Next, by constructing Lyapunov functional, we prove a blow-up of the solution with a negative initial energy, and establish a sufficient condition for the exponential decay of weak solutions

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

EXISTENCE, BLOW-UP AND EXPONENTIAL DECAY FOR KIRCHHOFF-LOVE EQUATIONS WITH DIRICHLET

CONDITIONS

NGUYEN ANH TRIET, VO THI TUYET MAI LE THI PHUONG NGOC, NGUYEN THANH LONG

Communicated by Dung Le

Abstract. The article concerns the initial boundary value problem for a non- linear Kirchhoff-Love equation. First, by applying the Faedo-Galerkin, we prove existence and uniqueness of a solution. Next, by constructing Lyapunov functional, we prove a blow-up of the solution with a negative initial energy, and establish a sufficient condition for the exponential decay of weak solutions.

1. Introduction

In this article, we consider the initial boundary value problem with homogeneous Dirichlet boundary conditions

utt− ∂

∂x

B x, t, u,kuk²,kuxk²,kutk²,kuxtk²

ux+λ1uxt+uxtt

+λut

=F x, t, u, ux, ut, uxt,ku(t)k²,kux(t)k²,kut(t)k²,kuxt(t)k²

− ∂

∂x

G x, t, u, u_x, u_t, u_xt,ku(t)k²,ku_x(t)k²,ku_t(t)k²,ku_xt(t)k² +f(x, t), x∈Ω = (0,1), 0< t < T,

(1.1)

u(0, t) =u(1, t) = 0, (1.2)

u(x,0) = ˜u₀(x), u_t(x,0) = ˜u₁(x), (1.3) where λ > 0,λ1 >0 are constants and ˜u0,u˜1 ∈ H₀¹∩H²; f, F and Gare given functions that assumptions stated later.

This problem has the so called model of Kirchhoff-Love type because it connects Kirchhoff and Love equation, this type is also introduced in [17]. More precisely (1.1) has its origin in the nonlinear vibration of an elastic string (Kirchhoff [5]), for which the associated equation is

ρhutt=

P0+Eh 2L

Z L

0

|∂u

∂y(y, t)|²dy

uxx, (1.4)

2010Mathematics Subject Classification. 35L20, 35L70, 35Q74, 37B25.

Key words and phrases. Nonlinear Kirchhoff-Love equation; blow-up; exponential decay.

c

2018 Texas State University.

Submitted May 21, 2018. Published October 4, 2018.

1

hereuis the lateral deflection,Lis the length of the string,his the cross sectional area,E is Young’s modulus,ρis the mass density, andP0is the initial tension. On the other hand, (1.1) arises from the Love equation

utt−E

ρuxx−2µ²ω²uxxtt= 0, (1.5) presented by Radochov´a [14]. This equation describes the vertical oscillations of a rod, which was established from Euler’s variational equation of an energy functional

Z T

0

Z L

0

1

2F ρ(u²_t+µ²ω²u²_tx)−1

2F(Eu²_x+ρµ²ω²uxuxtt)

dx dt , (1.6) whereuis the displacement,Lis the length of the rod,Fis the area of cross-section, ω is the cross-section radius,E is the Young modulus of the material andρis the mass density.

It is well known that the existence, global existence, decay properties and blow-up of solutions to the initial boundary value problem for Kirchhoff type models under different types of hypotheses in have been extensively studied by many authors, for example, we refer to [2, 3, 4, 13, 15, 18, 19], and references therein.

In [3], the authors studied the existence of global solutions and exponential decay for a Kirchhoff-Carrier model with viscosity.

In [15], the authors discussed the global well-posedness and uniform exponential stability for the Kirchhoff equation in Rⁿ. Here, the global solvability is proved when the initial data is taken small enough and the exponential decay of the energy is obtained in the strong topologyH²(Rⁿ)×H²(Rⁿ).

In [13], the author investigated the global existence, decay properties, and blow- up of solutions to the initial boundary value problem for the nonlinear Kirchhoff type.

In [18], the viscoelastic equation of Kirchhoff type was considered and the authors established a new blow-up result for arbitrary positive initial energy, by using simple analysis techniques.

The purpose of this paper is establishing the existence, blow up and exponential decay of weak solutions for(1.1)–(1.3). To our knowledge, there is no decay or blow up result for equations of this type. However, the existence and exponential decay of solutions or blow up results for Love equation were studied in [12]. Here, by combining the linearization method for the nonlinear term, the Faedo-Galerkin method and the weak compactness method, the existence of a unique weak solution of a Dirichlet problem for the nonlinear Love equation

u_tt−u_xx−u_xxtt−λ₁u_xxt+λu_t

=F(x, t, u, ux, ut, uxt)− ∂

∂x[G(x, t, u, ux, ut, uxt)] +f(x, t), (1.7) for 0 < x < 1 and t > 0, has been proved. When F = F(u) = a|u|^p−2u, G = G(ux) = b|ux|^p−2ux, a, b ∈ R, p > 2, the blow up and exponential decay of solutions were established. For details, in case ofa > 0, b >0;f(x, t)≡0, with negative initial energy, the solution of (1.7) blows up in finite time. In case of a >0, b <0, ifku˜0xk²−ak˜u0k^p_Lp >0 andf ∈L²((0,1)×R+), kf(t)k ≤Ce^−γ0t, such that f(t) decays exponentially ast→+∞, the energy of the solution decays exponentially as t → +∞. Finally, in case of a < 0, b < 0 and kf(t)k is small

enough, (1.7) has a unique global solution with energy decaying exponentially as t→+∞, without the initial data (˜u0,u˜1) small enough.

