation in (3.9) converges in L (Ω) uniformly in t ∈ [0 , T ], we get u ∈ C ([0 , T ]; L (Ω)) or(2.1). Furthermore, by the definition of D ( − L ), we see k u ( · , t ) k D ( − L ) = ∞ X n =1 (cid:16) λ n (cid:12)(cid:12)(cid:12) − λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12)(cid:17) | ( a, ϕ n ) | . In order to treat the term 1 − λ n t α E ( n ) α ′ , α ( t ), we substitute β = 1 , β = α , z = − λ n t α , β j = α − α j and z j = − q j t α − α j ( j = 2 , . . . , m )in Lemma 3.1 and then utilize Lemma 3.2 to deduce (cid:12)(cid:12)(cid:12) − λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( n ) α ′ , ( t ) + m X j =2 q j t α − α j E ( n ) α ′ , α − α j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E ( n ) α ′ , ( t ) (cid:12)(cid:12)(cid:12) + C m X j =2 t α − α j (cid:12)(cid:12)(cid:12) E ( n ) α ′ , α − α j ( t ) (cid:12)(cid:12)(cid:12) ≤ C m X j =1 t α − α j λ n t α . Therefore, for 0 < t ≤ T , we estimate k u ( · , t ) k D ( − L ) = ∞ X n =1 (cid:12)(cid:12)(cid:12) λ − γn (cid:16) − λ n t α E ( n ) α ′ , α ( t ) (cid:17)(cid:12)(cid:12)(cid:12) | λ γn ( a, ϕ n ) | ≤ C ∞ X n =1 m X j =1 λ − γn t α − α j λ n t α | λ γn ( a, ϕ n ) | ≤ C ∞ X n =1 m X j =1 ( λ n t α ) − γ λ n t α t α γ − α j | λ γn ( a, ϕ n ) | ≤ C m X j =1 t α γ − α j ∞ X n =1 | λ γn ( a, ϕ n ) | ≤ (cid:16) C k a k D (( − L ) γ ) t α ( γ − (cid:17) , where we use the fact ( λ n t α ) − γ λ n t α ≤ 11 + λ n t α if λ n t α ≤ λ n t α λ n t α if λ n t α ≥ ≤ D ( − L ) ⊂ H (Ω), yield the estimate (2.2).Furthermore, it follows immediately from (2.2) and α < u ∈ L − γ (0 , T ; H (Ω) ∩ H (Ω)).(b) In order to investigate the asymptotic behavior near t = 0, first we have k u ( · , t ) − a k D (( − L ) γ ) = ∞ X n =1 (cid:12)(cid:12)(cid:12) λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) | λ γn ( a, ϕ n ) | ≤ (cid:0) C k a k D (( − L ) γ ) (cid:1) < ∞ ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 11 + λ n t α if λ n t α ≤ λ n t α λ n t α if λ n t α ≥ ≤ D ( − L ) ⊂ H (Ω), yield the estimate (2.2).Furthermore, it follows immediately from (2.2) and α < u ∈ L − γ (0 , T ; H (Ω) ∩ H (Ω)).(b) In order to investigate the asymptotic behavior near t = 0, first we have k u ( · , t ) − a k D (( − L ) γ ) = ∞ X n =1 (cid:12)(cid:12)(cid:12) λ n t α E ( n ) α ′ , α ( t ) (cid:12)(cid:12)(cid:12) | λ γn ( a, ϕ n ) | ≤ (cid:0) C k a k D (( − L ) γ ) (cid:1) < ∞ ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations for 0 ≤ t ≤ T by a direct calculation and Lemma 3.2. On the other hand, in view of Lemma3.1, the term λ n t α E ( n ) α ′ , α ( t ) can be rewritten as λ n t α E ( n ) α ′ , α ( t ) = − ( E ( n ) α ′ , ( t ) − − m X j =2 q j t α − α j E ( n ) α ′ , α − α j ( t ) . Thanks to the fact that lim t → ( E ( n ) α ′ , ( t ) − 1) = 0 and the boundedness of E ( n ) α ′ , α − α j ( j =2 , . . . , m ) by Lemma 3.2 for each n = 1 , , . . . , the above observation implieslim t → ( λ n t α E ( n ) α ′ , α ( t )) = 0 , ∀ n = 1 , , . . . . Therefore, (2.3) follows immediately from Lebesgue’s dominated convergence theorem.(c) In order to deal with ∂ t u , we make use of Lemma 3.3 to obtain ∂ t u ( · , t ) = − t α − ∞ X n =1 λ n E ( n ) α ′ ,α ( t )( a, ϕ n ) ϕ n . Then a similar argument to that for (2.2) indicates k ∂ t u ( · , t ) k L (Ω) = t α − ∞ X n =1 (cid:12)(cid:12)(cid:12) λ − γn E ( n ) α ′ ,α ( t ) (cid:12)(cid:12)(cid:12) | λ γn ( a, ϕ n ) | ≤ C t α − ∞ X n =1 (cid:18) ( λ n t α ) − γ λ n t α t α ( γ − (cid:19) | λ γn ( a, ϕ n ) | ≤ (cid:0) C k a k D (( − L ) γ ) t α γ − (cid:1) , < t ≤ T or (2.4). This implies ∂ t u ∈ C ((0 , T ]; L (Ω)) immediately.(d) Finally, to give estimates for ∂ βt u with 0 < β < γ > 0, we employ (2.4) and turnto the definition of the Caputo derivative to obtain k ∂ βt u ( · , t ) k L (Ω) = 1Γ(1 − β ) (cid:13)(cid:13)(cid:13)(cid:13)Z t ∂ s u ( · , s )( t − s ) β d s (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C Z t k ∂ s u ( · , s ) k L (Ω) ( t − s ) β d s ≤ C k a k D (( − L ) γ ) Z t s α γ − ( t − s ) − β d s ≤ C k a k D (( − L ) γ ) t α γ − β , < t ≤ T or (2.5), where the first inequality follows from Minkowski’s inequality for integrals. Especially,as long as β ≤ α , there holds α γ − β > γ − ∂ βt v ∈ L − γ (0 , T ; L (Ω)).Collecting all the results above, we complete the proof of Theorem 2.1.Next we turn to the proof of Theorem 2.2, that is, the case of vanishing initial value a . Toconstruct an explicit solution, we apply the eigenfunction expansion method. In other words,we seek for a solution to (1.1)–(1.3) of the particular form u ( · , t ) = ∞ X n =1 T n ( t ) ϕ n , < t ≤ T, (3.11)where ϕ n is the n -th eigenfunction of − L . The substitution of (3.11) into (1.1) yields ∞ X n =1 m X j =1 q j ∂ α j t T n ( t ) ϕ n = − ∞ X n =1 λ n T n ( t ) ϕ n + ∞ X n =1 ( F ( · , t ) , ϕ n ) ϕ n . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 0, we employ (2.4) and turnto the definition of the Caputo derivative to obtain k ∂ βt u ( · , t ) k L (Ω) = 1Γ(1 − β ) (cid:13)(cid:13)(cid:13)(cid:13)Z t ∂ s u ( · , s )( t − s ) β d s (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C Z t k ∂ s u ( · , s ) k L (Ω) ( t − s ) β d s ≤ C k a k D (( − L ) γ ) Z t s α γ − ( t − s ) − β d s ≤ C k a k D (( − L ) γ ) t α γ − β , < t ≤ T or (2.5), where the first inequality follows from Minkowski’s inequality for integrals. Especially,as long as β ≤ α , there holds α γ − β > γ − ∂ βt v ∈ L − γ (0 , T ; L (Ω)).Collecting all the results above, we complete the proof of Theorem 2.1.Next we turn to the proof of Theorem 2.2, that is, the case of vanishing initial value a . Toconstruct an explicit solution, we apply the eigenfunction expansion method. In other words,we seek for a solution to (1.1)–(1.3) of the particular form u ( · , t ) = ∞ X n =1 T n ( t ) ϕ n , < t ≤ T, (3.11)where ϕ n is the n -th eigenfunction of − L . The substitution of (3.11) into (1.1) yields ∞ X n =1 m X j =1 q j ∂ α j t T n ( t ) ϕ n = − ∞ X n =1 λ n T n ( t ) ϕ n + ∞ X n =1 ( F ( · , t ) , ϕ n ) ϕ n . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Therefore, it is readily seen from the orthogonality of { ϕ n } and the homogeneous initial condi-tion (1.3) that T n satisfies an initial value problem for an ordinary differential equation m X j =1 q j ∂ α j t T n ( t ) + λ n T n ( t ) = ( F ( · , t ) , ϕ n ) , < t ≤ T,T n (0) = 0 . Then it follows from [19, Theorem 4.1] that T n ( t ) = Z t s α − E ( n ) α ′ ,α ( s )( F ( · , t − s ) , ϕ n ) d s, implying that the solution takes the form of a convolution u ( · , t ) = Z t U ( s ) F ( · , t − s ) d s, (3.12)where U ( t ) f := t α − ∞ X n =1 E ( n ) α ′ ,α ( t )( f, ϕ n ) ϕ n . (3.13)Before proceeding to the proof, we introduce a key lemma for showing Theorem 2.2(a). Lemma 3.4 (see [3, Theorem 3.2]) The function t α − E ( n ) α ′ ,α ( t ) is positive for t > .Proof of Theorem 2.2. Let F ∈ L p (0 , T ; D (( − L ) γ )) with 1 ≤ p ≤ ∞ and 0 ≤ γ ≤ 1. In thesequel, by C we refer to a general positive constant independent of F and τ .(a) Let p = 2. According to the expression (3.12)–(3.13), formally we write k u ( · , t ) k D ( − L ) = ∞ X n =1 λ n (cid:18)Z t s α − E ( n ) α ′ ,α ( s )( F ( · , t − s ) , ϕ n ) d s (cid:19) . Using Young’s inequality for convolutions, we estimate k u k L (0 ,T ; D ( − L )) = ∞ X n =1 λ n (cid:13)(cid:13)(cid:13)(cid:13)Z t s α − E ( n ) α ′ ,α ( s )( F ( · , t − s ) , ϕ n ) d s (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ) ≤ ∞ X n =1 λ n Z T t α − | E ( n ) α ′ ,α ( t ) | d t ! k ( F ( · , t ) , ϕ n ) k L (0 ,T ) . By Lemma 3.4, we can remove the absolute value of E ( n ) α ′ ,α ( t ) and apply Lemma 3.3 to derive Z T t α − | E ( n ) α ′ ,α ( t ) | d t = Z T t α − E ( n ) α ′ ,α ( t ) d t = T α E ( n ) α ′ , α ( T ) . Consequently, we use Lemma 3.2 to conclude k u k L (0 ,T ; H (Ω)) ≤ C k u k L (0 ,T ; D ( − L )) ≤ C ∞ X n =1 (cid:16) λ n T α E ( n ) α ′ , α ( T ) (cid:17) k ( F ( · , t ) , ϕ n ) k L (0 ,T ) ≤ C ∞ X n =1 k ( F ( · , t ) , ϕ n ) k L (0 ,T ) = (cid:0) C k F k L (Ω × (0 ,T )) (cid:1) . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 1. In thesequel, by C we refer to a general positive constant independent of F and τ .(a) Let p = 2. According to the expression (3.12)–(3.13), formally we write k u ( · , t ) k D ( − L ) = ∞ X n =1 λ n (cid:18)Z t s α − E ( n ) α ′ ,α ( s )( F ( · , t − s ) , ϕ n ) d s (cid:19) . Using Young’s inequality for convolutions, we estimate k u k L (0 ,T ; D ( − L )) = ∞ X n =1 λ n (cid:13)(cid:13)(cid:13)(cid:13)Z t s α − E ( n ) α ′ ,α ( s )( F ( · , t − s ) , ϕ n ) d s (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ) ≤ ∞ X n =1 λ n Z T t α − | E ( n ) α ′ ,α ( t ) | d t ! k ( F ( · , t ) , ϕ n ) k L (0 ,T ) . By Lemma 3.4, we can remove the absolute value of E ( n ) α ′ ,α ( t ) and apply Lemma 3.3 to derive Z T t α − | E ( n ) α ′ ,α ( t ) | d t = Z T t α − E ( n ) α ′ ,α ( t ) d t = T α E ( n ) α ′ , α ( T ) . Consequently, we use Lemma 3.2 to conclude k u k L (0 ,T ; H (Ω)) ≤ C k u k L (0 ,T ; D ( − L )) ≤ C ∞ X n =1 (cid:16) λ n T α E ( n ) α ′ , α ( T ) (cid:17) k ( F ( · , t ) , ϕ n ) k L (0 ,T ) ≤ C ∞ X n =1 k ( F ( · , t ) , ϕ n ) k L (0 ,T ) = (cid:0) C k F k L (Ω × (0 ,T )) (cid:1) . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations (b) Fix τ ∈ (0 , 1] arbitrarily. First we give an estimate for (3.13) with f ∈ D (( − L ) γ ).Similarly to the proof of Theorem 2.1, we apply Lemma 3.2 to deduce k U ( t ) f k D (( − L ) γ +1 − τ ) = t α − ∞ X n =1 (cid:12)(cid:12)(cid:12) λ − τn E ( n ) α ′ ,α (cid:12)(cid:12)(cid:12) | λ γn ( f, ϕ n ) | ≤ C t α − ∞ X n =1 (cid:18) ( λ n t α ) − τ λ n t α t α ( τ − (cid:19) | λ γn ( f, ϕ n ) | ≤ (cid:0) C k f k D (( − L ) γ ) t α τ − (cid:1) , < t ≤ T. Using (3.12) and Minkowski’s inequality for integrals, formally we have k u ( · , t ) k D (( − L ) γ +1 − τ ) = (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( s ) F ( · , t − s ) d s (cid:13)(cid:13)(cid:13)(cid:13) D (( − L ) γ +1 − τ ) ≤ Z t k U ( s ) F ( · , t − s ) k D (( − L ) γ +1 − τ ) d s ≤ C Z t k F ( · , t − s ) k D (( − L ) γ ) s α τ − d s, < t ≤ T. (3.14)Finally, it follows from Young’s inequality for convolutions that k u k L p (0 ,T ; D (( − L ) γ +1 − τ )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)Z t k F ( · , t − s ) k D (( − L ) γ ) s α τ − d s (cid:13)(cid:13)(cid:13)(cid:13) L p (0 ,T ) ≤ C k F k L p (0 ,T ; D (( − L ) γ )) Z T t α τ − d t ≤ Cτ k F k L p (0 ,T ; D (( − L ) γ )) . This completes the verification of (2.7).(c) Assume α p > τ ∈ ( α p , 1] arbitrarily. To investigate the asymptotic behaviornear t = 0, we apply H¨older’s inequality to (3.14) to see k u ( · , t ) k D (( − L ) γ +1 − τ ) ≤ C k F k L p (0 ,t ; D (( − L ) γ )) (cid:18)Z t s ( α τ − p ′ d s (cid:19) /p ′ , where p ′ is the conjugate number of p , i.e. 1 /p +1 /p ′ = 1. Since τ > α p , we see ( α τ − p ′ > − t → R t s ( α τ − p ′ d s = 0, indicating (2.8) immediately.As a direct application of Theorems 2.1–2.2, it is straightforward to show the Lipschitzstability of the solution with respect to various coefficients. Proof of Theorem 2.3. Let γ, τ ∈ (0 , a ∈ D (( − L ) γ ) and C > a , A , Q and U . First, a direct application of Theorem 2.1 immediately yields u ∈ L p (0 , T ; H (Ω) ∩ H (Ω)) and ∂ βt u ∈ L p (0 , T ; L (Ω)) for 0 < β ≤ α , where we abbreviate p := − γ . More precisely, there exists C > k u k L p (0 ,T ; H (Ω)) ≤ C, k ∂ βt u k L p (0 ,T ; L (Ω)) ≤ C (0 < β ≤ α ) . (3.15) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Let γ, τ ∈ (0 , a ∈ D (( − L ) γ ) and C > a , A , Q and U . First, a direct application of Theorem 2.1 immediately yields u ∈ L p (0 , T ; H (Ω) ∩ H (Ω)) and ∂ βt u ∈ L p (0 , T ; L (Ω)) for 0 < β ≤ α , where we abbreviate p := − γ . More precisely, there exists C > k u k L p (0 ,T ; H (Ω)) ≤ C, k ∂ βt u k L p (0 ,T ; L (Ω)) ≤ C (0 < β ≤ α ) . (3.15) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations On the other hand, by taking the difference of systems (2.10) and (2.9), it turns out that thesystem for v := e u − u reads m X j =1 e q j ∂ e α j t v = L e D v + F in Ω × (0 , T ] ,v = 0 on ∂ Ω × (0 , T ] ,v | t =0 = 0 in Ω , where F := m X j =1 e q j ( ∂ α j t u − ∂ e α j t u ) + m X j =2 ( q j − e q j ) ∂ α j t u + L e D − D u. Without loss of generality, we assume α ≥ e α , or otherwise we investigate v := u − e u instead.Therefore, together with D, e D ∈ C (Ω), we see F ∈ L p (0 , T ; L (Ω)) from (3.15). Now it isstraightforward to employ Theorem 2.2(b) to obtain k u − e u k L p (0 ,T ; D (( − L ) − τ )) = k v k L p (0 ,T ; D (( − L ) − τ )) ≤ Cτ k F k L p (0 ,T ; L (Ω)) . (3.16)Especially, if γ ≥ , we see p = − γ ≥ L p (Ω × (0 , T )) ⊂ L (0 , T ; L (Ω)). It thenfollows from Theorem 2.2(a) that k u − e u k L (0 ,T ; H (Ω)) ≤ C k F k L (Ω × (0 ,T )) ≤ C k F k L p (0 ,T ; L (Ω)) . (3.17)Therefore, it suffices to dominate k F k L p (0 ,T ; L (Ω)) by the difference of coefficients.To this end, first it is readily seen from (3.15) that k F k L p (0 ,T ; L (Ω)) ≤ m X j =1 e q j k ∂ α j t u − ∂ e α j t u k L p (0 ,T ; L (Ω)) + m X j =2 | q j − e q j |k ∂ α j t u k L p (0 ,T ; L (Ω)) + C k D − e D k C (Ω) k u k L p (0 ,T ; H (Ω)) ≤ C m X j =1 k ∂ α j t u − ∂ e α j t u k L p (0 ,T ; L (Ω)) + m X j =2 | q j − e q j | + k D − e D k C (Ω) . To give an estimate for ∂ α j t u − ∂ e α j t u by | α j − e α j | , we adopt a similar treatment as that in [13,Proposition 1] and decompose it by definition as ∂ α j t u ( · , t ) − ∂ e α j t u ( · , t ) = 1Γ(1 − α j ) Z t ∂ s u ( · , s )( t − s ) α j d s − − e α j ) Z t ∂ s u ( · , s )( t − s ) e α j d s = I j ( · , t ) + I j ( · , t ) , where I j ( · , t ) := Γ(1 − e α j ) − Γ(1 − α j )Γ(1 − e α j ) ∂ α j t u ( · , t ) ,I j ( · , t ) := 1Γ(1 − e α j ) Z t { ( t − s ) − α j − ( t − s ) − e α j } ∂ s u ( · , s ) d s. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Let γ, τ ∈ (0 , a ∈ D (( − L ) γ ) and C > a , A , Q and U . First, a direct application of Theorem 2.1 immediately yields u ∈ L p (0 , T ; H (Ω) ∩ H (Ω)) and ∂ βt u ∈ L p (0 , T ; L (Ω)) for 0 < β ≤ α , where we abbreviate p := − γ . More precisely, there exists C > k u k L p (0 ,T ; H (Ω)) ≤ C, k ∂ βt u k L p (0 ,T ; L (Ω)) ≤ C (0 < β ≤ α ) . (3.15) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations On the other hand, by taking the difference of systems (2.10) and (2.9), it turns out that thesystem for v := e u − u reads m X j =1 e q j ∂ e α j t v = L e D v + F in Ω × (0 , T ] ,v = 0 on ∂ Ω × (0 , T ] ,v | t =0 = 0 in Ω , where F := m X j =1 e q j ( ∂ α j t u − ∂ e α j t u ) + m X j =2 ( q j − e q j ) ∂ α j t u + L e D − D u. Without loss of generality, we assume α ≥ e α , or otherwise we investigate v := u − e u instead.Therefore, together with D, e D ∈ C (Ω), we see F ∈ L p (0 , T ; L (Ω)) from (3.15). Now it isstraightforward to employ Theorem 2.2(b) to obtain k u − e u k L p (0 ,T ; D (( − L ) − τ )) = k v k L p (0 ,T ; D (( − L ) − τ )) ≤ Cτ k F k L p (0 ,T ; L (Ω)) . (3.16)Especially, if γ ≥ , we see p = − γ ≥ L p (Ω × (0 , T )) ⊂ L (0 , T ; L (Ω)). It thenfollows from Theorem 2.2(a) that k u − e u k L (0 ,T ; H (Ω)) ≤ C k F k L (Ω × (0 ,T )) ≤ C k F k L p (0 ,T ; L (Ω)) . (3.17)Therefore, it suffices to dominate k F k L p (0 ,T ; L (Ω)) by the difference of coefficients.To this end, first it is readily seen from (3.15) that k F k L p (0 ,T ; L (Ω)) ≤ m X j =1 e q j k ∂ α j t u − ∂ e α j t u k L p (0 ,T ; L (Ω)) + m X j =2 | q j − e q j |k ∂ α j t u k L p (0 ,T ; L (Ω)) + C k D − e D k C (Ω) k u k L p (0 ,T ; H (Ω)) ≤ C m X j =1 k ∂ α j t u − ∂ e α j t u k L p (0 ,T ; L (Ω)) + m X j =2 | q j − e q j | + k D − e D k C (Ω) . To give an estimate for ∂ α j t u − ∂ e α j t u by | α j − e α j | , we adopt a similar treatment as that in [13,Proposition 1] and decompose it by definition as ∂ α j t u ( · , t ) − ∂ e α j t u ( · , t ) = 1Γ(1 − α j ) Z t ∂ s u ( · , s )( t − s ) α j d s − − e α j ) Z t ∂ s u ( · , s )( t − s ) e α j d s = I j ( · , t ) + I j ( · , t ) , where I j ( · , t ) := Γ(1 − e α j ) − Γ(1 − α j )Γ(1 − e α j ) ∂ α j t u ( · , t ) ,I j ( · , t ) := 1Γ(1 − e α j ) Z t { ( t − s ) − α j − ( t − s ) − e α j } ∂ s u ( · , s ) d s. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Since α j , e α j ∈ [ α, α ] and the Gamma function is Lipschitz continuous in [1 − α, − α ], it followsfrom (3.15) that k I j k L p (0 ,T ; L (Ω)) = | Γ(1 − e α j ) − Γ(1 − α j ) | Γ(1 − e α j ) k ∂ α j t u k L p (0 ,T ; L (Ω)) ≤ C | α j − e α j | . (3.18)In order to treat I j , we recall the estimate (2.4) for ∂ t u and utilize Minkowski’s inequality forintegrals to deduce k I j ( · , t ) k L (Ω) = 1Γ(1 − e α j ) (cid:13)(cid:13)(cid:13)(cid:13)Z t { ( t − s ) − α j − ( t − s ) − e α j } ∂ s u ( · , s ) d s (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ Z t | ( t − s ) − α j − ( t − s ) − e α j |k ∂ s u ( · , s ) k L (Ω) d s ≤ C Z t | ( t − s ) − α j − ( t − s ) − e α j | s α γ − d s = C Z t | s − α j − s − e α j | ( t − s ) α γ − d s. Using the mean value theorem, we have | s − α j − s − e α j | = | ln s | s − b α j ( s ) | α j − e α j | , where b α j ( s ) is a parameter depending on s such thatmin { α j , e α j } ≤ b α j ( s ) ≤ max { α j , e α j } ≤ α by the assumption α ≥ e α . Henceforth, we assume T > < t ≤ < t ≤ T . First, let 0 < t ≤ 1. Then there holds0 < s < s − b α j ( s ) ≤ s − α = s ε s − α − ε , where ε > α (1 − γ ) + ε < − γ . Since | ln s | s ε ≤ C for0 < s < 1, we obtain k I j ( · , t ) k L (Ω) ≤ C | α j − e α j | Z t | ln s | s − b α j ( s ) ( t − s ) α γ − d s ≤ C | α j − e α j | Z t ( | ln s | s ε ) s − α − ε ( t − s ) α γ − d s ≤ C | α j − e α j | t − α (1 − γ ) − ε , < t ≤ , (3.19)where we apply the boundedness of the Beta function B (1 − α − ε, α γ ) and γ > 0. Second,let 1 < t ≤ T . Then it is readily seen that t − s > − s for 0 < s < | ln s | s − b α j ( s ) ≤ C for1 ≤ s < t . These observation, together with the inequality (3.19) for t = 1, indicate k I j ( · , t ) k L (Ω) ≤ C | α j − e α j | (cid:18)Z + Z t (cid:19) | ln s | s − b α j ( s ) ( t − s ) α γ − d s ≤ C | α j − e α j | (cid:18)Z | ln s | s − b α j ( s ) (1 − s ) α γ − d s + C Z t ( t − s ) α γ − d s (cid:19) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 0. Second,let 1 < t ≤ T . Then it is readily seen that t − s > − s for 0 < s < | ln s | s − b α j ( s ) ≤ C for1 ≤ s < t . These observation, together with the inequality (3.19) for t = 1, indicate k I j ( · , t ) k L (Ω) ≤ C | α j − e α j | (cid:18)Z + Z t (cid:19) | ln s | s − b α j ( s ) ( t − s ) α γ − d s ≤ C | α j − e α j | (cid:18)Z | ln s | s − b α j ( s ) (1 − s ) α γ − d s + C Z t ( t − s ) α γ − d s (cid:19) ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations ≤ C | α j − e α j | , < t ≤ T. (3.20)The combination of (3.19) and (3.20) immediately yields k I j ( · , t ) k L (Ω) ≤ C | α j − e α j | t − α (1 − γ ) − ε , < t ≤ T and thus k I j k L p (0 ,T ; L (Ω)) ≤ C | α j − e α j | because α (1 − γ ) + ε < − γ = 1 /p . Consequently,collecting the estimate (3.18) for I j , we conclude k F k L p (0 ,T ; L (Ω)) ≤ C m X j =1 k I j + I j k L p (0 ,T ; L (Ω)) + m X j =2 | q j − e q j | + k D − e D k C (Ω) ≤ C m X j =1 | α j − e α j | + m X j =2 | q j − e q j | + k D − e D k C (Ω) , implying (2.12) and (2.13) with the aid of (3.16) and (3.17) respectively. In this subsection, we study the long-time asymptotic behavior of the solution u to (1.1)–(1.3) with F = 0 by a Laplace transform argument. In the sequel, by C we refer to a genericconstant which is independent of the initial value a and u but may depend on d , Ω, α j , q j ( j = 1 , . . . , m ) and the operator − L .Although an explicit representation (3.9) is available in this case, we write the solution inform of u ( · , t ) = ∞ X n =1 u n ( t ) ϕ n , t > { λ n , ϕ n } of − L , where a direct calculation and the orthogonality of { ϕ n } yield m X j =1 q j ∂ α j t u n ( t ) + λ n u n ( t ) = 0 , t > ,u n (0) = ( a, ϕ n ) , n = 1 , , . . . . (3.22)The proof of Theorem 2.4 relies on the following lemma. Lemma 3.5 Let u n ( n = 1 , , . . . ) solve the initial value problem (3.22) . Then there existsa constant C > such that (cid:12)(cid:12)(cid:12)(cid:12) u n ( t ) − q m ( a, ϕ n ) λ n Γ(1 − α m ) t α m (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ( a, ϕ n ) | λ n t α m − , t ≫ . (3.23) Proof. We abbreviate a n := ( a, ϕ n ) for simplicity. Applying the Laplace transform to (3.22)and using the formula L ( ∂ αt f )( s ) = s α L ( f )( s ) − s α − f (0+) , we are led to the transformed algebraic equation L ( u n )( s ) = a n w ( s ) m X j =1 q j s α j − , w ( s ) := m X j =1 q j s α j + λ n . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations We abbreviate a n := ( a, ϕ n ) for simplicity. Applying the Laplace transform to (3.22)and using the formula L ( ∂ αt f )( s ) = s α L ( f )( s ) − s α − f (0+) , we are led to the transformed algebraic equation L ( u n )( s ) = a n w ( s ) m X j =1 q j s α j − , w ( s ) := m X j =1 q j s α j + λ n . ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Noting that the Laplace transform of u n has a branch point zero, we should cut off the negativepart of the real axis so that the function w ( s ) has no zero in the main sheet of the Riemannsurface including its boundaries on the cut. In fact, for s = r e i θ , we see that sin( α j θ ) ( j =1 , · · · , m ) have the same signal and thus Im( w ( s )) = P mj =1 q j r α j sin( α j θ ) = 0 since q j > L ( u n ) can be represented by an integral on theHankel path Ha(0+) (i.e., the loop constituted by a small circle | s | = ε with ε → π i Z C L ( u n )( s ) e st d s (3.24)and estimate each H ℓ ( t ; R ) := Z C ℓ L ( u n )( s ) e st d s, ℓ = 1 , · · · , , where the loop C and its partitions C ℓ ( ℓ = 1 , . . . , 5) are illustrated in Figure 2. C xy ABCD γε i R C C C C C Figure 2. The loop C and its partition.For H ( t ; R ), noting that | s | = R > | H ( t ; R ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z C L ( u n )( s ) e st d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | a n | Z ππ/ R α m e Rt cos θ d θ = C | a n | R α m Z − e Rtη p − η d η, t > . Furthermore, we break up the above integral in [ − , 0] into two parts and calculate their boundsrespectively as R α m Z − e Rtη p − η d η = R α m Z − / − + Z − / ! e Rtη p − η d η ≤ R α m e − Rt/ Z − / − d η p − η + CR α m Z − / e Rtη d η ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 0] into two parts and calculate their boundsrespectively as R α m Z − e Rtη p − η d η = R α m Z − / − + Z − / ! e Rtη p − η d η ≤ R α m e − Rt/ Z − / − d η p − η + CR α m Z − / e Rtη d η ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations ≤ CR α m e − Rt/ + CR α m − − e − Rt/ t → R → ∞ , t > . Therefore, for any t > 0, we see that H ( t ; R ) → R → ∞ . Similarly to the calculation of H ( t ; R ), we have H ( t ; R ) → R → ∞ for any t > 0. On the other hand, since R cos θ ≤ γ for all θ ∈ [ θ R , π/ 2] where θ R denotes the argument of point A , we have | H ( t ; R ) | ≤ C | a n | R α m Z π/ θ R e Rt cos θ d θ ≤ C | a n | R α m e γt (cid:16) π − θ R (cid:17) = C | a n | R α m e γt (cid:18) π − arccos 1 R (cid:19) → R → ∞ . Therefore, since w ( s ) has no zero in the main sheet of the Riemann surface including theboundaries on the cut, the integral in (3.24) vanishes. By Fourier-Mellin formula (see e.g. [27]),we have u n ( t ) = lim M →∞ π i Z γ +i Mγ − i M L ( u n )( s ) e st d s = 12 π i Z Ha( ε ) L ( u n )( s ) e st d s. Here the integral is taken on the segment from γ − i M to γ + i M , and Ha( ε ) denotes the Hankelpath in C defined asHa( ε ) := { s ∈ C ; arg s = ± π, | s | ≥ ε } ∪ { s ∈ C ; − π ≤ arg s ≤ π, | s | = ε } . By a similar argument as above, we