Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 3 1-13; http://www.math.u-szeged.hu/ejqtde/

On a nonlinear system containing nonlocal terms related to a fluid flow model

Ádám Besenyei^{∗}

Abstract We consider a nonlinear system of differential equations where the main parts may contain nonlocal dependence on the unknowns. This system is a generalization of a model describing fluid flow in porous medium. Existence of weak solutions, boundedness and stabilization of solutions as t → ∞ is shown by using the theory of monotone operators, and some examples are given.

1 Introduction

This paper was motivated by the work [21]. There the authors investigated fluid flow in porous media. A porous medium is a solid medium with lots of tiny holes (e.g., limestone). The flow of a fluid through the medium is determined by the large surface of the solid matrix and the closeness of the holes. For a detailed introduction to this topic, see [3]. If the fluid carries dissolved chemical species, chemical reactions can occur, see [16]. Among these include reactions that can change the porosity. In the cited paper the following model was derived for such flow in one dimension:

ω(t, x)u_{t}(t, x) = α · (|v(t, x)|u_{x}(t, x))_{x} + K(ω(t, x))p_{x}(t, x)u_{x}(t, x) − ku(t, x)g(ω(t, x))  (1)
ω_{t}(t, x) = bu(t, x)g(ω(t, x))  (2)
(K(ω(t, x))p_{x}(t, x))_{x} = bu(t, x)g(ω(t, x)),  (3)
v(t, x) = −K(ω(t, x))p_{x}(t, x),  t > 0, x ∈ (0, 1),  (4)

with some initial and boundary conditions where ω is the porosity, u is the concentration of the dissolved chemical solute carried by the fluid, p is the pressure, v is the velocity, further, α, k, b are given constants, K and g are given real functions. Observe that after elimintaion of the fourth equation one obtains a system that contains three different types of differential equations: an ordinary, a parabolic and an elliptic one, see [11, 21]. Similar model was studied in [11] by using the method of Rothe. Some numerical experiments were done in [21] concering the above system, however, correct proof on existence of solutions was not made (and one can hardly find papers dealing with such kind of systems in rigorous mathematical way). In the following, we consider a generalization of the above system. Namely, we admit also nonlocal dependence on the unknowns. Such nonlocality is especially reasonable for diffusion processes (heat or population) where the diffusion coefficient may depend on terms which depend on the unknowns in a nonlocal way (e.g., on the integral of the density). Furthermore, nonlocal models arise also in climatology, see the papers [2, 12, 13, 14] where a climatology model containing functional differential equations was studied. For some other problems involving nonlocal differential equations, such as transmission problems, see [17, 18, 19], and as nonlocal boundary conditions, see [26, 25, 23].

In the following we show existence and properties of weak solutions (such as boundedness and stabilization as t → ∞) to a nonlocal generalization of the above system by using the theory of operators of monotone type. Our assumptions will be the generalizations of the classical conditions. However these are strict assumptions, the examples given after each statement will show that the results apply in a large class of problems.

1.1 Notation

Let Ω ⊂ ℝ^{n} be a bounded domain with the uniform C^{1} regularity property (see [1]), further, let 0 < T < ∞, 2 ≤ p_{1}, p_{2} < ∞ be real numbers. In the following, Q_{T} := (0, T) × Ω, Q_{∞} := (0, ∞) × Ω. Denote by W^{1,p_{i}}(Ω) the usual Sobolev space with the norm

∥v∥_{W^{1,p_{i}} (Ω)} = ( ∫_{Ω}(|v|^{p_{i}} + ∑_{j=1}^{n} |D_{j}v|^{p_{i}}))^{1/p_{i}}

∗This work was supported by the Hungarian National Foundation for Scientific Research under grant OTKA T 049819. This paper is in final form and no version of it is submitted for publication elsewhere.

