We consider the final-value problem of a system of strongly-damped wave equations

Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 149, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

FINAL-VALUE PROBLEM FOR A WEAKLY-COUPLED SYSTEM OF STRUCTURALLY DAMPED WAVES

NGUYEN HUY TUAN, VO VAN AU, NGUYEN HUU CAN, MOKHTAR KIRANE Communicated by Vicentiu D. Radulescu

Abstract. We consider the final-value problem of a system of strongly-damped wave equations. First of all, we find a solution of the system, then by an ex- ample we show the problem is ill-posed. Next, by using a filter method, we propose stable approximate (regularized) solutions. The existence, unique- ness of the corresponding regularized solutions are obtained. Furthermore, we show that the corresponding regularized solutions converge to the exact solu- tions inL²uniformly with respect to the space coordinate under some a priori assumptions on the solutions.

1. Introduction

Let T be a positive number and Ω ⊂Rⁿ, n ≥1, be an open bounded domain with a smooth boundary Γ. SetDT = Ω×(0, T), Σ = Γ×(0, T). In this article, we consider the question of finding a couple of functions (u, v)(x, t), (x, t)∈ Ω× [0, T], satisfying the Cauchy problem for the weakly-coupled system of nonlinear structurally damped wave equations

u_tt−∆u+ 2a(−∆)^γu_t=F(u, v), inD_T, v_tt−∆v+ 2a(−∆)^γv_t=G(u, v), inD_T,

u=v= 0, on Σ,

(1.1) subject to the final observation

u(x, T) =uT(x), ut(x, T) =euT(x), in Ω,

v(x, T) =v_T(x), v_t(x, T) =ev_T(x), in Ω, (1.2) where γ > 1/2 anda >0 is a damping constant, the functions u_T,eu_T, v_T,ev_T are given inL²(Ω). The source functions F and Gwill be defined later. The damped wave equations and systems occur in a wide range of applications modelling the mo- tion of viscoelastic materials. Some more physical applications of strongly damped waves can be found in [13]. The initial-value problem for damped wave equations (or pseudo-hyperbolic equations) have been widely studied, see for example Pata et al [12, 13], Thomee et al [14], Liu et al [10], Guo [6], Zelik et al [7], Yang et al [18].

However, studies of the initial-value problem for strongly damped wave systems are

2010Mathematics Subject Classification. 35K05, 35K99, 47J06, 47H10.

Key words and phrases. Ill-posed problems; regularization; systems of wave equations;

error estimate.

c

2018 Texas State University.

Submitted February 22, 2018. Published August 7, 2018.

1

limited. Recently, Hayashi et al [4] studied the existence of small global solutions to the initial-value problem for system (1.1) inRⁿ, n≥4, assuming that

a= 1

2, γ= 1, F(u, v) =F(v), G(u, v) =F(u).

D’Abbicco [16] studied the system of structurally damped waves (1.1) inRⁿassum- ing that

γ=1

2, F(u, v) =|v|^p, G(u, v) =|u|^q, wherep, q are chosen suitably.

To the best of our knowledge, the final-value (backward) problem for the system (1.1) has not been studied yet. The final-value problem for systems of partial differential equations play an important role in engineering areas, which aims to obtain the previous data of a physical field from a given state. The first work on the regularization result for the strongly damped wave equation seems the one by Lesnic et al [17].

In practice, the exact datauT,ueT, vT,evT can only be measured with errors, and we thus would have as data some functionu^δ_T,eu^δ_T, v_T^δ,ve^δ_T that belong toL²(Ω), for which

ku^δ_T −uTkL²(Ω)+keu^δ_T −ueTkL²(Ω)+kv_T^δ −vTkL²(Ω)+kve^δ_T−veTkL²(Ω)≤δ, where the constantδ >0 represents a bound on the measurement errors.

It is well-known that problem (1.1)-(1.2) is ill-posed in the sense of Hadamard for data u^δ_T,eu^δ_T, v^δ_T,ev_T^δ in any reasonable topology (see [4]). More details of ill- posedness of the solution are given in Section 2.2. In general, no solution which satisfies the system with final data and the boundary conditions exists. Even if a solution exists, it does not depend continuously on the final data and any small perturbation in the given data may cause large change to the solution. So we need some regularization methods to deal with this problem.

This article is organized as follows. In Section 2, we present the mild solution and the ill-posedness of system (1.1)-(1.2). In Section 3, we establish a regularized solution in the case of a global Lipschitz source function F, G. In Section 4, we extend Section 3 to the situation of the locally Lipschitz sources. Furthermore, we also obtain the convergence rate between the regularized solution and the exact solution inL² norm.

2. Solution of the initial inverse problem (1.1)-(1.2)

We begin by introducing some notation needed for our analysis throughout this paper.

2.1. Notation. We denote byh·,·iL²(Ω)the inner product inL²(Ω).

• Forw∈C([0, T];L²(Ω)), we define kwkC([0,T];L²(Ω))= sup

0≤t≤T

kw(t)kL²(Ω).