find1Γ(1 − α m ) t α m = lim M →∞ π i Z γ +i Mγ − i M s α m − e st d s = 12 π i Z Ha( ε ) s α m − e st d s. It is now straightforward to show that the contribution from the Hankel path Ha( ε ) as ε → u n ( t ) − q m a n λ n Γ(1 − α m ) t α m = a n Z ∞ H ( r, λ n ) e − rt d r, where (3.25) H ( r, λ n ) := − π Im w ( s ) m X j =1 q j s α j − − q m λ n s α m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = r e i π . To give the desired estimate (3.23), we observe that | w ( s ) | ≥ Cλ n as long as r = | s | ≤ ε λ n ,where ε > | H ( r, λ n ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n P m − j =1 q j s α j − − P mj =1 q j q m s α j + α m − λ n ( P mj =1 q j s α j + λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | a n | λ n m − X j =1 | s | α j − + m X j =1 | s | α j + α m − , ∀ | s | ≤ ε λ n . Meanwhile, for any s = r e ± i π with r ≥ ε λ n , we know that | H ( r, λ n ) | ≤ P m − j =1 q j r α j − | Im P mj =1 q j s α j | + P mj =1 q j q m r α j + α m − λ n | Im P mj =1 q j s α j | ≤ C. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 2] where θ R denotes the argument of point A , we have | H ( t ; R ) | ≤ C | a n | R α m Z π/ θ R e Rt cos θ d θ ≤ C | a n | R α m e γt (cid:16) π − θ R (cid:17) = C | a n | R α m e γt (cid:18) π − arccos 1 R (cid:19) → R → ∞ . Therefore, since w ( s ) has no zero in the main sheet of the Riemann surface including theboundaries on the cut, the integral in (3.24) vanishes. By Fourier-Mellin formula (see e.g. [27]),we have u n ( t ) = lim M →∞ π i Z γ +i Mγ − i M L ( u n )( s ) e st d s = 12 π i Z Ha( ε ) L ( u n )( s ) e st d s. Here the integral is taken on the segment from γ − i M to γ + i M , and Ha( ε ) denotes the Hankelpath in C defined asHa( ε ) := { s ∈ C ; arg s = ± π, | s | ≥ ε } ∪ { s ∈ C ; − π ≤ arg s ≤ π, | s | = ε } . By a similar argument as above, we find1Γ(1 − α m ) t α m = lim M →∞ π i Z γ +i Mγ − i M s α m − e st d s = 12 π i Z Ha( ε ) s α m − e st d s. It is now straightforward to show that the contribution from the Hankel path Ha( ε ) as ε → u n ( t ) − q m a n λ n Γ(1 − α m ) t α m = a n Z ∞ H ( r, λ n ) e − rt d r, where (3.25) H ( r, λ n ) := − π Im w ( s ) m X j =1 q j s α j − − q m λ n s α m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = r e i π . To give the desired estimate (3.23), we observe that | w ( s ) | ≥ Cλ n as long as r = | s | ≤ ε λ n ,where ε > | H ( r, λ n ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n P m − j =1 q j s α j − − P mj =1 q j q m s α j + α m − λ n ( P mj =1 q j s α j + λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | a n | λ n m − X j =1 | s | α j − + m X j =1 | s | α j + α m − , ∀ | s | ≤ ε λ n . Meanwhile, for any s = r e ± i π with r ≥ ε λ n , we know that | H ( r, λ n ) | ≤ P m − j =1 q j r α j − | Im P mj =1 q j s α j | + P mj =1 q j q m r α j + α m − λ n | Im P mj =1 q j s α j | ≤ C. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations Using these estimates, we break up the integral in (3.25) into two parts and give respectivebounds as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ε λ n H ( r, λ n ) e − rt d r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ n Z ∞ m − X j =1 r α j − + m X j =1 r α j + α m − e − rt d r ≤ Cλ n m − X j =1 t α j + m X j =1 t α j + α m , (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ ε λ n H ( r, λ n ) e − rt d r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z ∞ ε λ n e − rt d r = Ct e ε λ n t ≤ Cλ n t . Collecting the above two estimates, we obtain (3.23) for sufficiently large t . Proof of Theorem 2.4. Let u take the form of (3.21) which solves (1.1)–(1.3) with a ∈ L (Ω)and F = 0, and fix any T > t ≥ T , it immediately follows fromLemma 3.5 and the eigenfunction expansion that (cid:13)(cid:13)(cid:13)(cid:13) u ( · , t ) − ( − L ) − ( q m a )Γ(1 − α m ) t α m (cid:13)(cid:13)(cid:13)(cid:13) H (Ω) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 (cid:18) u n ( t ) − q m ( a, ϕ n ) λ n Γ(1 − α m ) t α m (cid:19) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D ( − L ) = C ∞ X n =1 (cid:12)(cid:12)(cid:12)(cid:12) λ n u n ( t ) − q m ( a, ϕ n )Γ(1 − α m ) t α m (cid:12)(cid:12)(cid:12)(cid:12) ! / ≤ Ct α m − ∞ X n =1 | ( a, ϕ n ) | ! / = C k a k L (Ω) t α m − , implying u ∈ C ([ T, ∞ ); H (Ω) ∩ H (Ω)). On the other hand, since Theorem 2.1(a) guarantees u ∈ C ([0 , T ]; L (Ω)) ∩ C ((0 , T ]; H (Ω) ∩ H (Ω)), the proof is finished by combining the regularityresults in the finite and infinite time spans. Remark 3.1 If some q j is negative, then we cannot obtain the asymptotic estimate for thesolution u of the initial-boundary value problem (1.1)–(1.3). In fact, for some n ∈ N sufficientlylarge, we study the following problem ∂ / t u ( x, t ) − λ n ∂ / t u ( x, t ) = Lu ( x, t ) , x ∈ Ω , t > ,u ( x, t ) = 0 , x ∈ ∂ Ω , t > .u ( x, 0) = a n ϕ n = ( a, ϕ n ) ϕ n , x ∈ Ω , where ( λ n , ϕ n ) is the n -th pair in the eigensystem of the elliptic operator − L , and a ∈ L (Ω).The Laplace transform of the solution reads L ( u )( s ) = a n w ( s ) (cid:16) s − / − λ n s − / (cid:17) ϕ n , w ( s ) := s / − λ n s / + λ n . We see that { s ; w ( s ) = 0 } is a finite set with all of the zero points having finite multiplicity inthe main sheet of the Riemann surface, and there is no zero point on the negative part of thereal axis since λ n is sufficiently large. Furthermore, we can prove that there exist zeros of w ( s )having positive real parts. In fact, obviously r ± := 3 λ n ± p λ n − λ n > ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations 0) = a n ϕ n = ( a, ϕ n ) ϕ n , x ∈ Ω , where ( λ n , ϕ n ) is the n -th pair in the eigensystem of the elliptic operator − L , and a ∈ L (Ω).The Laplace transform of the solution reads L ( u )( s ) = a n w ( s ) (cid:16) s − / − λ n s − / (cid:17) ϕ n , w ( s ) := s / − λ n s / + λ n . We see that { s ; w ( s ) = 0 } is a finite set with all of the zero points having finite multiplicity inthe main sheet of the Riemann surface, and there is no zero point on the negative part of thereal axis since λ n is sufficiently large. Furthermore, we can prove that there exist zeros of w ( s )having positive real parts. In fact, obviously r ± := 3 λ n ± p λ n − λ n > ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations solves w ( r ± ) = 0, and we have w ′ ( r ± ) = 12 r − / ± − λ n r − / ± = 0 . Note that 12 π i Z C L ( u )( s ) e st d s = X Res {L ( u )( s ) e st , C} , where C is defined in Figure 2, Res { f, C} denotes the residue of function f in the domainenclosed by C , and the sum is taken over all the poles of L ( u )( s ) e st in this domain. Repeatingthe argument in the proof of Lemma 3.5, we deduce u ( t ) = lim M →∞ π i Z γ +i Mγ − i M L ( u )( s ) e st d s = X Res {L ( u )( s ) e st } + 12 π i Z Ha(0+) L ( u )( s ) e st d s. Here the sum is taken over all the poles of L ( u )( s ) e st lying on the left-hand side of the line { z = γ +i M ; M ∈ R } with γ > r + , and there are only finite terms in this summation since w ( s )only has finite number of zero points including multiplicity in the main sheet of the Riemannsurface cutting of the negative axis. We can easily see thatRes { ( L ( u )( s ) e st ) | s = r ± } = r − / ± − λ n r − / ± w ′ ( r ± ) e r ± t ϕ n . Of course e r ± t tend to infinity as t → ∞ since r ± > 0, indicating that the asymptotic behaviorin Theorem 2.4 does not hold for this case. We summarize this paper by providing several concluding remarks. Concerning the initial-boundary value problem (1.1)–(1.3) for multi-term time-fractional diffusion equations, we mainlyinvestigate the well-posedness and the long-time asymptotic behavior of the solution, which turnout to be mostly parallel to those of the single-term prototype. On the basis of the representa-tion of solutions and a careful analysis of multinomial Mittag-Leffler functions, we succeed indominating the solutions by the initial value a and the source term F . Although uniquenessand stability also follow from the maximum principle developed in [18], we carry out variousestimates so that regularity and short-time asymptotic behaviors of the solutions are directlyconnected with the regularity of a and F (see Theorems 2.1–2.2). Furthermore, in Theorem2.3 we establish the Lipschitz stability of the solution with respect to α j , q j and the diffusioncoefficient, which is not only important by itself but also applicable to the corresponding inversecoefficient problem when treated by a minimization approach (see [13, Theorem 5]).Simultaneously, we also obtain an extended version of [26, Corollary 2.6] in Theorem 2.4,which asserts that, if the solution does not vanish identically, then its decay rate cannot exceed t − α m , where α m is the minimum order of fractional time-derivative. It is a remarkable propertyof fractional diffusion equations because the classical diffusion equation admits non-zero solu-tions decaying exponentially. This characterizes the slow diffusion in contrast to the classicalone. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations We summarize this paper by providing several concluding remarks. Concerning the initial-boundary value problem (1.1)–(1.3) for multi-term time-fractional diffusion equations, we mainlyinvestigate the well-posedness and the long-time asymptotic behavior of the solution, which turnout to be mostly parallel to those of the single-term prototype. On the basis of the representa-tion of solutions and a careful analysis of multinomial Mittag-Leffler functions, we succeed indominating the solutions by the initial value a and the source term F . Although uniquenessand stability also follow from the maximum principle developed in [18], we carry out variousestimates so that regularity and short-time asymptotic behaviors of the solutions are directlyconnected with the regularity of a and F (see Theorems 2.1–2.2). Furthermore, in Theorem2.3 we establish the Lipschitz stability of the solution with respect to α j , q j and the diffusioncoefficient, which is not only important by itself but also applicable to the corresponding inversecoefficient problem when treated by a minimization approach (see [13, Theorem 5]).Simultaneously, we also obtain an extended version of [26, Corollary 2.6] in Theorem 2.4,which asserts that, if the solution does not vanish identically, then its decay rate cannot exceed t − α m , where α m is the minimum order of fractional time-derivative. It is a remarkable propertyof fractional diffusion equations because the classical diffusion equation admits non-zero solu-tions decaying exponentially. This characterizes the slow diffusion in contrast to the classicalone. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations In the formulation of the initial-boundary value problem, we emphasize that the coefficients q j of the time derivatives are positive constants because this assumption is obligatory not onlyto acquire explicit solutions but also to apply the Laplace transform in time, which are essentialin the discussions of well-posedness and asymptotic behavior, respectively. On the other hand,if q j are space-dependent, then explicit solutions are not available so that one should rely ona fixed point argument for the unique existence of solution, and the improvement of regularityin space is strictly less than 2 orders (see [4, Theorem 2]). On the other hand, if some q j isnegative, then one may construct a counterexample in which the asymptotic property fails (seeRemark 3.1).However, in view of practical applications and theoretical interests, the linear non-symmetricdiffusion equation with positive variable coefficients of Caputo derivatives in time can be re-garded as a more feasible model equation than that we have studied in the current paper, butit will be definitely more challenging. Though still under consideration, we expect to establishparallel results for this more generalized case. Acknowledgement The second author appreciates the invaluable discussions with Pro-fessor William Rundell, Professor Raytcho Lazarov, Dr. Lihua Zuo and Mr. Zhi Zhou (TexasA&M University). References [1] E.E. Adams, L.W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis,Water Resources Res. 28 (1992) 3293–3307.[2] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.[3] E. Bazhlekova, Properties of the fundamental and the impulse-response solutions of multi-term fractionaldifferential equations, in: V. Kiryakova (Ed.), Complex Analysis and Applications ’13 (Proc. Intern. Conf.,Sofia, 2013), Bulg. Acad. Sci. Sofia, 2013, pp. 55–64.[4] S. Beckers, M. Yamamoto, Regularity and unique existence of solution to linear diffusion equation withmultiple time-fractional derivatives, in: K. Bredies, C. Clason, K. Kunisch, G. von Winckel (Eds.), Controland Optimization with PDE Constraints, Birkh¨auser, Basel, 2013, pp. 45–56.[5] C. Berge, Principles of Combinatorics, Academic Press, New York, 1971.[6] M. Ginoa, S. Cerbelli, H.E. Roman, Fractional diffusion equation and relaxation in complex viscoelasticmaterials, Phys. A 191 (1992) 449–453.[7] R. Gorenflo, Y. Luchko, P.P. Zabrejko, On solvability of linear fractional differential equations in Banachspaces, Frac. Calc. Appl. Anal. 2 (1999) 163–176.[8] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A.Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag,New York, 1997, pp. 223–276.[9] Y. Hatano, N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailedprofiles, Water Resources Res. 34 (1998) 1027–1033.[10] H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Rieszfractional advection-diffusion equations on a finite domain, J. Math. Anal. Appl. 389 (2012) 1117–1127.[11] B. Jin, R. Lazarov, Z. Zhou, Error estimates for a semidiscrete finite element method for fractional orderparabolic equations, SIAM J. Numer. Anal. 51 (2013) 445–466.[12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations,Elsevier, Amsterdam, 2006.[13] G. Li, D. Zhang, X. Jia, M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficientand the fractional order in the time-fractional diffusion equation, Inverse Problems 29 (2013) 065014. ulti-Term Time-Fractional Diffusion Equationsulti-Term Time-Fractional Diffusion Equations