2000 Mathematics Subject Classification: 35K60, 35J60 Key words and phrases: flow in porous medium, functional differential equation, monotone operators

where D_{j} denotes the distributional derivative with respect to the j-th variable (briefly D = (D_{1}, …, D_{n})). In addition, let V_{i} be a closed linear subspace of W^{1,p_{i}}(Ω) which contains W_{0}^{1,p_{i}}(Ω) (the closure of C_{0}^{∞}(Ω) in W^{1,p_{i}}(Ω)), and let L^{p_{i}}(0, T; V_{i}) be the Banach space of measurable functions u: (0, T) → V_{i} such that ∥u∥_{V_{i}}^{p_{i}} is integrable and the norm is given by

∥u∥_{L^{p_{i}} (0,T;V_{i})} = ( ∫_{0}^{T} ∥u(t)∥_{V_{i}}^{p_{i}}dt)^{1/p_{i}}.

The dual space of L^{p_{i}}(0, T; V_{i}) is L^{q_{i}}(0, T; V_{i}^{∗}) where \frac{1}{p_{i}} + \frac{1}{q_{i}} = 1 and V_{i}^{∗} is the dual of V_{i}. We write briefly X_{i} := L^{p_{i}}(0, T; V_{i}). The pairing between V_{i}^{∗}, V_{i} and X_{i}^{∗}, X_{i} is denoted by ⟨·, ·⟩ and [·, ·], respectively, further, D_{t}u stands for the derivative of a function u ∈ L^{p_{i}}(0, T; V_{i}). It is well known (see [27]) that if u ∈ X_{i}, D_{t}u ∈ X_{i}^{∗} then u ∈ C([0, T], L^{2}(Ω)) so that u(0) makes sense.

1.2 Formulation of the problem

Let us consider the following system of equations:

D_{t}ω(t, x) = f(t, x, ω(t, x), u(t, x); u),  ω(0, x) = ω_{0}(x),  (5)
D_{t}u(t, x) − ∑_{i=1}^{n} D_{i} [a_{i}(t, x, ω(t, x), u(t, x), Du(t, x), p(t, x), Dp(t, x); ω, u, p)] +  (6)
+a_{0}(t, x, ω(t, x), u(t, x), Du(t, x), p(t, x), Dp(t, x); ω, u, p) = g(t, x), u(0, x) = 0,
−∑_{i=1}^{n} D_{i}[b_{i}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); ω, u, p)] +  (7)
+b_{0}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); ω, u, p) = h(t, x)

with some boundary conditions. This system is a generalization of the model (1)–(4), functions f, a_{i}, b_{i} may contain nonlocal dependence on the unknown functions ω, u, p which are written after the symbol ”;”. In the next section we formulate some assumptions on these functions then we may define the weak form of the above system and prove existence of weak solutions.

1.3 Assumptions

In what follows, ξ, (ζ_{0}, ζ), (η_{0}, η) refer for the variables ω, (u, Du) and (p, Dp), respectively, further, w, v_{1} and v_{2} for the nonlocal dependence on ω, u and p.

(A1) For fixed (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2} functions a_{i}: Q_{T} × ℝ × ℝ^{n+1} × ℝ^{n+1} × L^{∞}(Q_{T}) × X_{1} × X_{2} → ℝ (i = 0, …, n) have the Carathéodory property, i.e., they are measurable in (t, x) ∈ Q_{T} for every (ξ, ζ_{0}, ζ, η_{0}, η) ∈ ℝ×ℝ^{n+1} ×ℝ^{n+1} and continuous in (ξ, ζ_{0}, ζ, η_{0}, η) ∈ ℝ×ℝ^{n+1} ×ℝ^{n+1} for a.a. (t, x) ∈ Q_{T}.

(A2) There exist a continuous function c_{1} : ℝ → ℝ^{+} and bounded operators 𝔠_{1} : L^{∞}(Q_{T}) × X_{1} × X_{2} → ℝ^{+}, k_{1} : L^{∞}(Q_{T}) × X_{1} × X_{2} → L^{q_{1}}(Q_{T}) such that

|a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2})| ≤

≤ 𝔠_{1}(w, v_{1}, v_{2})c_{1}(ξ) (|ζ_{0}|^{p_{1}−1} + |ζ|^{p_{1}−1} + |η_{0}|^{\frac{p_{2}}{q_{1}}} + |η|^{\frac{p_{2}}{q_{1}}} + [k_{1}(w, v_{1}, v_{2})](t, x)) ,

for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}, ζ, η_{0}, η) ∈ ℝ × ℝ^{n+1} × ℝ^{n+1} and (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2} (i = 0, …, n).

