On some properties of a system of nonlinear partial functional differential equations

2016, No.22, 1–12; doi: 10.14232/ejqtde.2016.8.22 http://www.math.u-szeged.hu/ejqtde/

On some properties of a system of nonlinear partial functional differential equations

László Simon

Eötvös Loránd University, Pázmány P. sétány 1/C, Budapest, H-1117, Hungary Appeared 11 August 2016

Communicated by Tibor Krisztin

Abstract. We consider a system of a semilinear hyperbolic functional differential equa- tion (where the lower order terms contain functional dependence on the unknown func- tion) with initial and boundary conditions and a quasilinear elliptic functional differ- ential equation (containing tas a parameter) with boundary conditions. Existence and some qualitative properties of weak solutions fort∈(0,∞)are proved.

Keywords: partial functional-differential equations, nonlinear systems of partial differ- ential equations, nonlinear systems of mixed type, qualitative properties.

2010 Mathematics Subject Classification: 35M33, 35R10.

1 Introduction

In the present paper we consider weak solutions of the following system of equations:

u⁰⁰(t) +Q(u(t)) +ϕ(x)h⁰(u(t)) +H(t,x;u,z) +ψ(x)u⁰(t) = F₁(t,x;z), (1.1)

−

∑

D_j[a_j(t,x,Dz(t),z(t);u)] +a₀(t,x,Dz(t),z(t);u,z) =F₂(t,x;u), (1.2) (t,x)∈ Q_T = (0,T)×_Ω

where Ω⊂_Rⁿ is a bounded domain and we use the notationsu(t) = u(t,x),u⁰ = Dtu, u⁰⁰ = D²_tu,z(t) =z(t,x),Dz= _∂x^∂z

1, . . ._∂x^∂z

n

, Qmay be e.g. a linear second order symmetric elliptic differential operator in the variable x; h is a C² function having certain polynomial growth, H contains nonlinear functional (non-local) dependence on u and z, with some polynomial growth and F₁ contains some functional dependence on z. Further, the functions a_j define a quasilinear elliptic differential operator in x (for fixed t) with functional dependence on u for i = 1, . . . ,n and on u, z fori = 0, respectively. Finally, F₂ may non-locally depend on u.

The system (1.1), (1.2) consists of a semilinear hyperbolic functional equation and an elliptic functional equation (containing the timetas a parameter).

This paper was motivated by some problems which were modelled by systems consist- ing of (functional) differential equations of different types. In [4] S. Cinca investigated a

BEmail: [email protected]

model, consisting of an elliptic, a parabolic and an ordinary nonlinear differential equation, which arise when modelling diffusion and transport in porous media with variable poros- ity. In [6] J. D. Logan, M. R. Petersen and T. S. Shores considered and numerically studied a similar system which describes reaction-mineralogy-porosity changes in porous media with one-dimensional space variable. J. H. Merkin, D. J. Needham and B. D. Sleeman considered in [7] a system, consisting of a nonlinear parabolic and an ordinary differential equation, as a mathematical model for the spread of morphogens with density dependent chemosensitivity.

In [3,8,9] the existence of solutions of such systems were studied.

In [12] existence of weak solutions was proved for t ∈ (0,T). In this paper existence and some qualitative properties of weak solutions fort ∈(0,∞)are proved.

In Section 2 the existence theorem in (_0,T) will be formulated and in Section 3 we shall prove existence and certain properties of solutions fort ∈(0,∞).

2 Solutions in ( 0, T )

Denote byΩ ⊂ _Rⁿ a bounded domain having the uniform C¹ regularity property (see [1]), QT = (0,T)×Ω. Denote byW^1,p(_Ω)the Sobolev space of real valued functions with the norm

kuk=

"

Z

Ω

∑

|D_ju|^p+|u|^p

! dx

#1/p

2≤ p<_∞, D_ju= _∂x^∂u

j

.

The numberqis defined by 1/p+1/q = 1. Further, letV₁ ⊂W^1,2(_Ω)andV₂ ⊂ W^1,p(_Ω)be closed linear subspaces containingC₀^∞(_Ω), V_j^? the dual spaces of V_j, the duality betweenV_j^? andV_j will be denoted byh·_,·i, the scalar product in L²(_Ω)will be denoted by(·_,·)_{. Finally,} denote byL^p(0,T;V_j)the Banach space of the set of measurable functionsu:(0,T)→V_j with the norm

kuk_Lp(0,T;V_j) = _{Z T}

0

ku(t)k_V^p

jdt 1/p

andL^∞(0,T;V_j), L^∞(0,T;L²(_Ω))the set of measurable functionsu :(0,T)→ V_j,u :(0,T)→ L²(_Ω), respectively, with the L^∞(0,T) norm of the functions t 7→ ku(t)k_Vj, t 7→ ku(t)k_L2(_Ω), respectively.