Our model was inspired in the above mentioned works and motivated by the results in [12], we study the existence, blow-up and exponential decay estimates for (1.1)–(1.3). This article is organized as follows. Section 2 is devoted to preliminaries and an existence result for (1.1)–(1.3) in caseF,G∈C¹([0,1]×[0, T]×R⁴×R⁴+);

B∈C¹([0,1]×[0, T]×R×R⁴+) withB(x, t, y, z)≥b0>0,∀(x, t)∈[0,1]×[0, T], for all y ∈ R, for all z ∈ R⁴+. Since f, G, B are arbitrary, we need to combine the linearization method, the Faedo-Galerkin method and the weak compactness method.

In Sections 3, 4, Problem (1.1)–(1.3) is considered in the case B =B(x, t) and F =F(u, ux),G=G(u, ux) such that (F, G) = (^∂_∂u^F, ^∂F_∂v). More details, in Section 3, with f(x, t) ≡ 0 and a negative initial energy, we prove that the solution of (1.1)–(1.3) blows up in finite time. In Section 4, we give a sufficient condition, in which the initial energy is positive and small, to guarantee the global existence and exponential decay of weak solutions. In the proof, a suitable Lyapunov functional is constructed. The results obtained here may be considered as the generalizations of those in [7, 12, 17], based on the main tool in [17] and the techniques in [12].

2. Existence of a weak solution First, we set the preliminary as follows.

Leth·,·ibe either the scalar product in L² or the dual pairing of a continuous linear functional and an element of a function space,k · kbe the norm in L² and k · k_X be the norm in the Banach spaceX. Let L^p(0, T;X), 1≤ p≤ ∞ be the Banach space of the real functionsu: (0, T)→X measurable, with

kuk_Lp(0,T;X)=Z T 0

ku(t)k^p_Xdt1/p

<∞ for 1≤p <∞, and

kuk_L^∞_(0,T;X)= ess sup0<t<T ku(t)kX forp=∞.

Denote u(t) = u(x, t), u⁰(t) = ut(t) = ^∂u_∂t(x, t), u⁰⁰(t) = utt(t) = ^∂_∂t^2u2(x, t), ux(t) = ^∂u_∂x(x, t),uxx(t) = ^∂_∂x^2u2(x, t).

With F ∈ C^k([0,1]×R+ ×R⁴×R⁴+), F = F(x, t, y1, . . . , y4, z1, . . . , z4), we put D1F = ^∂F_∂x, D2F = ^∂F_∂t, Di+2F = _∂y^∂F

i, Di+6F = _∂z^∂F

i, with i = 1, . . . ,4 and D^αF = D^α₁¹. . . D₁₀^α10F, α = (α₁, . . . , α₁₀) ∈ Z¹⁰+, |α| = α₁+· · ·+α₁₀ ≤ k, D^(0,...,0)F=F.

Similarly, withB∈C^k([0,1]×[0, T]×R×R⁴+),B=B(x, t, y, z1, . . . , z4), we put D1B = ^∂B_∂x,D2B = ^∂B_∂t, D3B = ^∂B_∂y,Di+3B = _∂z^∂B

i, withi= 1, . . . ,4 and D^βB = D₁^β1. . . D^β₇⁷B,β= (β1, . . . , β7)∈Z⁷+, |β|=β1+· · ·+β7≤k, D^(0,...,0)B=B.

We recall the following properties related to the usual spacesC([0,1]),H¹, and H₀¹={v∈H¹:v(1) =v(0) = 0}.

Lemma 2.1. (i) The imbedding =H¹,→C([0,1]) is compact and kvkC[0,1]≤√

2 kvk²+kvxk²1/2

, ∀v∈H¹. (2.1)

(ii) OnH₀¹, the norms kvxk andkvkH¹= kvk²+kvxk²1/2

are equivalent. On the other hand

kvkC([0,1])≤ kvxk for allv∈H₀¹. (2.2) Now, we consider the existence of a local solution for (1.1)–(1.3), with λ, λ1 ∈ R, λ1 >0. Without loss of generality, by the fact thatF contains the variableut

and λ is arbitrary, we can suppose that λ = 0. The weak formulation of (1.1)–

(1.3) can be given in as follows: Find u ∈ fW = {u ∈ L^∞(0, T_∗;H₀¹∩H²) : u⁰, u⁰⁰∈L^∞(0, T_∗;H₀¹∩H²)}, such thatusatisfies the variational equation

hu⁰⁰(t), wi+hB[u](t)(ux(t) +λ1u⁰_x(t) +u⁰⁰_x(t)), wxi

=hf(t), wi+hF[u](t), wi+hG[u](t), wxi, (2.3) for allw∈H₀¹, a.e.,t∈(0, T), with the initial conditions

u(0) = ˜u₀, u_t(0) = ˜u₁, (2.4) where

B[u](x, t) =B x, t, u(x, t),ku(t)k²,kux(t)k²,ku⁰(t)k²,ku⁰_x(t)k² , F[u](x, t) =F

x, t, u(x, t), ux(x, t), u⁰(x, t), u⁰_x(x, t),ku(t)k²,kux(t)k², ku⁰(t)k²,ku⁰_x(t)k²

, G[u](x, t) =G

x, t, u(x, t), ux(x, t), u⁰(x, t), u⁰_x(x, t),ku(t)k²,kux(t)k², ku⁰(t)k²,ku⁰_x(t)k²

.