• LetX,Y be Banach spaces;X × Y is also a Banach space and its norm is defined as

k(w1, w₂)kX ×Y =kw1kX+kw2kY, for any (w₁, w₂)∈ X × Y.

When Ω is bounded, the system (1.1) can also be solved by a decomposition in a Hilbert basis ofL²(Ω).

• For this purpose, it is very convenient to choose a basis{ξp}_p∈N^∗ ofL²(Ω) composed of eigenfunctions of−∆ (with zero Dirichlet condition), i.e.,

−∆ξp(x) =λpξp(x), in Ω,

ξ_p(x) = 0, on Γ, (2.1)

which admits a family of eigenvalues 0< λ1≤λ2≤λ3≤ · · · ≤λp· · · and λp→ ∞asp→ ∞, see [3, p. 335].

• Via the spectral decomposition ofw∈L²(Ω), for eachγ > ¹₂, we define the fractional Laplacian using the spectral theorem as follows

(−∆)^γw=

∞

X

p=1

λ^γ_p w, ξp

L²(Ω)ξp(x). (2.2) More details on this fractional Laplacian can be found in [8].

In addition, we introduce the abstract Gevrey class of functions of indexm, n >

0, see e.g., [1], defined by G^γm,n=n

w∈L²(Ω) :

∞

X

p=1

(λ^γ_p)^2mexp(2nλ^γ_p)hw, ξpi²_L2(Ω)<∞o , for 2γ >1, which is a Hilbert space equipped with the inner product

hw₁, w₂i_G^γ_m,n :=

((−∆)^γ/2)^mexp n(−∆)^γ/2

w1,((−∆)^γ/2)^mexp n(−∆)^γ/2 w2

L²(Ω), for allw1, w2∈G^γm,n, and its corresponding norm

kwk²

G^γm,n =

∞

X

p=1

(λ^γ_p)^2mexp(2nλ^γ_p)hw, ξ_pi²_L2(Ω)<∞.

2.2. Mild solution of (1.1)-(1.2). We look for a solution of problem (1.1)-(1.2) of the form

u(x, t) =

∞

X

p=1

u_p(t)ξ_p(x), v(x, t) =

∞

X

p=1

v_p(t)ξ_p(x), (2.3) whereu_p(t) =hu(x, t), ξp(x)iL²(Ω),v_p(t) =hv(x, t), ξp(x)iL²(Ω).

Put F_p(u, v)(t) = hF(u, v), ξ_p(x)iL²(Ω). We consider the problem of finding a functionup(t) satisfying

d²

dt²u_p(t) + 2aλ^γ_p d

dtu_p(t) +λ_pu_p(t) =F_p(u, v)(t), t∈(0, T), up(T) =hu(x, T), ξp(x)i_L2(Ω), d

dtup(T) =heu(x, T), ξp(x)i_L2(Ω).

(2.4)

The quadratic characteristic polynomial of (2.4) is Z²+ 2aλ^γ_pZ+λ_p= 0.

With the notation α_p=a²λ^2γ_p −λ_p, for anya >0 andγ >1/2, we consider three cases

Case 1. p∈N1={p∈N^∗:λ_p> a^1−2γ2 }. We putµ_j =µ_j,p :=aλ^γ_p+ (−1)^j√ α_p, j = 1,2. Multiplying the first equation in (2.4) by ^{exp µ1(s−t)}

−exp µ₂(s−t)

2√α_p and

integrating both sides fromt toT, we obtain up(t) =µ2exp µ1(T −t)

−µ1exp µ2(T−t) 2√

α_p hu(x, T), ξp(x)i_L2(Ω)

+exp µ1(T −t)

−exp µ2(T−t) 2√

αp

hu(x, Te ), ξp(x)i_L2(Ω)

+ Z T

t

exp µ1(s−t)

−exp µ2(s−t) 2√

αp

Fp(u, v)(s)ds.

(2.5)

Case 2. p∈N2 ={p∈N^∗:λp =a^1−2γ2 }. Multiplying the first equation in (2.4) by (s−t) exp a^1−2γ1 (s−t)

, and integrating both sides fromt toT, we have u_p(t) = exp aλ^γ_p(T−t)

(1−aλ^γ_p(T−t))hu(x, T), ξ_p(x)iL²(Ω)

−exp aλ^γ_p(T−t)

(T−t)heu(x, T), ξ_p(x)i_L2(Ω)

− Z T

t

(s−t) exp aλ^γ_p(s−t)

Fp(u, v)(s)ds.

(2.6)

Case 3. p∈N3 ={p∈N^∗:λp < a^1−2γ2 }. Multiplying the first equation in (2.4) by

exp aλ^γ_p

√−αp

sin(√

α_p(s−t)), and integrating both sides fromttoT, we have

up(t) = exp aλ^γ_p(T−t)

√−αp

aλ^γ_psin(p

−αp(T−t))−cos(p

−αp(T−t))

× hu(x, T), ξp(x)iL²(Ω)

+exp aλ^γ_p(T−t)

√−α_p sin(p

−αp(T−t))heu(x, T), ξp(x)i_L2(Ω)

− Z T

t

exp aλ^γ_p

√−α_p sin(√

αp(s−t))Fp(u, v)(s)ds.