First we formulate the existence theorem fort∈ (0,T)which was proved in [12], by using the results of [11], the theory of monotone operators (see, e.g., [14,15]) and Schauder’s fixed point theorem.

Now we formulate the assumptions on the functions in (1.1), (1.2).

(A₁) Q:V₁→V₁^? is a linear continuous operator such that

hQu,vi=hQv,ui, hQu,ui ≥c₀kuk²_V

1

for allu,v∈V₁with some constantc₀>0.

(A₂) ϕ,ψ:Ω→_Rare measurable functions satisfying

c₁ ≤ ϕ(x)≤ c₂, c₁≤ψ(x)≤c₂ for a.a. x∈_Ω with some positive constantsc₁,c₂.

(A₃) h :R→_Ris a twice continuously differentiable function satisfying h(η)≥0, |h⁰⁰(η)| ≤const|η|^λ−1 for|η|>1 where 1<λ≤λ₀= ⁿ

n−2 ifn≥3, 1<λ<_∞ ifn=2.

(A₄) H : Q_T× L²(Q_T)×L^p(_0,T;V₂) → _R is a function for which (t,x) 7→ H(t,x;u,z) _is measurable for all fixedu∈ L²(_Ω),z∈ L^p(0,T;V₂), Hhas the Volterra property, i.e. for allt∈ [0,T],H(t,x;u,z)depends only on the restriction ofuandzto(0,t). Further, the following inequality holds for all t∈[_0,T]_andu,u_j ∈ L²(_Ω)_,z∈ L^p(_0,T;V₂)_:

Z

Ω|H(t,x;u,z)|²dx≤consth

kzk²_Lp(0,T;V2)+1i_{Z t}

0

Z

Ωh(u)dxdτ+

Z

Ωh(u)dx+1

; Z _t

0

_Z

Ω|H(τ,x;u₁,z)−H(τ,x;u₂,z)|²dx

dτ≤ M(K,z)

Z _t

0

_Z

Ω|u₁−u₂|²dx

dτ if ku_jk_L∞(0,T;V1) ≤K

where for all fixed numberK>0,z7→ M(K,z)∈_R⁺is a bounded (nonlinear) operator.

Finally, (z_k)→z inL^p(0,T;V₂)implies

H(t,x;u_k,z_k)−H(t,x;u_k,z)→0 inL²(Q_T)uniformly ifku_kk_L2(QT)≤const.

(A₅) F₁ : Q_T×L^p(0,T;V₂) → _R is a function satisfying (t,x) 7→ F₁(t,x;z) ∈ L²(Q_T) for all fixed z∈ L^p(0,T;V₂)and(z_k)→ zin L^p(0,T;V₂)implies that F₁(t,x;z_k)→ F₁(t,x;z)in L²(Q_T).

Further,

Z _T

0

kF₁(τ,x;z)k²_L2(_Ω)dτ≤consth

1+kzk^β_L¹_p(0,T;V

2)

i

with some constant β₁>0.

(B₁) The functions

a_j :Q_T×_Rⁿ⁺¹×L²(Q_T)→_R (j=_{1, . . .}n)_, a₀ :Q_T×_Rⁿ⁺¹×L²(Q_T)×L^p(0,T;V₂)→_R

are such that a_j(t,x,ξ;u), a0(t,x,ξ;u,z)are measurable functions of variable(t,x)∈ QT

for all fixed ξ ∈_Rⁿ⁺¹,u∈ L²(Q_T),z∈ L^p(0,T;V₂)and continuous functions of variable ξ ∈ _Rⁿ⁺¹for all fixedu ∈L²(Q_T), z∈ L^p(0,T;V₂)and a.a. fixed(t,x)∈ Q_T.

Further, if(u_k)→uinL²(Q_T)then for allz∈ L^p(0,T;V₂),ξ ∈ _Rⁿ⁺¹, a.a.(t,x)∈Q_T, for a subsequence

a_j(t,x,ξ;u_k)→a_j(t,x,ξ;u) (j=1, . . . ,n), a0(t,x,ξ;u_k,z_k)−a0(t,x,ξ;u,z_k)→0.