(2.5)

We use the following assumptions:

(H1) ˜u0,u˜1∈H₀¹∩H²;

(H2) f, f⁰∈L²(QT),QT = (0,1)×(0, T);

(H3) B∈C¹([0,1]×[0, T]×R×R⁴+) and there exists a constantb0>0 such that B(x, t, y, z)≥b₀, for all (x, t)∈[0,1]×[0, T], for ally∈R, for allz∈R⁴+; (H4) F ∈C¹([0,1]×[0, T]×R⁴×R⁴+);

(H5) G∈C¹([0,1]×[0, T]×R⁴×R⁴+).

Theorem 2.2. Let (H1)–(H5) hold. Then Problem (1.1)–(1.3) has a unique local solution uand

u∈L^∞(0, T_∗;H₀¹∩H²), u⁰∈L^∞(0, T_∗;H₀¹∩H²),

u⁰⁰∈L^∞(0, T_∗;H₀¹∩H²), (2.6) forT_∗>0 small enough.

Remark 2.3. Thanks to the regularity obtained by (2.6), Problem (1.1)–(1.3) has a unique strong solution

u∈C¹([0, T_∗];H₀¹∩H²), u⁰⁰∈L^∞(0, T_∗;H₀¹∩H²). (2.7) Proof of Theorem 2.2. We have two steps. Using linearization, step 1 constructs a linear recurrent sequence {um}. Step 2 shows that {um} converges to uand uis exactly a unique local solution of (1.1)–(1.3).

Step 1. Consider T >0 fixed, letM >0, we put

KM(f) = (kfk²_L2(QT)+kf⁰k²_L2(QT))^1/2, (2.8)

kBk_C0( ˜A_M)= sup

(x,t,y,z1,...,z4)∈A˜M

|B(x, t, y, z1, . . . , z4)|, with

A˜M = [0,1]×[0, T]×[−M, M]×[0, M²]⁴, B¯M =kBk_C1( ˜AM)=kBk_C0( ˜AM)+

7

X

i=1

kDiBk_C0( ˜AM),

kFk_C0(AM)= sup

x

, t, y1, . . . , y4, z1, . . . , z4)∈AM|F(x, t, y1, . . . , y4, z1, . . . , z4)|, with

A_M = [0,1]×[0, T]×[−M, M]⁴×[0, M²]⁴, F¯_M =kFkC¹(A_M)=kFkC⁰(A_M)+

10

X

i=1

kDiFkC⁰(A_M),

G¯_M =kGkC¹(A_M)=kGkC⁰(A_M)+

10

X

i=1

kDiGkC⁰(A_M). For eachT_∗∈(0, T] andM >0, we put

W(M, T_∗) =n

v∈L^∞(0, T_∗;H₀¹∩H²) :v⁰∈L^∞(0, T_∗;H₀¹∩H²), v⁰⁰∈L^∞(0, T∗;H₀¹), withkvk_L∞(0,T∗;H₀¹∩H²), kv⁰k_L^∞_(0,T∗;H¹

0∩H²), kv⁰⁰k_L^∞_(0,T∗;H¹

0)≤Mo , W1(M, T_∗) ={v∈W(M, T_∗) :v⁰⁰∈L^∞(0, T_∗;H₀¹∩H²)},

(2.9)

whereQT_∗= Ω×(0, T_∗).

We establish the linear recurrent sequence{um}as follows. We choose the first termu0≡0, suppose that

u_m−1∈W1(M, T_∗), (2.10)

and associate with problem (1.1)–(1.3) the following problem.

Findum∈W1(M, T∗) (m≥1) which satisfies the linear variational problem hu⁰⁰_m(t), wi+hBm(t)(umx(t) +λ1u⁰_mx(t) +u⁰⁰_mx(t)), wxi

=hf(t), wi+hFm(t), wi+hGm(t), wxi, ∀w∈H₀¹, u_m(0) = ˜u₀, u⁰_m(0) = ˜u₁,

(2.11)

where

B_m(x, t) =B[u_m−1](x, t)

=B

x, t, um−1(x, t),kum−1(t)k²,k∇um−1(t)k²,ku⁰_m−1(t)k²,k∇u⁰_m−1(t)k² , Fm(x, t) =F[um−1](x, t)

=F

x, t, u_m−1(x, t),∇u_m−1(x, t), u⁰_m−1(x, t),∇u⁰_m−1(x, t), kum−1(t)k²,k∇um−1(t)k²,ku⁰_m−1(t)k²,k∇u⁰_m−1(t)k²

,

(2.12)

Gm(x, t) =G[u_m−1](x, t)

=G

x, t, u_m−1(x, t),∇u_m−1(x, t), u⁰_m−1(x, t),∇u⁰_m−1(x, t), ku_m−1(t)k²,k∇u_m−1(t)k²,ku⁰_m−1(t)k²,k∇u⁰_m−1(t)k²

.

Lemma 2.4. Let (H1)–(H5) hold. Then there exist positive constantsM, T∗ >0 such that, foru0≡0, there exists a recurrent sequence{um} ⊂W1(M, T∗)defined by (2.10)–(2.12).

Proof. The proof consists of several steps.