(A3) There exists a constant C > 0 such that for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}, ζ, η_{0}, η), (ξ, ζ_{0}, ζ̃, η_{0}, η) ∈ ℝ × ℝ^{n+1} × ℝ^{n+1} and (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2}

∑_{i=1}^{n} (a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) − a_{i}(t, x, ξ, ζ_{0}, ζ̃, η_{0}, η; w, v_{1}, v_{2})) (ζ_{i} − ζ̃_{i}) ≥ C · |ζ − ζ̃|^{p_{1}}.

(A4) There exist a constant c_{2} > 0, a continuous function γ: ℝ → ℝ and bounded operators Γ: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), k_{2} : X_{1} → L^{1}(Q_{T}) such that

∑_{i=0}^{n} a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2})ζ_{i} ≥ c_{2} (|ζ_{0}|^{p_{1}} + |ζ|^{p_{1}}) − γ(ξ)[Γ(w)](t, x)[k_{2}(v_{1})](t, x)

for a.a. (t, x) ∈ Q_{T} and every (ξ, ζ_{0}, ζ, η_{0}, η) ∈ ℝ×ℝ^{n+1} ×ℝ^{n+1}, (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T})×X_{1} ×X_{2}. Further,

lim_{∥v_{1}∥_{X_{1}} →+∞} \frac{∥k_{2}(v_{1})∥_{L^{1}(Q_{T})}}{∥v_{1}∥_{X_{1}}^{p_{1}}} = 0.  (8)

(A5) If (ω_{k}) is bounded in L^{∞}(Q_{T}), ω_{k} → ω a.e. in Q_{T} and u_{k} → u weakly in X_{1}, strongly in L^{p_{1}}(Q_{T}), further, p_{k} → p strongly in X_{2} then

lim_{k→∞} ∥a_{i}(·, ω_{k}, u_{k}, Du_{k}, p_{k}, Dp_{k}; ω_{k}, u_{k}, p_{k}) − a_{i}(·, ω_{k}, u_{k}, Du_{k}, p_{k}, Dp_{k}; ω, u, p)∥_{L^{q_{1}} (Q_{T})} = 0.

(B1) For fixed (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2} functions b_{i}: Q_{T} × ℝ × ℝ × ℝ^{n+1} × L^{∞}(Q_{T}) × X_{1} × X_{2} → ℝ (i = 0, …, n) have the Carathéodory property, i.e., they are measurable in (t, x) ∈ Q_{T} for every (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1} and continuous in (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1} for a.a. (t, x) ∈ Q_{T}.

(B2) There exist a continuous function ĉ_{1} : ℝ → ℝ^{+} and bounded operators 𝔠̂_{1} : L^{∞}(Q_{T}) × X_{1} × X_{2} → ℝ^{+}, k̂_{1} : L^{∞}(Q_{T}) × X_{1} × X_{2} → L^{q_{2}}(Q_{T}) such that

|b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2})| ≤ 𝔠̂_{1}(w, v_{1}, v_{2})ĉ_{1}(ξ) (|η_{0}|^{p_{2}−1} + |η|^{p_{2}−1} + |ζ_{0}|^{\frac{p_{1}}{q_{2}}} + [k̂_{1}(w, v_{1}, v_{2})](t, x))

for a.a. (t, x) ∈ Q_{T} and every (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1}, (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2} (i = 0, …, n).

(B3) There exists a constant Ĉ > 0 such that for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}, η_{0}, η), (ξ, ζ_{0}, η̃_{0}, η̃) ∈ ℝ×ℝ×ℝ^{n+1} and (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2}

∑_{i=0}^{n} (b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2}) − b_{i}(t, x, ξ, ζ_{0}, η̃_{0}, η̃; w, v_{1}, v_{2})) (η_{i} − η̃_{i}) ≥ Ĉ · (|η_{0} − η̃_{0}|^{p_{2}} + |η − η̃|^{p_{2}}) .