(B₂) Forj=1, . . . ,n

|a_j(t,x,ξ;u)| ≤g₁(u)|ξ|^p−1+ [k₁(u)](t,x),

where g₁: L²(Q_T)→_R⁺is a bounded, continuous (nonlinear) operator, k₁: L²(Q_T)→L^q(Q_T)is continuous andkk₁(u)k_Lq(QT)≤const(1+kuk^γ

L²(QT));

|a₀(t,x,ξ;u,z)| ≤g₂(u,z)|ξ|^p−1+ [k₂(u,z)](t,x) where

g2: L²(QT)×L^p(0,T;V2)→_R⁺ and k2 :L²(QT)×L^p(0,T;V2)→L^q(QT) are continuous bounded operators such that

kk2(u,z)k_Lq(Q_T) ≤consth

1+kuk^γ_L2(Q

T)

i

with some constantγ>0.

(B3) The following inequality holds for all t ∈ [0,T]with some constants c2 > 0, β> 0 (not depending ont,u):

Z

QT

∑

[a_j(t,x,Dz(t),z(t);u)−a_j(t,x,Dz^?(t),z^?(t);u)][D_jz(t)−D_jz^?(t)]dxdt +

Z

QT

[a₀(t,x,Dz(t),z(t);u,z)−a₀(t,x,Dz^?(t),z^?(t);u,z^?)][z(t)−z^?(t)]dxdt

≥ ^c2 1+kuk^β

L²(QT)

kz−z^?k^p

L^p(0,T;V2). (B₄) For all fixedu∈ L²(Q_T)the function

F2: QT×L²(QT)→_Rsatisfies(t,x)7→ F2(t,x;u)∈ L^q(QT), kF₂(t,x;u)k_Lq(QT) ≤consth

1+kuk^γ

L²(QT)

i

(see (B₂)) and

(u_k)→uin L²(Q_T)implies F₂(t,x;u_k)→ F₂(t,x;u)in L^q(Q_T). Finally,

β₁ 2

β+γ p−1 <1.

Theorem 2.1. Assume(A₁)–(A₅) and(B₁)–(B₄). Then for all u₀ ∈ V₁, u₁ ∈ L²(_Ω)there exists u∈ L^∞(0,T;V₁)such that

u⁰ ∈ L^∞(0,T;L²(_Ω)), u⁰⁰ ∈ L²(0,T;V₁^?) and z ∈L^p(0,T;V₂) such that u,z satisfy(1.1)in the sense: for a.a. t∈[_0,T], all v∈V₁

hu⁰⁰(t),vi+hQ(u(t)),vi+

Z

Ωϕ(x)h⁰(u(t))vdx+

Z

ΩH(t,x;u,z)vdx+

Z

Ωψ(x)u⁰(t)vdx

=

Z

ΩF₁(t,x;z)v)dx (2.1)

and the initial conditions

u(0) =u0, u⁰(0) =u₁. (2.2)

Further, u,z satisfy(1.2)in the sense: for a.a. t∈ (0,T), all w∈V₂ Z

Ω

"

∑

a_j(t,x,Dz(t),z(t);u)

#

D_jwdx+

Z

Ωa₀(t,x,Dz(t),z(t);u,z)wdx

=

Z

ΩF₂(t,x;u)wdx. (2.3)

Remark 2.2. Examples, satisfying the assumptions of Theorem2.1 can be found in [12].

Main steps of the proof

Now we formulate the main steps in the proof in Theorem 2.1 which will be applied in the next section. (For the detailed proof , see [12].)

Consider the problem (2.1), (2.2) for u with an arbitrary fixed z = z˜ ∈ L^p(0,T;V₂). Ac- cording to [11] assumptions (A₁)–(A₅) imply that there exists a unique solution u = u˜ ∈ L^∞(_0,T;V₁) with the properties ˜u⁰ ∈ L^∞(_0,T;L²(_Ω))_{, ˜}u⁰⁰ ∈ L²(_0,T;V₁^?) satisfying (2.1) and the initial condition (2.2). Then consider problem (2.3) for z with the above u = u. Ac-˜ cording to the theory of monotone operators (see, e.g., [14,15]) there exists a unique solu- tion z ∈ L^p(_0,T;V₂) of (2.3). By using the notation S(z˜) = z, it is shown that the operator S: L^p(0,T;V₂)→ L^p(0,T;V₂)satisfies the assumptions of Schauder’s fixed point theorem: it is continuous, compact and there exists a closed ballB₀(R)⊂ L^p(0,T;V₂)such that

S(B₀(R))⊂B₀(R). (2.4)

Then Schauder’s fixed point theorem implies thatShas a fixed pointz^? ∈ L^p(0,T;V₂). Defin- ing u^? by the solution of (2.1), (2.2) withz =z^?, functionsu^?,z^? satisfy (2.1)–(2.3).