(i)The Faedo-Galerkin approximation (introduced by Lions [6]). Consider a special orthonormal basis {wj} on H₀¹ : wj(x) = √

2 sin(jπx), j ∈ N, formed by the eigenfunctions of the Laplacian −∆ = −_∂x^∂22. It is clear to see that there exists c^(k)_mj(t), 1≤j≤k, on interval [0, T] such that if we have expression in form

u^(k)_m(t) =

k

X

j=1

c^(k)_mj(t)wj, (2.13)

thenu^(k)m(t) satisfies

h¨u^(k)_m(t), wji+hBm(t) u^(k)_mx(t) +λ1u˙^(k)_mx(t) + ¨u^(k)_mx(t) , wjxi

=hf(t), wji+hFm(t), wji+hGm(t), wjxi, 1≤j ≤k, u^(k)_m(0) = ˜u0k, u˙^(k)_m(0) = ˜u1k,

(2.14)

in which

˜ u0k =

k

X

j=1

α^(k)_j wj →u˜0 strongly inH₀¹∩H²,

˜ u_1k =

k

X

j=1

β_j^(k)w_j →u˜₁ strongly inH₀¹∩H².

(2.15)

Indeed, (2.14) leads to an equivalent form of system (2.14) as follows

¨ c^(k)_mi(t) +

k

X

j=1

b^(m)_ij (t)(¨c^(k)_mj(t) +λ1c˙^(k)_mj(t) +c^(k)_mj(t)) =fmi(t), c^(k)_mi(0) =α^(k)_i , c˙^(k)_mi(0) =β^(k)_i , 1≤i≤k,

(2.16)

where

fmj(t) =hf(t), wji+hFm(t), wji+hGm(t), wjxi,

b^(m)_ij (t) =hBm(t)w_ix, w_jxi, 1≤i, j≤k. (2.17) System (2.16), (2.17) has a unique solutionc^(k)_mj(t), 1≤j ≤kon interval [0, T], the proof is obtained through (2.10) and normal argument (see [1]).

(ii) A priori estimates. We shall give a priori estimates to show that there exist positive constantsM,T_∗>0 such thatu^(k)m ∈W(M, T_∗), for allm andk. Put

S_m^(k)(t) =kp

Bm(t)u^(k)_mx(t)k²+kp

Bm(t)∆u^(k)_m (t)k² +ku˙^(k)_m (t)k²+ku˙^(k)_mx(t)k²

+ 2kp

Bm(t) ˙u^(k)_mx(t)k²+kp

Bm(t)∆ ˙u^(k)_m (t)k² +ku¨^(k)_m (t)k²+kp

Bm(t)¨u^(k)_mx(t)k² + 2λ1

Z t

0

hkp

Bm(s) ˙u^(k)_mx(s)k²+kp

Bm(s)∆ ˙u^(k)_m(s)k² +kp

Bm(s)¨u^(k)_mx(s)k²i ds.

(2.18)

It follows from (2.14) and (2.18) that S_m^(k)(t)

=S_m^(k)(0) + 2 Z t

0

hf(s),u˙^(k)_m (s)ids−2 Z t

0

hf(s),∆ ˙u^(k)_m (s)ids + 2

Z t

0

hf⁰(s),u¨^(k)_m (s)ids+ 2 Z t

0

hFm(s),u˙^(k)_m(s)ids

−2 Z t

0

hFm(s),∆ ˙u^(k)_m(s)ids+ 2 Z t

0

hGm(s),u˙^(k)_mx(s)ids + 2

Z t

0

hF_m⁰ (s),¨u^(k)_m (s)ids+ 2 Z t

0

hG⁰_m(s),u¨^(k)_mx(s)ids + 2

Z t

0

hGmx(s),4u˙^(k)_m (s)ids+ Z t

0

ds Z 1

0

B_m⁰ (x, s)h

|u^(k)_mx(x, s)|²+|∆u^(k)_m (x, s)|² + 2|u˙^(k)_mx(x, s)|²+|∆ ˙u^(k)_m(x, s)|²− |¨u^(k)_mx(x, s)|²i

dx

−2 Z t

0

hB⁰_m(s)(u^(k)_mx(s) +λ1u˙^(k)_mx(s)),u¨^(k)_mx(s)ids

−2 Z t

0

hBmx(s)(u^(k)_mx(s) +λ1u˙^(k)_mx(s) + ¨u^(k)_mx(s)),∆ ˙u^(k)_m(s)ids

=S_m^(k)(0) +

12

X

j=1

Ij.

(2.19) First, we need to estimateξ^(k)m =k¨u^(k)m (0)k²+kp

Bm(0)¨u^(k)mx(0)k². Lettingt→ 0₊ in (2.14)₁, multiplying the result by ¨c^(k)_mj(0), it gives

k¨u^(k)_m (0)k²+kp

B_m(0)¨u^(k)_mx(0)k² +hBm(0)(λ₁u˜_1kx+ ˜u_0kx),u¨^(k)_mx(0)i

=hf(0),u¨^(k)_m (0)i+hFm(0),u¨^(k)_m (0)i+hGm(0),u¨^(k)_mx(0)i.