(B4) There exist a constant ĉ_{2} > 0, a continuous function γ̂: ℝ → ℝ and bounded operators Γ̂: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), k̂_{2} : X_{2} → L^{1}(Q_{T}) such that

∑_{i=0}^{n} b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2})η_{i} ≥ ĉ_{2} (|η_{0}|^{p_{2}} + |η|^{p_{2}}) − γ̂(ξ)[Γ̂(w)](t, x) (|ζ_{0}|^{p_{1}} + [k̂_{2}(v_{2})](t, x))

for a.a. (t, x) ∈ Q_{T}, and every (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1}, (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{T}) × X_{1} × X_{2}. Further,

lim_{∥v_{2}∥_{X_{2}} →∞} \frac{∥k̂_{2}(v_{2})∥_{L^{1}(Q_{T})}}{∥v_{2}∥_{X_{2}}^{p_{2}}} = 0.  (9)

(B5) If (ω_{k}) is bounded in L^{∞}(Q_{T}), ω_{k} → ω a.e. in Q_{T} and u_{k} → u weakly in X_{1}, strongly in L^{p_{1}}(Q_{T}), further, p_{k} → p weakly in X_{2} then

lim_{k→∞} ∥b_{i}(·, ω_{k}, u_{k}, p_{k}, Dp_{k}; ω_{k}, u_{k}, p_{k}) − b_{i}(·, ω_{k}, u_{k}, p_{k}, Dp_{k}; ω, u, p)∥_{L^{q_{2}} (Q_{T})} = 0.

(F1) For fixed v ∈ X_{1} function f: Q_{T} ×ℝ^{2}×L^{∞}(Q_{T})×X_{1} → ℝ is a Carathéodory function, i.e., it is measurable in (t, x) ∈ Q_{T} for every (ξ, ζ_{0}) ∈ ℝ^{2} and continuous in (ξ, ζ_{0}) ∈ ℝ^{2} for a.a. (t, x) ∈ Q_{T}. Further, there exists a bounded operator 𝒦_{1} : X_{1} → ℝ^{+} such that

(i) for every bounded set I ⊂ ℝ there is a continuous function K_{1} : ℝ → ℝ^{+} satisfying |K_{1}(ζ_{0})| ≤ d_{1}|ζ_{0}|^{\frac{p_{1}}{q_{2}}} + d_{2} for every ζ_{0} ∈ ℝ, with some nonnegative constants d_{1}, d_{2} (depending on I),

(ii) for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}), (ξ̃, ζ_{0}) ∈ I × ℝ and every v ∈ X_{1},

|f(t, x, ξ, ζ_{0}; v) − f(t, x, ξ̃, ζ_{0}; v)| ≤ 𝒦_{1}(v)K_{1}(ζ_{0}) · |ξ − ξ̃|.

(F2) There exist a bounded operator 𝒦_{2} : X_{1} → ℝ^{+} and a continuous function K_{2} : ℝ → ℝ^{+} such that for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}), (ξ, ζ̃_{0}) ∈ ℝ^{2} and v ∈ X_{1}

|f(t, x, ξ, ζ_{0}; v) − f(t, x, ξ, ζ̃_{0}; v)| ≤ 𝒦_{2}(v)K_{2}(ξ) · |ζ_{0} − ζ̃_{0}|.

(F3) There exists ω^{∗} ∈ L^{∞}(Ω) such that for a.a. (t, x) ∈ Q_{T}, every (ξ, ζ_{0}) ∈ ℝ^{2} and v ∈ X_{1},

(ξ − ω^{∗}(x)) · f(t, x, ξ, ζ_{0}; v) ≤ 0.

(F4) If (ω_{k}) is bounded in L^{∞}(Q_{T}) and u_{k} → u strongly in L^{p_{1}}(Q_{T}) then

lim_{k→∞} ∥f(·, ω_{k}, u_{k}; u_{k}) − f(·, ω_{k}, u_{k}; u)∥_{L^{1}(Q_{T})} = 0.