Now we formulate some details of the proof which will be used in the next section.

According to [11] the solution ˜u of (2.1), (2.2) with z = z˜ we obtain as the weak limit in L^p(0,T;V₁)of Galerkin approximations

˜ u_m(t) =

∑

g_lm(t)w_l where g_lm ∈W^2,2(0,T)

and w₁,w₂, . . . is a linearly independent system inV₁ such that the linear combinations are dense inV₁, further, the functions ˜um satisfy (forj=1, . . . ,m)

hu˜⁰⁰_m(t),w_ji+hQ(u˜_m(t)),w_ji+

Z

Ωϕ(x)h⁰(u˜_m(t))w_jdx +

Z

ΩH(t,x; ˜um, ˜z)w_jdx+

Z

Ωψ(x)u˜⁰_m(t)w_jdx=

Z

ΩF₁(t,x; ˜z)w_jdx, (2.5)

˜

u_m(₀) =u_m0, u˜⁰_m(₀) =u_m1 (2.6) where um0,um1 (m=1, 2, . . . ) are linear combinations ofw₁,w2, . . . ,wm, satisfying(um0)→u0

inV₁and(u_m1)→u₁ in L²(_Ω)as m→_∞.

Multiplying (2.5) by (g_jm)⁰(t), summing with respect to j and integrating over (0,t), by Young’s inequality we find

1

2ku˜⁰_m(t)k²_L2(_Ω)+ ¹

2hQ(u˜m(t)), ˜um(t)i+

Z

Ωϕ(x)h(u˜m(t))dx +

Z _t

0

Z

ΩH(_τ,x; ˜u_m, ˜z_k)u˜⁰_m(τ)dx

dτ+

Z _t

0

Z

Ωψ(x)|u˜⁰_m(τ)|²dx

dτ

=

Z _t

0

Z

ΩF₁(τ,x; ˜z)u˜⁰_m(τ)dx

dτ+¹

2ku˜⁰_m(0)k²_L2(_Ω)+¹

2hQ(u˜m(0)), ˜um(0)i +

Z

Ωϕ(x)h(u˜_m(0))dx≤ ¹ 2

Z _T

0

kF₁(τ,x; ˜z)k²_L2(Ω)dτ+ ¹ 2

Z _T

0

ku˜⁰_m(τ)k²_L2(Ω)+const (2.7)

where the constant is not depending onm,k,t. (See [11].)

By using (A2), (A₄), (A5) and the Cauchy–Schwarz inequality, we obtain from (2.7) 1

2ku˜⁰_m(t)k²_L2(Ω)dτ+²+^c0

2ku˜_m(t)k²_V

1+c₁ Z

Ωh(u˜_m(t))dx (2.8)

≤

Z _T

0

kF₁(_τ,x; ˜z)k²_L2(Ω)dτ+_const

1+

Z _t

0

ku˜⁰_m(τ)k²_L2(Ω)dτ+

Z _t

0

Z

Ωh(u˜_m(τ))dx

dτ

where the constants are not depending onm,t, ˜z. Hence, by Gronwall’s lemma one obtains ku˜⁰_m(t)k²_L2(Ω)+

Z

Ωh(u˜_m(t))dx

≤_const

Z _T

0

kF₁(_τ,x; ˜z)k²_L2(Ω)dτ+_const

Z _t

0

_{Z T}

0

h

1+kF₁(_τ,x; ˜z)k²_L2(Ω)dτi

·_e^t−s

ds

=const Z _T

0

kF₁(τ,x; ˜z)k²_L2(_Ω)dτ (2.9)

where the constants are independent ofm,t, ˜z. Thus by (2.8) and (A5) we find ku˜_m(t)k²_V

1 ≤const Z _T

0

kF₁(τ,x; ˜z)k²_L2(Ω)dτ≤consth

1+kz˜k^β1

L^p(0,T;V2)

i

which implies (for the limit of(u˜m))

ku˜k²_L2(QT) ≤consth

1+kz˜k^β_L¹_p(0,T;V

2)

i

. (2.10)

On the other hand, by (B₃), (B₄) we have for the solutionz of (2.3) withu=u˜ c₂

1+ku˜k^β

L²(QT)

kzk^p

L^p(0,T;V2)