Then

ξ_m^(k)=k¨u^(k)_m(0)k²+kp

B_m(0)¨u^(k)_mx(0)k²

≤ λ₁kp

B_m(0)˜u_1kxk+kp

B_m(0)˜u_0kxk kp

B_m(0)¨u^(k)_mx(0)k

+ [kf(0)k+kFm(0)k]ku¨^(k)_m (0)k+kGm(0)kk¨u^(k)_mx(0)k

≤[λ1kp

Bm(0)˜u1kxk+kp

Bm(0)˜u0kxk] q

ξ^(k)m

+ [kf(0)k+kFm(0)k]

q

ξm^(k)+kGm(0)k s

ξm^(k)

b0

≤[λ1kp

Bm(0)˜u1kxk+kp

Bm(0)˜u0kxk+kf(0)k+kFm(0)k+ 1

√b0

kGm(0)k]². On the other hand, Bm(x,0) =B(x,0,u˜0,k˜u0k²,ku˜0xk²,k˜u1k²,ku˜1xk²) is inde- pendent ofmand the constantkFm(0)k+kGm(0)k/√

b0 is also independent ofm, because

kFm(0)k+kGm(0)k

√b₀

=kF ·,0,u˜0,u˜0x,u˜1,u˜1x,k˜u0k²,k˜u0xk²,k˜u1k²,ku˜1xk² k

+ 1

√b0

kG ·,0,u˜0,u˜0x,u˜1,u˜1x,ku˜0k²,k˜u0xk²,ku˜1k²,k˜u1xk² k.

Therefore,

ξ_m^(k)≤S¯0, for allm, k, (2.20) where ¯S0 is a constant depending only onf, ˜u0, ˜u1,B,F,Gandλ1.

Equations (2.15), (2.18) and (2.20) imply that S_m^(k)(0) =kp

Bm(0)˜u0kxk²+kp

Bm(0)∆˜u0kk²+k˜u1kk²+k˜u1kxk² + 2kp

Bm(0)˜u1kxk²+kp

Bm(0)∆˜u1kk²+ξ_m^(k)

≤S₀, for allm, k∈N,

whereS0 is also a constant depending only onf, ˜u0, ˜u1,B,F,Gandλ1.

We estimate the terms Ij of (2.19). By the Cauchy - Schwartz inequality, we obtain

I1= 2 Z t

0

hf(s),u˙^(k)_m (s)ids≤ kfk²_L2(QT)+ Z t

0

ku˙^(k)_m (s)k²ds;

I₂=−2 Z t

0

hf(s),∆ ˙u^(k)_m (s)ids≤ kfk²_L2(QT)+ Z t

0

k∆ ˙u^(k)_m (s)k²ds;

I3= 2 Z t

0

hf⁰(s),u¨^(k)_m(s)ids≤ kf⁰k²_L2(Q_T)+ Z t

0

ku¨^(k)_m (s)k²ds.

Note that

S^(k)_m (t)≥ kp

B_m(t) ˙u^(k)_mx(t)k²+kp

B_m(t)∆ ˙u^(k)_m (t)k²+kp

B_m(t)¨u^(k)_mx(t)k²

≥b0 ku˙^(k)_mx(t)k²+k∆ ˙u^(k)_m (t)k²+ku¨^(k)_mx(t)k² , so

I1+I2+I3≤2K_M² (f) + 1 b0

Z t

0

S_m^(k)(s)ds. (2.21) Because

|Fm(x, t)| ≤F¯_M, |Gm(x, t)| ≤G¯_M, (2.22)

we have

I4= 2 Z t

0

hFm(s),u˙^(k)_m (s)ids≤T∗F¯_M² + Z t

0

ku˙^(k)_m (s)k²ds;

I₅=−2 Z t

0

hF_m(s),∆ ˙u^(k)_m (s)ids≤T_∗F¯_M² + Z t

0

k∆ ˙u^(k)_m(s)k²ds;

I6= 2 Z t

0

hGm(s),u˙^(k)_mx(s)ids≤T_∗G¯²_M + Z t

0

ku˙^(k)_mx(s)k²ds.

By

S_m^(k)(t)≥2kp

Bm(t) ˙u^(k)_mx(t)k²+kp

Bm(t)∆ ˙u^(k)_m (t)k²

≥b0(ku˙^(k)_m (t)k²+ku˙^(k)_mx(t)k²+k∆ ˙u^(k)_m(t)k²), we have

I4+I5+I6≤2T_∗( ¯F_M² + ¯G²_M) + 1 b0

Z t

0

S_m^(k)(s)ds. (2.23) We remark that

F_m⁰ (t) =D2F[um−1] +D3F[um−1]u⁰_m−1+D4F[um−1]∇u⁰_m−1 +D5F[u_m−1]u⁰⁰_m−1+D6F[u_m−1]∇u⁰⁰_m−1

+ 2D7F[u_m−1]hu_m−1(t), u⁰_m−1(t)i+ 2D8F[u_m−1]h∇u_m−1(t),∇u⁰_m−1(t)i + 2D9F[u_m−1]hu⁰_m−1(t), u⁰⁰_m−1(t)i+ 2D10F[u_m−1]h∇u⁰_m−1(t),∇u⁰⁰_m−1(t)i yields

kF_m⁰ (t)k ≤(1 + 4M+ 8M²) ¯F_M ≡F˜_M. (2.24) Thus

I7= 2 Z t

0

hF_m⁰ (s),u¨^(k)_m (s)ids≤T_∗F˜_M² + Z t

0

ku¨^(k)_m (s)k²ds. (2.25) In a similar way, we obtain the estimate

I₈= 2 Z t

0

hG⁰_m(s),u¨^(k)_mx(s)ids≤T_∗G˜²_M + Z t

0

k¨u^(k)_mx(s)k²ds, (2.26) with ˜GM = (1 + 4M+ 8M²) ¯GM. From

Gmx(t) =D1G[u_m−1] +D3G[u_m−1]∇u_m−1+D4G[u_m−1]∆u_m−1 (2.27) +D5G[u_m−1]∇u⁰_m−1+D6G[u_m−1]∆u⁰_m−1, (2.28) we obtain