1.4 Weak form

If the above assumptions are satisfied we may define operators A: L^{∞}(Q_{T}) × X_{1} × X_{2} → X_{1}^{∗}, B: L^{∞}(Q_{T}) × X_{1} × X_{2} → X_{2}^{∗} by:

[A(ω, u, p), v_{1}] := ∫_{Q_{T}} ∑_{i=1}^{n} a_{i}(t, x, ω(t, x), u(t, x), Du(t, x), p(t, x), Dp(t, x); ω, u, p)D_{i}v_{1}(t, x) dt dx+

+ ∫_{Q_{T}} a_{0}(t, x, ω(t, x), u(t, x), Du(t, x), p(t, x), Dp(t, x); ω, u, p)v_{1}(t, x) dt dx, (10)

[B(ω, u, p), v_{2}] := ∫_{Q_{T}} ∑_{i=1}^{n} b_{i}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); ω, u, p)D_{i}v_{2}(t, x) dt dx+

+ ∫_{Q_{T}} b_{0}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); ω, u, p)v_{2}(t, x) dt dx, (11)

for v_{1} ∈ X_{1} and v_{2} ∈ X_{2}. In addition, let us introduce the linear operator L : D(L) → X_{1}^{∗} by

D(L) = {u ∈ X_{1} : D_{t}u ∈ X_{1}^{∗}, u(0) = 0},  Lu = D_{t}u.  (12)

By the operators above we may define the weak form of system (5)–(7) as

ω(t, x) = ω_{0}(x) + ∫_{0}^{t} f(s, x, ω(s, x), u(s, x); u) ds for a.a. (t, x) ∈ Q_{T}  (13)
Lu + A(ω, u, p) = G  (14)
B(ω, u, p) = H  (15)

where G ∈ X_{1}^{∗} and H ∈ X_{2}^{∗} are given by

[G, v_{1}] = ∫_{Q_{T}} g(t, x)v_{1}(t, x) dt dx, [H, v_{2}] = ∫_{Q_{T}} h(t, x)v_{2}(t, x) dt dx

where v_{i} ∈ X_{i} (i = 1, 2). It is well-known (see, e.g., [20]) that one obtains the above weak form by taking sufficiently smooth solutions, using Green’s theorem and finally considering the whole system in the space L^{p}(0, T; V). Clearly, if the boundary condition is homogeneous Neumann then V = W^{1,p}(Ω) (since the boundary term vanishes in Green’s theorem) and if we have homogeneous Dirichlet boundary condition then V = W_{0}^{1,p}(Ω) (in order to eliminate the boundary term in Green’s theorem). Further, if we have a partition, for example in one dimension with homogenous Dirichlet and Neumann boundary conditions then V = {v ∈ W^{1,p_{1}}(0, 1) : v(0) = 0, D_{x}v(1) = 0}.

2 Existence of solutions

In this section we prove

Theorem 1. Suppose that conditions (A1)–(A5), (B1)–(B5), (F1)–(F4) are fulfilled. Then for every ω_{0} ∈ L^{∞}(Ω), G ∈ X_{1}^{∗} and H ∈ X_{2}^{∗} there exists a solution ω ∈ L^{∞}(Q_{T}), u ∈ D(L), p ∈ L^{p_{2}}(0, T; V_{2}) of problem (13)–(15).

First we formulate some statements related to the solvability of the above equations (13)–(15).

Proposition 2. Assume that conditions (F1), (F3) are satisfied. Then for every fixed u ∈ L^{p_{1}}(Q_{T}) and ω_{0} ∈ L^{∞}(Q_{T}) there exists a unique solution ω ∈ L^{∞}(Q_{T}) of the integral equation (13), further, for the solution ω, estimate ∥ω∥_{L^{∞}(Q_{T})} ≤ ∥ω_{0}∥_{L^{∞}(Ω)} + ∥ω^{∗}∥_{L^{∞}(Ω)} holds.

Proof. Immediately follows from Proposition 2.3 in [6] since for fixed nonlocal variable u, condition (F1) is the same as in the cited paper.

Proposition 3. Assume (F1)–(F4) and let (u_{k}) ⊂ L^{p_{1}}(Q_{T}), further, let ω_{k} be the solution of (13) corresponding to u_{k}. If u_{k} → u in L^{p_{1}}(Q_{T}) then ω_{k} → ω a.e. in Q_{T} where ω is the solution of (13) corresponding to u.