≤ kF2(t,x; ˜u)k_L2(QT)kzk_Lp(0,T;V₂)+consth

kk₁(u˜)k_Lq(Q_T)+c(u˜)ⁱkzk_Lp(0,T;V₂) (2.11) where the constant is not depending on ˜u, further, by (B₂)

kk₁(u˜)k_Lq(QT)≤consth

1+ku˜k^γ

L²(QT)

i

and c(u˜)≤consth

1+ku˜k^γ

L²(QT)

i

. (2.12) The inequalities (2.11), (2.12) imply

kzk^p−1

L^p(0,T;V2)≤consth

1+ku˜k^β

L²(QT)

i·^hkF₂(t,x; ˜u)k_L2(Q_T)+1+ku˜k^γ

L²(QT)

i

(2.13) thus by (2.10) and (B₄)

kzk_Lp(0,T;V₂)≤const

1+ku˜k

β+_γ p−1

L²(Q_T)

≤const

"

1+kz˜k

β1(β+γ) 2(p−1) L^p(0,T;V2)

#

(2.14) where the constants are not depending on ˜uand ˜z.

According to the assumption (B₄)

β₁(β+γ)

2(p−₁) <1, (2.15)

so (2.14) implies that there is a closed ballB₀(R)⊂ L^p(_0,T;V₂)_{such that}S(B₀(R))⊂ B₀(R)_.

3 Solutions in ( 0, ∞ )

Now we formulate an existence theorem with respect to solutions for t ∈ (0,∞). Denote by L_loc^p (0,∞;V₁)the set of functions u : (0,∞) → V₁ such that for each fixed finite T > 0, their restrictions to(0,T)satisfyu|_(0,T) ∈ L^p(0,T;V₁)and letQ_∞ = (0,∞)×_Ω, L^α_loc(Q_∞)the set of functionsu:Q_∞ →_Rsuch thatu|_QT ∈ L^α(QT)for any finiteT.

Now we formulate assumptions on H,F₁,a_j, F₂.

( ˜A₄) The function H : Q_∞×L²_loc(Q_∞)×L_loc^p (_0,∞;V₂) → _R is such that for all fixed u ∈ L²_loc(Q_∞), z ∈ L^p_loc(0,∞;V₂) the function (t,x) 7→ H(t,x;u,z) is measurable, H has the Volterra property (see (A₄)) and for each fixed finite T > 0, the restriction H_T of H to QT×L²(_QT)×L^p(_0,T;V2)satisfies (A4).

Remark 3.1. Since H has the Volterra property, this restriction H_T is well defined by the formula

HT(_{t,x; ˜u, ˜z}) = _H(_t,x;u,z)_, (_t,x)∈QT, u˜ ∈ L²(_QT)_, z˜ ∈ L^p(_0,T;V2)

where u ∈ L²_loc(Q_∞), z ∈ L_loc^p (0,∞;V2) may be any functions satisfying u(t,x) = u˜(t,x), z(t,x) =z˜(t,x)for(t,x)∈Q_T.

( ˜A5) F₁ : Q_∞×L_loc^p (0,∞;V2) → _Rhas the Volterra property and for each fixed finite T > 0, the restriction of F₁ to(0,T)satisfies (A₅).

( ˜B) a_j : Q_∞×_Rⁿ⁺¹×L²_loc(Q_∞) → _R (j = 1, . . . ,n) and a₀ : Q_∞×_Rⁿ⁺¹× L²_loc(Q_∞)× L_loc^p (0,∞;V₂) → _R have the Volterra property and for each finite T > 0, their restric- tions to(_0,T)satisfy (B₁)–(B₃).

( ˜B₄) F₂ : Q_∞×L²_loc(Q_∞) → _R has the Volterra property and for each fixed finite T > 0, the restriction ofF₂to (0,T)satisfies (B₄).

Theorem 3.2. Assume(A₁)–(A₃),(A^˜₄),(A^˜₅), (B^˜), (B^˜₄). Then for all u₀ ∈ V₁, u₁ ∈ L²(_Ω)there exist

u∈ L^∞_loc(0,∞;V₁), z∈ L_loc^p (0,∞;V₂) such that u⁰ ∈ L^∞_loc(0,∞;L²(_Ω)), u⁰⁰∈ L²_loc(0,∞;V₁^?),

(2.1)and(2.3)hold for a.a. t ∈(0,∞)and the initial condition(2.2)is fulfilled.