kGmx(t)k ≤(1 + 4M) ¯GM ≤G˜M. (2.29) Hence

I9= 2 Z t

0

hGmx(s),4u˙^(k)_m (s)ids≤T_∗G˜²_M+ Z t

0

k4u˙^(k)_m (s)k²ds. (2.30) On the other hand

2S_m^(k)(t)≥2kp

Bm(t)¨u^(k)_mx(t)k²+ 2kp

Bm(t)∆ ˙u^(k)_m(t)k²

≥b₀(2k¨u^(k)_mx(t)k²+ 2k∆ ˙u^(k)_m (t)k²)

≥b₀(ku¨^(k)_m (t)k²+k¨u^(k)_mx(t)k²+k∆ ˙u^(k)_m(t)k²).

We have verified that

I7+I8+I9≤2T_∗( ˜F_M² + ˜G²_M) +

Z t

0

[k¨u^(k)_m(s)k²+k¨u^(k)_mx(s)k²+k4u˙^(k)_m (s)k²]ds

≤2T_∗( ˜F_M² + ˜G²_M) + 2 b0

Z t

0

S_m^(k)(s)ds.

(2.31)

It is known that

B_m⁰ (t) =D2B[u_m−1] +D3B[u_m−1]u⁰_m−1 + 2D4B[u_m−1]hu_m−1(t), u⁰_m−1(t)i + 2D5B[u_m−1]h∇u_m−1(t),∇u⁰_m−1(t)i + 2D6B[u_m−1]hu⁰_m−1(t), u⁰⁰_m−1(t)i + 2D₇B[u_m−1]h∇u⁰_m−1(t),∇u⁰⁰_m−1(t)i,

(2.32)

so

|B_m⁰ (x, t)| ≤(1 +M + 8M²) ¯BM ≡B˜M. (2.33) We also have

S_m^(k)(t)≥ kp

Bm(t)u^(k)_mx(t)k²+kp

Bm(t)∆u^(k)_m(t)k²+ 2kp

Bm(t) ˙u^(k)_mx(t)k² +kp

B_m(t)∆ ˙u^(k)_m(t)k²+kp

B_m(t)¨u^(k)_mx(t)k²

≥b0[ku^(k)_mx(t)k²+k∆u^(k)_m(t)k²+ 2ku˙^(k)_mx(t)k²+k∆ ˙u^(k)_m (t)k²+k¨u^(k)_mx(t)k²], hence

|I10|=

Z t

0

ds Z 1

0

B_m⁰ (x, s)[|u^(k)_mx(x, s)|²+|∆u^(k)_m (x, s)|²+ 2|u˙^(k)_mx(x, s)|² +|∆ ˙u^(k)_m (x, s)|²− |¨u^(k)_mx(x, s)|²]dx

≤B˜M

Z t

0

hku^(k)_mx(s)k²+k∆u^(k)_m(s)k²+ 2ku˙^(k)_mx(s)k²+k∆ ˙u^(k)_m (s)k² +ku¨^(k)_mx(s)k²i

ds

≤ B˜_M b0

Z t

0

S^(k)_m (s)ds.

(2.34)

Note that

S_m^(k)(t)≥ kp

Bm(t)u^(k)_mx(t)k²+ 2kp

Bm(t) ˙u^(k)_mx(t)k²+kp

Bm(t)¨u^(k)_mx(t)k²

≥b0[ku^(k)_mx(t)k²+ 2ku˙^(k)_mx(t)k²+k¨u^(k)_mx(t)k²], we deduce that

|I11|= 2

Z t

0

hB_m⁰ (s)(u^(k)_mx(s) +λ1u˙^(k)_mx(s)),u¨^(k)_mx(s)ids

≤2 ˜B_M Z t

0

(ku^(k)_mx(s)k+λ₁ku˙^(k)_mx(s)k)k¨u^(k)_mx(s)kds

≤ B˜M

b0

(2 +λ1) Z t

0

S_m^(k)(s)ds.

(2.35)

Because of

Bmx(x, t) =D1B[u_m−1] +D3B[u_m−1]∇u_m−1,

|B_mx(x, t)| ≤B¯_M(1 + 2M)≡Bˆ_M, S_m^(k)(t)≥ kp

Bm(t)u^(k)_mx(t)k²+ 2kp

Bm(t) ˙u^(k)_mx(t)k² +kp

Bm(t)¨u^(k)_mx(t)k²+kp

Bm(t)∆ ˙u^(k)_m (t)k²

≥b0

ku^(k)_mx(t)k²+ 2ku˙^(k)_mx(t)k²+k¨u^(k)_mx(t)k²+k∆ ˙u^(k)_m (t)k² , we have the estimate

I12= 2 Z t

0

hBmx(s)(u^(k)_mx(s) +λ1u˙^(k)_mx(s) + ¨u^(k)_mx(s)),∆ ˙u^(k)_m (s)ids

≤2 ˆBM

Z t

0

(ku^(k)_mx(s)k+λ1ku˙^(k)_mx(s)k+ku¨^(k)_mx(s)k)k∆ ˙u^(k)_m(s)kds

≤ Bˆ_M b0

(4 +λ1) Z t

0

S_m^(k)(s)ds.