(2.7)

Similar considerations apply tovp(t) that satisfies d²

dt²vp(t) + 2aλ^γ_p d

dtvp(t) +λpvp(t) =Gp(u, v)(t), t∈(0, T), v_p(T) =hv(x, T), ξ_p(x)iL²(Ω), d

dtv_p(T) =hev(x, T), ξ_p(x)iL²(Ω),

(2.8)

wherev_p =hv, ξpiL²(Ω), G_p(u, v) =hG(u, v), ξpiL²(Ω). We also have three cases.

Case 1. p∈N1. We obtain v_p(t) =µ2exp µ1(T−t)

−µ1exp µ2(T −t) 2√

α_p hv(x, T), ξ_p(x)i_L2(Ω)

+exp µ1(T−t)

−exp µ2(T −t) 2√

α_p hev(x, T), ξp(x)i_L2(Ω)

+ Z T

t

exp µ1(s−t)

−exp µ2(s−t) 2√

αp

Gp(u, v)(s)ds.

(2.9)

Case 2. p∈N2. We obtain v_p(t) = exp aλ^γ_p(T −t)

(1−aλ^γ_p(T−t))hv(x, T), ξ_p(x)iL²(Ω)

−exp aλ^γ_p(T−t)

(T−t)hev(x, T), ξ_p(x)i_L2(Ω)

− Z T

t

(s−t) exp aλ^γ_p(s−t)

Gp(u, v)(s)ds.

(2.10)

Case 3. p∈N3. We have vp(t) = exp aλ^γ_p(T−t)

√−α_p (aλ^γ_psin(p

−αp(T −t))

−cos(p

−αp(T−t)))hv(x, T), ξp(x)i_L2(Ω)

+exp aλ^γ_p(T−t)

√−αp

sin(p

−α_p(T−t))hv(x, Te ), ξ_p(x)i_L2(Ω)

− Z T

t

exp(aλ^γ_p)

√−α_p sin(√

αp(s−t))Gp(u, v)(s)ds.

(2.11)

Hence, the solution of (1.1) is u(x, t) = X

p∈N1

up(t)ξp+ X

p∈N2

up(t)ξp+ X

p∈N3

up(t)ξp, v(x, t) = X

p∈N1

v_p(t)ξ_p+ X

p∈N2

v_p(t)ξ_p+ X

p∈N3

v_p(t)ξ_p.

(2.12)

Letz∈(0, T), w∈L²(Ω),wp=hw, ξpi_L2(Ω), we define A(z)w=

∞

X

p∈N1

µ₂exp(zµ₁)−µ₁exp(zµ₂) 2√

αp

wpξp

+

∞

X

p∈N2

exp(aλ^γ_pz)(1−aλ^γ_pz)wpξp

+

∞

X

p∈N3

exp(aλ^γ_pz)

√−αp

[aλ^γ_psin(p

−αpz)−cos(p

−αpz)]wpξp,

(2.13)

B(z)w=

∞

X

p∈N1

exp(zµ1)−exp(zµ2) 2√

αp

wpξp+

∞

X

p∈N2

zexp(aλ^γ_pz)wpξp

+

∞

X

p∈N3

exp(aλ^γ_pz)

√−α_p sin(p

−αpz)wpξp.

(2.14)

Then, we can rewrite (2.12) as

u(x, t) =A(T−t)u_T+B(T−t)eu_T + Z T

t

B(s−t)F(u, v)(s)ds, v(x, t) =A(T−t)vT +B(T−t)veT+

Z T t

B(s−t)G(u, v)(s)ds.

(2.15)

We expressed the solution of problem (1.1) with the final observation in an integral formulation (2.15). In the next section, we indicate the reasons which make the solution (2.15) ill-posed in the Hadamard sense. For clarity, we give an example to show that the regularization method is necessary.

2.3. Ill-posedness of the inverse problem for(1.1)-(1.2). We first observe that if p ∈ N²∪N³ then λp ≤ a^1−2γ2 . It is obvious that the terms P

p∈N2up(t)ξp+ P

p∈N3u_p(t)ξ_p and P

p∈N2v_p(t)ξ_p+P

p∈N3v_p(t)ξ_p are bounded and stable inL² norm. However, since p∈N1 implies that λp > a^1−2γ2 then exponential functions in the right-hand sides of (2.5) and (2.9) tend to infinity as p tends to infinity.

Therefore, the terms P

p∈N1up(t)ξp and P

p∈N1vp(t)ξp are unbounded. From the above arguments, we take N2 =N3=∅ by assuming thata²λ^2γ−1₁ >1. Note that this also implies a²λ^2γ_p −λp >0 for all p∈ N^∗, and hence the root ofαp are real and distinct.