Assume that the following additional conditions are satisfied: there exist H^∞, F₁^∞ ∈ L²(_Ω), u_∞ ∈ V₁, a bounded functionβ, belonging to L˜ ²(0,∞;L²(_Ω))such that

Q(u_∞) =F₁^∞−H^∞, (3.1)

|H(t,x;u,z)−H^∞(x)| ≤ β^˜(t,x), |F₁(t,x;z)−F₁^∞(x)| ≤β^˜(t,x) (3.2) for all fixed u∈ L²_loc(Q_∞), z∈ L_loc^p (0,∞;V₂)). Further, there exist functions

a^∞_j :Ω×_Rⁿ⁺¹×V₁→_R, j=_{1, . . . ,}n a^∞₀ :Ω×_Rⁿ⁺¹×V₁×V₂→_R, F₂^∞ :Ω×V₁→_R such that for each fixed z₀ ∈V₂and w₀∈V₁ with the property

tlim→_∞ku(t)−w₀k_L2(_Ω) =0,

tlim→_∞ka_j(t,x,Dz₀,z₀;u)−a^∞_j (x,Dz₀,z₀;w₀)k_Lq(_Ω)=0, j=1, . . . ,n, (3.3)

tlim→_∞ka₀(t,x,Dz₀,z₀;u,z₀)−a^∞₀ (x,Dz₀,z₀;w₀,z₀)k_Lq(_Ω) =_{0, (3.4)}

tlim→_∞kF₂(t,x;u)−F₂^∞(x;w₀)k_Lq(_Ω) =0. (3.5) Finally, (B₃) is satisfied such that the following inequalities hold for all t> 0with some constants c₂>0, β>₀(not depending on t):

Z

Ω

∑

[a_j(t,x,Dz(t),z(t);u)−a_j(t,x,Dz^?(t),z^?(t);u)][D_jz−D_jz^?]dx +

Z

Ω[a₀(t,x,Dz(t),z(t);u,z)−a₀(t,x,Dz^?(t),z^?(t);u,z^?)][z(t)−z^?(t)]dx

≥ ^c2

1+kuk^β

L²(Qt\Qt−a)

kz−z^?k_V^p

1 (3.6)

with some fixed a>0(finite delay).

Then for any solution u, z of (2.1)–(2.3)in(0,∞)we have

u∈L^∞(0,∞;V₁), (3.7)

ku⁰(t)k_H ≤const e^−c1T (3.8)

where c₁is given in (A₂) and there exists w₀ ∈V₁such that

u(T)→w0in L²(_Ω)as T→_∞, ku(T)−w0k_H ≤const e^−c1T (3.9) and w₀satisfies

Q(w0) +ϕh⁰(w0) =F₁^∞−H^∞. (3.10) Finally, there exists a unique solution z₀∈V₂of

∑

Z

Ωa^∞_j (x,Dz₀,z₀;w₀)D_jvdx+

Z

Ωa^∞₀ (x,Dz₀,z₀;w₀,z₀)vdx

=

Z

ΩF₂^∞(x;w₀)vdx for all v∈V₂ (3.11) (where w₀is the solution of (3.10)) and

tlim→_∞kz(t)−z0k_V2 =0. (3.12) Proof. Let(T_k)_k∈Nbe a monotone increasing sequence, converging to+∞. According to The- orem2.1, there exist solutions u_k,z_k of (2.1)–(2.3) for t ∈ (0,T_k). The Volterra property of H, F₁, a_j, F₂ implies that the restrictions ofu_k,z_k to t ∈ (0,T_l)with T_l < T_k satisfy (2.1)–(2.3) for t∈ (_0,T_l)_.

Now consider the restrictionsu_k|_(0,T1), z_k|_(0,T1),k = 2, 3, . . . Applying (2.14) to T= T₁ and z=z˜= z_k|_(0,T1), by (2.15) we obtain that the sequence

z_k|_(0,T1)

k∈_N is bounded inL^p(0,T₁;V2). (3.13) The operatorS : L^p(0,T₁;V₂)→ L^p(0,T₁;V₂)is compact thus there is a subsequence(z_1k)_k∈N of(z_k)_k∈Nsuch that the sequence of restrictions(z_1k|_(0,T

1))_k∈Nis convergent in L^p(_0,T₁;V₂)_.