(2.36)

Consequently, estimates (2.19), (2), (2.21), (2.23), (2.31), (2.34), (2.35) and (2.36) show that

S_m^(k)(t)≤S0+ 2K_M² (f) + 4T∗( ¯F_M² + ¯G²_M) + 1

b₀[4 + (7 + 2λ1) ˜BM] Z t

0

S_m^(k)(s)ds.

(2.37) We chooseM >0 sufficiently large such that

S0+ 2K_M² (f)≤ 1

2M², (2.38)

and then chooseT_∗∈(0, T] small enough such that 1

2M²+ 4T_∗( ¯F_M² + ¯G²_M) exp[T_∗

b0

[4 + (7 + 2λ₁) ˜B_M]]≤M², (2.39) and

k_T∗ = 2p D¯_Mp

T_∗exp[T_∗(1 +B˜_M 2b0

)]<1, (2.40) with

D¯M = 1 b0

[4(1 + 2M)²( ¯FM + ¯GM)²+ (2 +λ1)²(1 + 4M)²M²B¯_M² ].

From (2.37)–(2.39), we have S_m^(k)(t)≤exp[−T_∗

b₀ [4 + (7 + 2λ1) ˜BM]]M² + 1

b0

[4 + (7 + 2λ1) ˜BM] Z t

0

S_m^(k)(s)ds.

(2.41)

Using Gronwall’s Lemma, (2.41) leads to S_m^(k)(t)≤exp[−T_∗

b0

[4 + (7 + 2λ₁) ˜B_M]]M²exp[−t b0

[4 + (7 + 2λ₁) ˜B_M]]≤M², (2.42) for allt∈[0, T_∗], for allmandk, so

u^(k)_m ∈W(M, T_∗), for allmandk. (2.43)

(iii)Limiting process. By (2.42), there exists a subsequence of{u^(k)m }with a same notation, such that

u^(k)_m →u_m inL^∞(0, T_∗;H₀¹∩H²) weakly*,

˙

u^(k)_m →u⁰_m inL^∞(0, T∗;H₀¹∩H²) weakly*,

¨

u^(k)_m →u⁰⁰_m in L^∞(0, T∗;H₀¹) weakly*, um∈W(M, T_∗).

(2.44)

Passing to limit in (2.14), (2.15), it is clear to see that um is satisfying (2.11), (2.12) inL²(0, T_∗). Furthermore, (2.11)1and (2.44)4imply that

Bm(t)∆u⁰⁰_m(t) =−Bm(t)[∆um(t) +λ1∆u⁰_m(t)]−Bmx(t) umx(t) +λ1u⁰_mx(t) +u⁰⁰_mx(t)

+u⁰⁰_m(t)−f(t)−Fm(t) +Gmx(t)

≡Ψm∈L^∞(0, T∗;L²).

We have

b0k∆u⁰⁰_m(t)k ≤ kBm(t)∆u⁰⁰_m(t)k=kΨm(t)k ≤ kΨmkL^∞(0,T_∗;L²).

Hence u⁰⁰_m ∈ L^∞(0, T∗;H₀¹∩H²), so we obtain um ∈ W1(M, T∗), Lemma 2.4 is

proved. It means that step 1 is done.

Step 2. We state the following lemma.

Lemma 2.5. Let (H1)–(H5)hold. Then

(i) Problem (1.1)–(1.3)has a unique weak solutionu∈W₁(M, T_∗), whereM >

0 andT_∗>0 are chosen constants as in Lemma 2.4.

(ii) The linear recurrent sequence {um} defined by (2.10)–(2.12) converges to the solutionuof (1.1)–(1.3)strongly in the space

W₁(T_∗) ={v∈L^∞(0, T_∗;H₀¹) :v⁰∈L^∞(0, T_∗;H₀¹)}. (2.45) Proof. We use the result obtained in Lemma 2.4 and the compact imbedding the- orems to prove Lemma 2.5. It means that the existence and uniqueness of a weak solution of Prob. (1.1)–(1.3) is proved.

(i)Existence. It is well known thatW₁(T_∗) is a Banach space (see Lions [6]), with respect to the norm

kvk_W1(T∗)=kvk_L∞(0,T∗;H₀¹)+kv⁰k_L∞(0,T∗;H₀¹). (2.46) It is clear that{um}is a Cauchy sequence inW₁(T_∗). Indeed, letw_m=u_m+1−um, we have

hw_m⁰⁰(t), wi+hBm+1(t)(wmx(t) +λ1w_mx⁰ (t) +w_mx⁰⁰ (t)), wxi

=hF_m+1(t)−F_m(t), wi+hG_m+1(t)−G_m(t), w_xi

− h[B_m+1(t)−B_m(t)](u_mx(t) +λ₁u⁰_mx(t) +u⁰⁰_mx(t)), w_xi, ∀w∈H₀¹, w_m(0) =w⁰_m(0) = 0.

(2.47)

Consider (2.47) withw=w_m⁰ , and then integrating int, we obtain Zm(t)

= 2 Z t

0

hFm+1(s)−Fm(s), w⁰_m(s)ids+ 2 Z t

0

hGm+1(s)−Gm(s), w⁰_mx(s)ids +

Z t

0

ds Z 1

0

B⁰_m+1(x, s)(w²_mx(x, s) +|w⁰_mx(x, s)|²)dx

−2 Z t

0

h(Bm+1(s)−Bm(s))(umx(s) +λ1u⁰_mx(s) +u⁰⁰_mx(s)), w⁰_mx(s)ids

=J1+J2+J3+J4,

(2.48)

with

Z_m(t) =kw⁰_m(t)k²+kp

B_m+1(t)w_mx⁰ (t)k²+kp

B_m+1(t)w_mx(t)k² + 2λ1

Z t

0

kp

Bm+1(s)w_mx⁰ (s)k²ds.