Next, we give an example which shows the the solution of problem (1.1) is not stable.

Leta=λ^1/2−γ₁ + 1,∂_tu^(k)(x, T) =∂_tv^(k)(x, T) = ^√1

λkξ_k(x) :=Ψ^(k), u(x, T) = v(x, T) = 0, for anyk∈N^∗. Let us define functions

F(w₁, w₂)(t) =

∞

X

p=1

exp(−2(λ^1/2−γ₁ + 1)T λ^γ_p) 2³√

2T²

×

hw1(t), ξ_piL²(Ω)ξ_p+hw2(t), ξ_piL²(Ω)ξ_p , G(w₁, w₂)(t) =

∞

X

p=1

exp(−2(λ^1/2−γ₁ + 1)T λ^γ_p) 2³√

2T²

×

hw1(t), ξ_piL²(Ω)ξ_p+hw2(t), ξ_piL²(Ω)ξ_p . Letu^(k), v^(k)satisfy the system

u^(k)(x, t) =B(T−t)Ψ^(k)+ Z T

t

B(s−t)F(u^(k)(x, s), v^(k)(x, s))ds, v^(k)(x, t) =B(T−t)Ψ^(k)+

Z T t

B(s−t)G(u^(k)(x, s), v^(k)(x, s))ds,

(2.16)

witha=λ^1/2−γ₁ + 1; recalling that B(z)w=

∞

X

p=1

exp(zµ1)−exp(zµ2) 2√

α_p hw, ξpi_L2(Ω)ξp, (2.17) and, forj= 1,2,

α_p= (λ^1/2−γ₁ + 1)²λ^2γ_p −λ_p, µ_j =µ_j,p:= (λ^1/2−γ₁ + 1)λ^γ_p+ (−1)^j√

α_p. (2.18)

Step 1. We show that (2.16) has a unique solution (u^(k), v^(k))∈[C([0, T];L²(Ω))]². Indeed, we consider for (r1, r2)∈[C([0, T];L²(Ω))]² the function

E(r1, r2)(t) =

E(r1, r2)(t),Ee(r1, r2)(t) , where

E(r1, r2)(t) =B(T−t)Ψ^(k)+ Z T

t

B(τ−t)F(r1(x, τ), r2(x, τ))dτ,

E(re 1, r2)(t) =B(T−t)Ψ^(k)+ Z T

t

B(τ−t)G(r1(x, τ), r2(x, τ))dτ.

Then for any (r1, r2),(s1, s2)∈[C([0, T];L²(Ω))]², we obtain kE(r1, r2)(t)− E(s1, s2)(t)k_L2(Ω)

≤ Z T

t

kB(τ−t)[F(r1, r2)−F(s1, s2)]kL²(Ω)dτ

= Z T

t

nX^∞

p=1

hexp((τ−t)µ2)−exp((τ−t)µ1)

√α_p

i2

× hF(r₁, r₂)−F(s₁, s₂), ξ_pi²_L2(Ω)

o1/2

dτ

≤ Z T

t

nX^∞

p=1

hexp((τ−t)µ2)−exp((τ−t)µ1)

√α_p

i2exp(−4(λ^1/2−γ₁ + 1)T λ^γ_p) 2⁷T⁴

×2hr1(x, τ)−s₁(x, τ), ξ_pi²_L2(Ω)+ 2hr2(x, τ)−s₂(x, τ), ξ_pi²_L2(Ω)

o^1/2 dτ.

(2.19)

Moreover, using the the inequality |exp(−b)−exp(−c)| ≤ |b−c| forb, c >0, we obtain the estimate

hexp((τ−t)µ2)−exp((τ−t)µ1)

√αp

i²exp(−4(λ^1/2−γ₁ + 1)T λ^γ_p) 2⁷T⁴

=h

exp (τ−t)(µ1+µ2)exp −(τ−t)µ1

−exp −(τ−t)µ2

√αp

i²

×exp(−4(λ^1/2−γ₁ + 1)T λ^γ_p) 2⁷T⁴

≤exp

4(λ^1/2−γ₁ + 1)(τ−t)λ^γ_ph(−2√

αp)(τ−t)

√α_p

i2exp(−4(λ^1/2−γ₁ + 1)T λ^γ_p) 2⁷T⁴

≤h(−2√

α_p)(τ−t)

√αp

i² 1 2⁷T⁴

= 4(τ−t)² 1

2⁷T⁴ ≤ 1 2⁵T²,

where we have used µ1+µ2 = 2aλ^γ_p = 2(λ^1/2−γ₁ + 1)λ^γ_p and µ2−µ1 = 2√ αp. According to the above observations, we deduce that for allt∈[0, T]

kE(r1, r2)(t)− E(s1, s2)(t)k_L2(Ω)≤1 4

Z T t

1

TkR(·, τ)−S(·, τ)k_[L2(Ω)]²dτ

≤1

4kR−Sk[C([0,T];L²(Ω))]², whereR:= (r1, r2),S:= (s1, s2)∈[C([0, T];L²(Ω))]². Whereupon

kE(r1, r2)− E(s1, s2)kL²(Ω)≤1

4kR−Sk[C([0,T];L²(Ω))]². (2.20) Similarly,

kE(re ₁, r₂)−E(se ₁, s₂)k_L2(Ω)≤1

4kR−Sk_[C([0,T];L2(Ω))]². (2.21) Combining (2.20) and (2.21), we obtain

kE(R)−E(S)k[C([0,T];L²(Ω))]² ≤ 1

2kR−Sk[C([0,T];L²(Ω))]².