Now consider the restrictions z_1k|_(0,T2) By using the above arguments, we find that there exists a subsequence(z_2k)_k∈Nof(z_1k)_k∈Nsuch that(z_2k|_(0,T2))_k∈Nis convergent inL^p(0,T2;V2). Thus for alll ∈ _Nwe obtain a subsequence(z_lk)_k∈N of (z_k)_k∈N such that(z_lk|_(0,Tl))_k∈N is convergent in L^p(0,T_l;V₂). Then the diagonal sequence(z_kk)_k∈N is a subsequence of (z_k)_k∈N such that for all fixed l ∈ _N, (z_kk|_(0,Tl))_k∈N is convergent in L^p(0,T_l;V2) to some z^? ∈ L_loc^p (0,∞;V2). Since z_ll is a fixed point of S = S_l : L^p(0,T_l;V2) → L^p(0,T_l;V2) and S_l is continuous thus the limitz^?|_(0,Tl)in L^p(0,T_l;V₂)of(z_kk|_(0,Tl))_k∈N is a fixed point ofS =S_l.

Consequently, the solutions u^?_l of (2.1), (2.2) when z is the restriction of z^? to (0,T_l) and the restriction of z^? to (0,T_l)satisfy (2.1)–(2.3) fort ∈ (0,T_l). Since for m < l, u^?_l|_(0,Tm) = u^?_m (by the Volterra property of H, F₁, a_j, F₂), we obtain u^? ∈ L²_loc(Q_∞) such that for all fixed l, u^?|_(0,T

l),z^?|_(0,T

l)satisfy (2.1)–(2.3) fort∈ (0,T_l), so the first part of Theorem3.2 is proved.

Now assume that the additional conditions (3.1), (3.2) are satisfied. Then we obtain (3.7)–

(3.10) for u = u^?, z = z^? by using the arguments of the proof of Theorem 3.2 in [11]. For convenience we formulate the main steps of the proof.

Let u, z be arbitrary solutions of (2.1)–(2.3) fort ∈ (0,∞)andz_kk = z|_(0,Tk), u_kk = u|_(0,Tk). Then z_kk, u_kk are solutions of (2.1)–(2.3) for t ∈ (0,T_l) if k ≥ l, hence the sequence (z_kk)|_k∈N is bounded in L^p(0,T_l;V₂) for each fixed l (see, e.g., (3.13)), consequently, from (2.7) (with

˜

z_k = z_kk) we obtain for the solutionsu_kk of (2.1), (2.2) with ˜z=z_kk (sinceu_kk is the limit of the Galerkin approximations)

1

2ku⁰_kk(t)k²_H+¹

2hQ(u_kk(t)),u_kk(t)i+

Z

Ωϕ(x)h(u_kk(t))dx +

Z _t

0

_Z

Ωψ(x)|u⁰_kk(τ)|²dx

dτ+

Z _t

0

_Z

ΩH(τ,x;u_kk,z_kk)u⁰_kk(τ)dx

dτ

=

Z _t

0

_Z

ΩF₁(τ,x;z_kk)u⁰_kk(τ)dx

dτ+¹

2ku⁰_kk(0)k²_H+ ¹

2hQ(u_kk(0)),u_kk(t)i +

Z

Ωϕ(x)h(u_kk(0))dx (3.14)

for all t>0. Hence we find by (3.1), (3.2) and Young’s inequality forw_kk =u_kk−u_∞ 1

2kw⁰_kk(t)k²_L2(Ω)+^c0

2ku_kk(t))k²_V

1+c₁ Z

Ωh(u_kk(t))dx+const Z _t

0

_Z

Ω|w⁰_kk|²dx

dτ

≤const _{Z t}

0

kF₁(τ,x;z_kk)−F₁^∞k²_L2(Ω)dτ+

Z _t

0

kH(τ,x;u_kkz_kk)−H^∞k²_L2(Ω)dτ

+ε Z _t

0

Z

Ω|w⁰_kk|²dx

dτ+¹

2ku⁰_kk(0)k²_L2(_Ω)+¹

2hQ(u_kk(0)),u_kk(0)i+c2

Z

Ωh(u_kk(0))dx

≤ε Z _t

0

_Z

Ω|w⁰_kk|²dx

dτ+const+C(ε)kβ^˜k²_L2(_0,∞;L²(_Ω)). (3.15) Choosing sufficiently small ε>0, we obtain

Z _t

0

Z

Ω|w⁰_kk|²dx

dτ≤_{const (3.16)}

and thus by (3.15)

ku⁰_kk(t)k²_L2(_Ω)+c˜ Z _t

0

ku⁰_kk(τ)k²_L2(_Ω)dτ≤c^?

with some positive constants ˜candc^? not depending onkandt∈(0,∞). Hence by Gronwall’s lemma we obtain (3.8) for the weak limit of the sequence(u_kk)and by (3.15) we find (3.7).