Proof. We may assume that for a.a. x ∈ Ω, u_{k}(·, x) → u(·, x). Fix such a point x ∈ Ω. Consider the following estimate:

|ω_{k}(t, x) − ω(t, x)| ≤ ∫_{0}^{t} |f(s, x, ω_{k}(s, x), u_{k}(s, x); u_{k}) − f(s, x, ω_{k}(s, x), u_{k}(s, x); u)| ds+

∫_{0}^{t} |f(s, x, ω_{k}(s, x), u_{k}(s, x); u) − f(s, x, ω(s, x), u(s, x); u)| ds.

The first integral converges to 0 for a.a. x ∈ Ω by condition (F4), further, by (F1), (F2) it is easy to show that the second integral is less then

const · (∫_{0}^{t} |ω_{k}(s, x) − ω(s, x)|^{p_{2}} ds)^{1/p_{2}} + const · ∫_{0}^{T} |u_{k}(s, x) − u(s, x)| ds.

Hence

|ω_{k}(t, x) − ω(t, x)|^{p_{2}} ≤ const · ∫_{0}^{t} |ω_{k}(s, x) − ω(s, x)|^{p_{2}} ds + r(u_{k}, ω_{k})

where the remainder term r(u_{k}, ω_{k}) tends to 0 as k → ∞. Thus Gronwall’s lemma yields |ω_{k}(t, x) − ω(t, x)| ≤ const · r(u_{k}) → 0 which implies the desired a.e. convergence of (ω_{k}).

Proposition 4. Assume (A1)–(A5). Then for every fixed ω ∈ L^{∞}(Q_{T}), p ∈ X_{2} and G ∈ X_{1}^{∗} there exists a solution u ∈ D(L) of problem Lu + A(ω, u, p) = G.

Proof. The proof follows from Theorem 1.1 in [22] (based on the theory of monotone type operators, see [4]) since for fixed ω ∈ L^{∞}(Q_{T}) and p ∈ X_{2} conditions (A1)–(A5) imply that operator A(ω, ·, p): X_{1} → X_{1}^{∗} fulfils conditions I–V of the mentioned theorem.

Proposition 5. Suppose that (B1)–(B5) hold. Then for every fixed ω ∈ L^{∞}(Q_{T}), u ∈ X_{1} and H ∈ X_{2}^{∗} there exists a solution p ∈ X_{2} of problem B(ω, u, p) = H.

Proof. The statement follows from the theory of monotone operators (see [27]) since conditions (B1)–(B5) imply the boundedness, demicontinuity, pseudomonotonicity and coerciveness of operator B(ω, u, ·): X_{2} → X_{2}^{∗} for fixed ω ∈ L^{∞}(Q_{T}), u ∈ X_{1}.

Proof of Theorem 1. The idea is similar as in [6]. We define sequences of approximate solutions of problem (13)– (15) and we show the boundedness of these sequences. By using the diagonal method we will choose wealky convergent subsequences and we verify that the weak limits of the subsequences are solutions of the problem. For simplicity, in the proof we omit the variable (t, x) of functions a_{i}, b_{i} if it is not confusing.

Step 1: approximation. Define the sequences (ω_{k}), (u_{k}), (p_{k}) as follows. Let ω_{0}(t, x) ≡ u_{0}(t, x) ≡ p_{0}(t, x) ≡ 0 ((t, x) ∈ Q_{T}) and for k = 0, 1, … let ω_{k+1}, u_{k+1}, p_{k+1} be a solution of the system:

ω_{k+1}(t, x) = ω_{0}(x) + ∫_{0}^{t} f(s, x, ω_{k+1}(s, x), u_{k}(s, x); u_{k}) ds  (16)
Lu_{k+1} + A(ω_{k}, u_{k+1}, p_{k}) = G  (17)
B(ω_{k}, u_{k}, p_{k+1}) = H.  (18)

By Propositions 2, 4, 5 we have solutions ω_{k+1} ∈ L^{∞}(Q_{T}), u_{k+1} ∈ X_{1}, p_{k+1} ∈ X_{2} so the above recurrence yields the sequences (ω_{k}) ⊂ L^{∞}(Q_{T}), (u_{k}) ⊂ X_{1}, (p_{k}) ⊂ X_{2}.

Step 2: boundedness. We show that the above defined sequences are bounded. By Proposition 2 for fixed ω_{0} ∈ L^{∞}(Ω) for the solution of equation (16) estimate ∥ω_{k+1}∥_{L^{∞}(Q_{T})} ≤ ∥ω_{0}∥_{L^{∞}(Ω)} +∥ω^{∗}∥_{L^{∞}(Ω)} holds thus (ω_{k}) is bounded in L^{∞}(Q_{T})

Now by choosing the test function v = u_{k+1} in (17) and by using condition (A4) and the monotonicity of operator L we obtain

[G, u_{k+1}] = [Lu_{k+1}, u_{k+1}] + [A(ω_{k}, u_{k+1}, p_{k}), u_{k+1}] ≥ c_{2} ∫_{Q_{T}} (|u_{k+1}|^{p_{1}} + |Du_{k+1}|^{p_{1}} − γ(ω_{k})Γ(ω_{k})k_{2}(u_{k+1})) ≥

≥ c_{2}∥u_{k+1}∥_{X_{1}} (∥u_{k+1}∥_{X_{1}}^{p_{1}−1} − ∥γ(ω_{k})Γ(ω_{k})∥_{L^{∞}(Q_{T})} · \frac{∥k_{2}(u_{k+1})∥_{L^{1}(Q_{T})}}{∥u_{k+1}∥_{X_{1}}} ) .

Whence by the boundedness of (ω_{k}) we conclude for some K > 0

∥u_{k+1}∥_{X_{1}}^{p_{1}−1} (1 − K · \frac{∥k_{2}(u_{k+1})∥_{L^{1}(Q_{T})}}{∥u_{k+1}∥_{X_{1}}^{p_{1}}} ) ≤ const.

Now (8) implies that (u_{k}) is bounded in X_{1}.

The boundedness of (p_{k}) in X_{2} follows by similar arguments as above by using condition (B4) and the boundedness of the sequences (ω_{k}), (u_{k}).

We need also the boundedness of the sequence (Lu_{k}) in X_{1}^{∗}. By Hölder’s inequality

|[A(ω_{k}, u_{k+1}, p_{k}), v]| ≤ (∑_{i=0}^{n} ∥a_{i}(ω_{k}, u_{k+1}, Du_{k+1}, p_{k}, Dp_{k}; ω_{k}, u_{k+1}, p_{k})∥_{L^{q_{1}} (Q_{T})}) · ∥v∥_{X_{1}}.

and from condition (A2) it follows that for all i

∥a_{i}(ω_{k}, u_{k+1}, Du_{k+1}, p_{k}, Dp_{k}; ω_{k}, u_{k+1}, p_{k})∥_{L^{q_{1}} (Q_{T})} ≤

≤ const · c_{1}(ω_{k})𝔠_{1}(ω_{k}, u_{k+1}, p_{k}) (∥u_{k+1}∥_{X_{1}}^{p_{1}} + ∥p_{k}∥_{X_{2}}^{p_{2}} + ∥k_{1}(ω_{k}, u_{k+1}, p_{k})∥_{L^{q_{1}} (Q_{T})}) .

Therefore by the boundedness of the sequences (ω_{k}), (u_{k}), (p_{k}) and the boundedness of operators c_{1}, 𝔠_{1}, k_{2} we conclude |[Lu_{k+1}, v]| = |[A(ω_{k}, u_{k+1}, p_{k}) + G, v]| ≤ const · ∥v∥_{X_{1}} so (Lu_{k}) is a bounded sequence in X_{1}^{∗}.