HenceEis a contraction. Using the Banach fixed-point theorem, we conclude that E(R) =Rhas a unique solution (u^(k), v^(k))∈[C([0, T];L²(Ω))]².

Step 2. Problem (2.16) is ill-posed in the sense of Hadamard. We have ku^(k)(t)kL²(Ω)≥ kB(T −t)Ψ^(k)kL²(Ω)− k

Z T t

B(s−t)F(w^(k))(s)dskL²(Ω), (2.22) wherew^(k)= (u^(k), v^(k))∈[C([0, T];L²(Ω))]².

Firstly, it is easy to see that (noting thatF(0,0) = 0 and using (2.20)) k

Z T t

B(s−t)F(w^(k)(s))dsk_L2(Ω)=kE(u^(k), v^(k))(t)− E(0,0)(t)k_L2(Ω)

≤ 1

4kw^(k)k[C([0,T];L²(Ω))]².

(2.23)

Hence

ku^(k)(t)k_L2(Ω)≥

B(T−t)Ψ^(k) L²(Ω)

−1

4kw^(k)k_[C([0,T];L2(Ω))]². This leads to

ku^(k)k_C([0,T];L2(Ω))≥ sup

0≤t≤T

B(T−t)Ψ^(k) _L2(Ω)−

4kw^(k)k_[C([0,T];L2(Ω))]². (2.24) By an argument analogous to the previous one. We get

kv^(k)kC([0,T];L²(Ω))≥ sup

0≤t≤T

B(T −t)Ψ^(k) _L2(Ω)−

4kw^(k)k[C([0,T];L²(Ω))]². (2.25) Combining (2.24) and (2.25) yields

kw^(k)k[C([0,T];L²(Ω))]² ≥4 3 sup

0≤t≤T

B(T−t)Ψ^(k)

_L2(Ω). (2.26) Secondly, we have

kB(T −t)Ψ^(k)k²_L2(Ω)=hexp (T−t)µ2k

−exp (T−t)µ1k 2√

α_k

i2 1 λ_k

= exp 2(T −t)µ2k

1−exp −2(T−t)√ αk

2

4λkαk

≥ exp 2(T −t)µ2k

1−exp −2(T−t)√ αk²

4λ_kα_k ,

where we setµ_j,k:= (λ^1/2−γ₁ +1)λ^γ_k+(−1)^j√

α_k,j= 1,2,k∈N^∗. Since the function Θ(t) = exp 2(T−t)µ_2k

1−exp −2(T−t)√ α_k²

is a decreasing function with respect to the variablet, noting thatµ2k≈2(λ^1/2−γ₁ + 1)λ^γ_k, we deduce that

sup

0≤t≤T

kB(T−t)Ψ^(k)k²_L2(Ω)

= sup

0≤t≤T

kB(T −t)Ψ^(k)k²_L2(Ω)

≥ sup

0≤t≤T

exp 2(T−t)µ2k

1−exp −2(T−t)√ αk² 4λ_kα_k

≥exp(2T µ2k) 1−exp −2T√ αk

2

4λkαk

≥exp(4T(λ^1/2−γ₁ + 1)λ^γ_k) 1−exp −2T√ αk²

4λ_kα_k .

(2.27)

Next we estimate the right-hand side of the latter inequality. Indeed, combining (2.26) and (2.27) yields

kw^(k)k_[C([0,T];L2(Ω))]²≥ 2 3

exp 2T(λ^1/2−γ₁ + 1)λ^γ_k

1−exp −2T√ α_k

√λ_kα_k . (2.28)

Ask→+∞, we see that

k→∞lim

ku^(k)(x, T)k_L2(Ω)+kv^(k)(x, T)k_L2(Ω)

= lim

k→∞

√2 λ_k = 0,

k→∞lim kw^(k)k_[C([0,T];L2(Ω))]²

≥ lim

k→∞

2 3

exp 2T(λ^1/2−γ₁ + 1)λk

1−exp −2T√ αk

√λkαk

=∞.

(2.29)

Thus, Problem (2.16) is ill-posed in the sense of Hadamard inL²-norm.