From

kFm+1(s)−Fm(s)k ≤2(1 + 2M) ¯FMkwm−1kW₁(T∗), kGm+1(s)−Gm(s)k ≤2(1 + 2M) ¯GMkw_m−1k_W1(T∗),

|B_m+1⁰ (x, s)| ≤(1 +M+ 8M²) ¯BM ≡B˜M,

|Bm+1(x, s)−Bm(x, s)| ≤(1 + 4M) ¯BMkw_m−1k_W1(T∗), ku_mx(s) +λ₁u⁰_mx(s) +u⁰⁰_mx(s)k ≤(2 +λ₁)M, we obtain the estimates

J1+J2= 2 Z t

0

hFm+1(s)−Fm(s), w_m⁰ (s)ids + 2

Z t

0

hG_m+1(s)−G_m(s), w⁰_mx(s)ids

≤ 4 b0

(1 + 2M)²( ¯FM + ¯GM)²T_∗kw_m−1k²_W

1(T∗)+ Z t

0

Zm(s)ds;

(2.49)

J3= Z t

0

ds Z 1

0

B⁰_m+1(x, s)(w_mx² (x, s) +|w_mx⁰ (x, s)|²)dx

≤B˜M

Z t

0

(kwmx(s)k²+kw⁰_mx(s)k²)ds≤ B˜_M b0

Z t

0

Zm(s)ds;

J₄=−2 Z t

0

h(Bm+1(s)−B_m(s))(u_mx(s) +λ₁u⁰_mx(s) +u⁰⁰_mx(s)), w⁰_mx(s)ids

≤2(2 +λ1)(1 + 4M)MB¯Mkw_m−1k_W1(T∗) Z t

0

kw⁰_mx(s)kds

≤ 1 b0

(2 +λ1)²(1 + 4M)²M²B¯_M² T_∗kw_m−1k²_W

1(T∗)+ Z t

0

Zm(s)ds.

From (2.48) and (2.49) we have

Z_m(t)≤T_∗D¯_Mkwm−1k²_W1(T∗)+ 2 + B˜_M b0

Z t

0

Z_m(s)ds, (2.50) with

D¯_M = 1 b0

[4(1 + 2M)²( ¯F_M + ¯G_M)²+ (2 +λ₁)²(1 + 4M)²M²B¯_M² ]. (2.51) Using Gronwall’s Lemma, (2.50) leads to

kwmk_W1(T∗)≤kT∗kw_m−1k_W1(T∗) ∀m∈N, (2.52) so

kum−um+pk_W1(T∗)≤M(1−kT∗)⁻¹k^m_T∗, ∀m, p∈N. (2.53) It follows that{um} is a Cauchy sequence inW₁(T_∗), so there existsu∈W₁(T_∗) such that

um→ustrongly inW1(T∗). (2.54) Note that u_m ∈W₁(M, T_∗), so there exists a subsequence{u_mj} of {u_m} such that

um_j →u in L^∞(0, T_∗;H₀¹∩H²) weakly*, u⁰_mj →u⁰ in L^∞(0, T_∗;H₀¹∩H²) weakly*,

u⁰⁰_m

j →u⁰⁰ in L^∞(0, T_∗;H₀¹) weakly*, u∈W(M, T_∗).

(2.55)

On the other hand, by (2.8), (2.10), (2.12) and (2.55)4, we obtain kFm(t)−F[u](t)k ≤2(1 + 2M) ¯FMku_m−1−uk_W1(T∗), kG_m(t)−G[u](t)k ≤2(1 + 2M) ¯G_Mku_m−1−uk_W1(T∗),

|Bm+1(x, t)−B[u](x, t)| ≤(1 + 4M) ¯BMkum−1−ukW₁(T∗).

(2.56) Then (2.54) and (2.56) imply

Fm→F[u] strongly inL^∞(0, T_∗;L²), Gm→G[u] strongly inL^∞(0, T_∗;L²), B_m→B[u] strongly in L^∞(Q_T∗).

(2.57)

Passing to limit in (2.11), (2.12) as m = mj → ∞, by (2.54), (2.55) and (2.57), there existsu∈W(M, T∗) satisfying

hu⁰⁰(t), wi+hB[u](t)(u⁰⁰_x(t) +λ₁u⁰_x(t) +u_x(t)), w_xi

=hf(t), wi+hF[u](t), wi+hG[u](t), wxi, ∀w∈H₀¹, (2.58) and satisfying the initial conditions

u(0) = ˜u0, u⁰(0) = ˜u1. (2.59) Furthermore, assumption (H2) implies, from (2.55)₄and (2.58) that

B[u]∆u⁰⁰=−B[u](∆u+λ1∆u⁰)− ∂

∂x(B[u])(ux+λ1u⁰_x+u⁰⁰_x) +u⁰⁰−F[u] + ∂

∂xG[u]−f ≡Ψ∈L^∞(0, T_∗;L²).

(2.60) From

b₀k∆u⁰⁰(t)k ≤ kB[u](t)∆u⁰⁰(t)k=kΨ(t)k ≤ kΨkL^∞(0,T∗;L²),