It is not difficult to show that

ku(T₂)−u(T₁)k_L2(_Ω)≤

Z _T2

T₁

ku⁰(t)k_L2(_Ω)dt (3.17) (see [11]), thus (3.8) implies (3.9) and byu∈ L^∞(0,∞;V₁), the limitw0 ofu(t)ast →_∞must belong toV₁.

In order to prove (3.10) we apply equation (1.1) to vχT_k(_t) with arbitrary fixed v ∈ V1

where lim_k→_∞(T_k) = +_∞and

χ_Tk(t) =χ(t−T_k), χ∈C₀^∞, suppχ⊂[0, 1], Z ₁

0 χ(t)dt=1.

Then by (3.8) one obtains (3.10) as k→_∞.

Now we show that there exists a unique solutionz₀ ∈V₂ of (3.11). This statement follows from the fact that the operator (applied toz₀∈ V₂) on the left-hand side of (3.11) is bounded, demicontinuous and uniformly monotone (see, e.g. [14,15]) by (B₁), (B2), (3.9) (3.3), (3.4), (3.6).

Finally, we show (3.12). By (3.6) we have c₂

1+kuk_L2(Qt\Qt−a)

kz(t)−z₀k_V^p

2

≤

Z

Ω

∑

[a_j(t,x,Dz,z;u)−a_j(t,x,Dz₀,z₀;u)](D_jz−D_jz₀)dx +

Z

Ω[a0(t,x,Dz,z;u,z)−a0(t,x,Dz0,z0;u,z0)](z−z0)dx

=

Z

Ω[F₂(t,x;u)−F₂^∞(x,w₀)](z−z₀)dx

−

Z

Ω

∑

[a_j(t,x,Dz₀,z₀;u)−a^∞_j (x,Dz₀,z₀;w₀)](D_jz−D_jz₀)dx

−

Z

Ω[a₀(t,x,Dz₀,z₀;u,z₀)−a^∞₀(t,x,Dz₀,z₀;w₀,z₀)](z−z₀)dx

≤ kF2(t,x;u)−F₂^∞(x,w0)k_Lq(_Ω)kz(t)−z0k_Lp(_Ω)

+

∑

ka_j(t,x,Dz0,z0;u)−a^∞_j (x,Dz0,z0;w0)k_Lq(_Ω)kD_jz(t)−D_jz0k_Lp(_Ω)

+ka₀(t,x,Dz₀,z₀;u,z₀)−a^∞₀(x,Dz₀,z₀;w₀,z₀)k_Lq(_Ω)kz(t)−z₀k_Lp(_Ω). (3.18) Since p > 1 and kuk^β

L²(Qt\Qt−a) is bounded for t ∈ (0,∞) by (3.9), thus (3.3)–(3.5), (3.18) imply (3.12).

Remark 3.3. Assume that the inequalities (3.3)–(3.5) hold such that forj=1, . . . ,n

|a_j(t,x,ξ;u)−a^∞_j (x,ξ;u)| ≤consth

ku(t)−w0k_Lp(Qt\Qt−a)+η(t)^{i h}1+|ξ|^p−1i,

|a₀(t,x,ξ;u,z₀)−a^∞₀ (x,ξ;u,z)| ≤consth

ku(t)−w₀k_Lp(Qt\Qt−a)+η(t)^{i h}1+|ξ|^p−1i,

|F₂(t,x;u)−F₂^∞(x;w₀)| ≤_const^hku(t)−w₀k_Lp(qt\Qt−a)+η(t)ⁱ_. Then

kz(t)−z₀k_V^p−1

2 ≤_conste^−c1t+η(t)_, t>_0.

The above inequalities are satisfied e.g. if a_j(t,x,ξ;u) =g_j(x,ξ)

1+

Z _t

t−a

|u(τ,x)|dτ+η(t)

, j=1, . . . ,n a₀(t,x,ξ;u,z) =g₀(x,ξ)

1+

Z _t

t−a

|u(τ,x)|dτ+η(t)

where

|g_j(x,ξ)| ≤const[|ξ|^p−1+g˜(x)], g˜∈ L^q(_Ω), η≥0, lim

∞ η=0,

∑

[g_j(x,ξ)−g_j(x,ξ^?)](ξ_j−ξ^?_j)≥c₂|ξ−ξ^?|^p with some constantc2 >0.

Acknowledgements

This work was supported by the Hungarian National Foundation for Scientific Research under grant OTKA K 81403.

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