3. Regularization and error estimate in for globally Lipschitz nonlinearities

Observe that when p → ∞ the operators A,B are unbounded; to establish a regularized solution, we need to find new operators which are bounded operators, more specifically,

A^Λ(t)w:=H^ΛA(t)w, (3.1) B^Λ(t)w:=H^ΛB(t)w, (3.2) H^Λw:=

∞

X

p=1

[1 + ΛC_pe^CpT]⁻¹hw, ξpiL²(Ω)ξ_p, (3.3) whereCp= 2aλ^γ_p. Here Λ := Λ(δ)>0 is a parameter regularization which satisfies lim_δ→0+Λ = 0. The function H^Λ is called the filter function. The regularized

problem is

U_δ^Λ(x, t) =A^Λ(T −t)u^δ_T(x) +B^Λ(T−t)ue^δ_T(x) +

Z T t

B^Λ(s−t)F(U_δ^Λ(x, s), V_δ^Λ(x, s))ds, V_δ^Λ(x, t) =A^Λ(T −t)v_T^δ(x) +B^Λ(T−t)ev_T^δ(x)

+ Z T

t

B^Λ(s−t)G(U_δ^Λ(x, s), V_δ^Λ(x, s))ds.

(3.4)

The following technical lemma plays a key role in our analysis.

Lemma 3.1. Let t∈[0, T],¹₂ ≤γ≤1 anda >0. Then kA^Λ(t)k_L(L2(Ω),L²(Ω))≤T_ah T

Λ log(^T_Λ) it/T

, (3.5)

kB^Λ(t)k_L(L2(Ω),L²(Ω))≤Ta

h T Λ log(^T_Λ)

it/T

, (3.6)

whereTa:= max{2, T,1 +T a^1−2γ1 }.

Proof. To show (3.5), lettingw∈L²(Ω), we have kA^Λ(t)wk²_L2(Ω)

= X

p∈N1

h µ₂e^µ1t−µ₁e^µ2t 2√

αp[1 +CpΛe^CpT] i²

hw, ξpi²_L2(Ω)

+ X

p∈N2

e^Cpt(1−^C₂^pt)²

[1 +CpΛe^CpT]²hw, ξpi²_L2(Ω)

+ X

p∈N3

e^Cpt

(−αp)[1 +CpΛe^CpT]²

×hC_p 2 sin(p

−αpt)−cos(p

−αpt)i²

hw, ξpi²_L2(Ω)

≤ X

p∈N1

hµ2e^−µ2t−µ1e^−µ1t 2√

αp

i²h 1

e^−Cpt+C_pΛe^Cp(T−t) i²

hw, ξpi²_L2(Ω)

+ X

p∈N2

e^−Cpt(1−C_p

2 t)²h 1

e^−Cpt+C_pΛe^Cp(T−t) i²

hw, ξpi²_L2(Ω)

+ X

p∈N3

h 1

e^−Cpt+C_pΛe^Cp(T−t) i²

e^−Cpt[^C₂^psin(√

−αpt)−cos(√

−αpt)]² (−αp)

× hw, ξpi²_L2(Ω).

(3.7)

Now, we continue estimating the terms in (3.7): First, we have 1

e^−Cpt+CpΛe^Cp(T−t) = e^−Cp(T−t)

[CpΛ +e^−CpT]^TT−t[CpΛ +e^−CpT]^t/T

≤ 1

[CpΛ +e^−CpT]^t/T.

(3.8)

On other hand, it is easy to see that h(y) = _by+e¹−yT ≤ _blog(^TT

b) for 0 < b < T e.

Hence if Λ< T e, then we obtain 1

C_pΛ +e^−CpT ≤ T Λ log(^T_Λ). It follows from (3.8) that

1

e^−Cpt+CpΛe^Cp(T−t) ≤ T Λ log(^T_Λ)

^t/T

. (3.9)

Ifp∈N2, thenλp=a^1−2γ2 impliesCp= 2a^1−2γ1 , [1−Cp

2 t]≤1 +a^1−2γ1 T. (3.10)

Ifp∈N3, thenλp< a^1−2γ2 ; using sin(z)≤z for anyz≥0, we have

Cp

2 sin(√

−αpt)−cos(√

−αpt)

√−αp

≤1 +a^1−2γ1 T. (3.11) Ifp∈N¹, thenλp> a^1−2γ2 ; using the inequalities 1−e^−z≤zand ze^−z ≤1 for z >0, we obtain (noting thatµ2−µ1= 2√

αp)

µ₂e^−tµ2−µ₁e^−tµ1 2√

αp

=e^−tµ2+µ₁|e^−tµ2−e^−tµ1| 2√

αp

≤e^−tµ2+µ1e^−tµ1|1−e^{−t(µ2−µ1)}| 2√

αp

≤e^−tµ2+µ1e^−tµ1t(µ2−µ1) 2√

α_p

≤e^−tµ2+tµ1e^−tµ1 ≤2.

(3.12)

Combining (3.7), (3.9), (3.10), (3.11), (3.12), we conclude that kA^Λ(t)wk²_L2(Ω)≤h T

Λ log(^T_Λ) i2t/T

T_a²kwk²_L2(Ω). (3.13) To show (3.6), lettingw∈L²(Ω), we have

kB^Λ(t)wk²_L2(Ω)

= X

p∈N1

h

e^t(µ1+µ2)e^−µ2t−e^−µ1t 2√

αp

1 1 +CpΛe^CpT

i²

hw, ξpi²_L2(Ω)

+ X

p∈N2

he^{Cp2 t}t 1 1 +CpΛe^CpT

i²

hw, ξpi²_L2(Ω)

+ X

p∈N3

h 1 1 +CpΛe^CpT

e^{Cp2 t}

√−αp

sin(p

−αpt)i²

hw, ξpi²_L2(Ω)

≤ X

p∈N1

h 1

e^−Cpt+CpΛe^Cp(T−t)

i2he^−µ2t−e^−µ1t 2√

αp

i2

hw, ξ_pi²_L2(Ω)

+ X

p∈N2

h 1

e^−Cpt+CpΛe^Cp(T−t) i2

e^−Cptt²hw, ξpi²_L2(Ω)

+ X

p∈N3

h 1

e^−Cpt+CpΛe^Cp(T−t)

i²e^−Cpt

(−αp)sin²(p

−αpt)hw, ξpi²_L2(Ω).

As |e^−c−e^−d| ≤ |c−d| forc, d >0, and noting thate^−Cpt≤1, for allt ∈[0, T] and sinz≤z, forz >0, we obtain

kB^Λ(t)wk²_L2(Ω)≤T² X

p∈N1

h 1

e^−Cpt+C_pΛe^Cp(T−t) i²

hw, ξpi²_L2(Ω)

+T² X

p∈N2

h 1

e^−Cpt+CpΛe^Cp(T−t) i²

hw, ξpi²_L2(Ω)

+T² X

p∈N3

h 1

e^−Cpt+CpΛe^Cp(T−t) i2

hw, ξpi²_L2(Ω)

≤T_a²h T Λ log(^T_Λ)

i^2t/T

kwk²_L2(Ω).

This completes the proof.

Now we are ready to state and prove the main results of this paper.

3.1. Existence and uniqueness for problem(2.16).

Theorem 3.2. The nonlinear integral system (2.16) has a solution (U_δ^Λ, V_δ^Λ) ∈ [C([0, T];L²(Ω))]².

Proof. For anyw₁, w₂∈[C([0, T];L²(Ω))]², we define the function L: [C([0, T];L²(Ω))]²→[C([0, T];L²(Ω))]² as

L(w1, w2)(t) := (X(w1, w2)(t),Y(w1, w2)(t)), where

X(w1, w2)(t) :=A^Λ(T−t)u^δ_T(x) +B^Λ(T −t)eu^δ_T(x) +

Z T t

B^Λ(s−t)F(w1(x, s), w2(x, s))ds,

(3.14)

Y(w1, w2)(t) :=A^Λ(T−t)v_T^δ(x) +B^Λ(T−t)ve^δ_T(x) +

Z T t

B^Λ(s−t)G(w1(x, s), w2(x, s))ds.

(3.15)

Then for anyW = (w₁, w₂), W = (w₁, w₂)∈[C([0, T];L²(Ω))]², we obtain kL(W)(t)−L(W)(t)k[L²(Ω)]²

≤ kX(W)(t)−X(W)(t)k_L2(Ω)+kY(W)(t)−Y(W)(t)k_L2(Ω). (3.16) Let the functionsF, G: [L²(Ω)]²→L²(Ω) satisfy the global Lipschitz condition

kF(W)−F(W)k_L2(Ω)≤KFkW−Wk_[L2(Ω)]², (3.17) kG(W)−G(W)kL²(Ω)≤K_GkW−Wk[L²(Ω)]², (3.18) where KF, KG are constants which are independent of W, W. We shall prove the estimate

kXⁿ(W)(t)−Xⁿ(W)(t)k_L2(Ω)≤Eⁿ_Λ,δ(t)

n! kW −Wk_[C([0,T];L2(Ω))]², (3.19)

whereEⁿ_Λ,δ(t) :=_KTa(T−t) Λ

n

, n≥1, andK= max{KF, KG}, by induction.

•Forn= 1, using Lemma (3.1) and the global Lipschitz condition of the function F, we obtain

kX(W)(t)−X(W)(t)k_L2(Ω)

≤ Z T

t

kB^Λ(s−t)(F(W)(s)−F(W)(s))kL²(Ω)ds

≤ Z T

t

kB^Λ(s−t)k_L(L2(Ω)×L²(Ω))kF(W)(s)−F(W)(s)k_L2(Ω)ds

≤ Z T

t

T_ah T Λ log(^T_Λ)

i^s−t_T

K_FkW(s)−W(s)k[L²(Ω)]²ds

≤KTaΛ⁻¹ Z T

t

kW(s)−W(s)k_[L2(Ω)]²ds

≤KTaΛ⁻¹(T−t)kW −Wk[C([0,T];L²(Ω))]²

=EΛ,δ(t)kW−Wk_[C([0,T];L2(Ω))]², whereEΛ,δ(t) := ^KTa(T_Λ^−t)·

•Assume that (3.19) holds forn=k. Then we obtain kX^k(W)(t)−X^k(W)(t)k_L2(Ω)≤E^k_Λ,δ(t)

k! kW −Wk_[C([0,T];L2(Ω))]². (3.20)

•We show that (3.19) holds for n=k+ 1. In fact, we have kX^k+1(W)(t)−X^k+1(W)(t)k_L2(Ω)

=

Z T t

B^Λ(s−t)[F(X^k(W)(s))−F(X^k(W)(s))]ds _L2(Ω)

≤ Z T

t

Ta

h T Λ log(^T_Λ)

i^s−t_T

KFkX^k(W)(s)−X^k(W)(s)k_L2(Ω)ds

≤KT_aΛ⁻¹ Z T

t

kX^k(W)(s)−X^k(W)(s)k_L2(Ω)ds

≤KT_aΛ⁻¹ Z T

t

KT_aΛ⁻¹(T−s)^k

k! kW −Wk_[C([0,T];L2(Ω))]²ds

= (KT_aΛ⁻¹)^k+1

k! kW −Wk[C([0,T];L²(Ω))]²

Z T t

(T−s)^kds

= E^k+1_Λ,δ (t)

(k+ 1)!kW −Wk_[C([0,T];L2(Ω))]². Therefore, (3.19) holds forn≥1.

Secondly, we estimatekY(W)(t)−Y(W)(t)kL²(Ω). Using similar arguments, we infer that ifW := (w₁, w₁), W := (w₁, w₂)∈[L²(Ω)]²then

kYⁿ(W)(t)−Yⁿ(W)(t)k_L2(Ω)≤Eⁿ_Λ,δ(t)

n! kW −Wk_[C([0,T];L2(Ω))]², (3.21) whereEⁿ_Λ,δ(t) :=_KT_a(T−t)

Λ

ⁿ .

Combining (3.16), (3.19) and (3.21), we obtain kLⁿ(W)(t)−Lⁿ(W)(t)k_[L2(Ω)]² ≤2Eⁿ_Λ,δ(t)

n! kW−Wk_[C([0,T];L2(Ω)]² (3.22) On the other hand,

Eⁿ_Λ,δ(t)≤Eⁿ_Λ,δ(0) =KT_aT Λ

ⁿ

, n≥1.

This implies

kLⁿ(W)−Lⁿ(W)k[C([0,T];L²(Ω)]² ≤2Eⁿ_Λ,δ(0)

n! kW−Wk[C([0,T];L²(Ω)]²,

n→∞lim

2Eⁿ_Λ,δ(0) n! = 0.

There exits a positive integern₀such thatLⁿ⁰is a contraction. Thus, the existence and uniqueness arguments are obtained by the Banach fixed-point theorem, i.e, L(w₁, w₂) = (w₁, w₂) has a unique solution (w₁, w₂) ∈ [C([0, T];L²(Ω)]². Hence, X(U_δ^Λ, V_δ^Λ) =U_δ^Λand Y(U_δ^Λ, V_δ^Λ) =V_δ^Λ. 3.2. Error estimate. Now, we shall state (and prove) some regularization results under some conditions on the exact solution (u, v) of system (2.15).

Theorem 3.3. Let Λ(δ) be a regularization parameter such that lim

δ→0⁺Λ = lim

δ→0⁺δ/Λ = 0. (3.23)

Form≥1, n≥2aT, we assume that system (2.16) has a unique solution S := (u, v)∈[C([0, T];L²(Ω))∩L^∞(0, T;G^γm,n)]² which satisfies

kS(t)k_[C([0,T];L2(Ω))]²+kS(t)k_[L∞(0,T;G^γm,n)]²≤Q. (3.24) Then we have the estimate

kS_δ^Λ(t)− S(t)k[L²(Ω)]² ≤

2aQ+TaδΛ⁻¹

e^2TaK(T−t)h T log(^T_Λ)

i1−_T^t

Λ^t/T, (3.25) whereS_δ^Λ:= (U_δ^Λ, V_δ^Λ)∈[C([0, T];L²(Ω))]² andK:= max{KF, K_G}.

Remark 3.4. In (3.25), the error estimate is of order Λ^t/T[_log(^TT

Λ)]^1−Tt. Ift ≈T, the first term Λ^t/T tends to zero quickly, and ift≈0, the second term [_log(^TT

Λ)]^1−Tt tends to zero asδ→0⁺. And ift= 0, the error (3.25) becomes

kS_δ^Λ(t)− S(t)k_[L2(Ω)]²≤C[log(T

Λ)]⁻¹. (3.26)

We also note that the right-hand side of (3.26) tends to zero whenδ→0⁺.