Step 3: convergence. Due to the boundedness of the sequences (u_{k}), (Lu_{k}), (p_{k}) (in reflexive Banach spaces) each has a weakly convergent subsequence, further, by applying a well known embedding theorem (see [20]) it follows that there exist subsequences (which will be denoted same as the original sequences) and functions ω ∈ L^{∞}(Q_{T}), u ∈ X_{1}, p ∈ X_{2} such that

u_{k} → u weakly in X_{1}, strongly in L^{p_{1}}(Q_{T}), a.e. in Q_{T};
Lu_{k} → Lu weakly in X_{1}^{∗};
p_{k} → p weakly in X_{2}.

In what follows, we show that ω, u, p are solutions of problem (13)–(15).

Since u_{k} → u in L^{p_{1}}(Q_{T}), further, ω_{k+1} is the solution of equation (16), by Proposition 3 it follows that ω_{k} → ω a.e. in Q_{T} and functions ω, u satisfy the integral equation (13).

Now let us consider equation (18). First we show that p_{k} → p in X_{2}. To this end, let us introduce operator B̃: L^{∞}(Q_{T}) × X_{1} × X_{2} × L^{∞}(Q_{T}) × X_{1} × X_{2} → X_{2}^{∗} by

[B̃(ω, u, p; w, v_{1}, v_{2}), z_{2}] := ∫_{Q_{T}} ∑_{i=1}^{n} b_{i}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); w, v_{1}, v_{2})D_{i}z_{2}(t, x) dt dx+

+ ∫_{Q_{T}} b_{0}(t, x, ω(t, x), u(t, x), p(t, x), Dp(t, x); w, v_{1}, v_{2})z_{2}(t, x) dt dx (19)

for z_{2} ∈ X_{2}. Observe B(ω, u, p) = B̃(ω, u, p; ω, u, p). By condition (B3) we have

[B̃(ω_{k}, u_{k}, p_{k+1}; ω, u, p) − B̃(ω_{k}, u_{k}, p; ω, u, p), p_{k+1} − p] ≥ Ĉ · ∥p_{k+1} − p∥_{X_{2}}^{p_{2}}.  (20)

On the left hand side of the above inequality we have the following decomposition:

[B̃(ω_{k}, u_{k}, p_{k+1}; ω, u, p) − B̃(ω_{k}, u_{k}, p; ω, u, p), p_{k+1} − p] = [B̃(ω_{k}, u_{k}, p_{k+1}; ω_{k}, u_{k}, p_{k+1}), p_{k+1} − p]+

+ [B̃(ω_{k}, u_{k}, p_{k+1}; ω, u, p) − B̃(ω_{k}, u_{k}, p_{k+1}; ω_{k}, u_{k}, p_{k+1}), p_{k+1} − p]+

+ [B̃(ω, u, p; ω, u, p) − B̃(ω_{k}, u_{k}, p; ω, u, p), p_{k+1} − p] − [B̃(ω, u, p; ω, u, p), p_{k+1} − p]. (21)

We show that each term on the right hand side tends to 0. By recurrence (18), B̃(ω_{k}, u_{k}, p_{k+1}; ω_{k}, u_{k}, p_{k+1}) = H, further, p_{k+1} → p weakly in X_{2} which implies the convergence of the first and the last term. The convergence of the second term follows from condition (B5). In order to verify the convergence of the third term, observe that

|[B̃(ω_{k}, u_{k}, p; ω, u, p) − B̃(ω, u, p; ω, u, p), p_{k+1} − p]| ≤

≤ ∑_{i=0}^{n} ∥b_{i}(ω_{k}, u_{k}, p, Dp; ω, u, p) − b_{i}(ω, u, p, Dp; ω, u, p)∥_{L^{q_{2}} (Q_{T})} · ∥p_{k+1} − p∥_{X_{2}} (22)

and by condition (B2)

|b_{i}(ω_{k}, u_{k}, p, Dp; ω, u, p) − b_{i}(ω, u, p, Dp; ω, u, p)|^{q_{2}} ≤

≤ const · 𝔠̂_{1}(ω, u, p) · (|ĉ_{1}(ω_{k})|^{q_{2}} + |ĉ_{1}(ω)|^{q_{2}}) (|p|^{p_{2}} + |Dp|^{p_{2}} + |u_{k}|^{p_{1}} + |u|^{p_{1}} + |k̂_{1}(ω, u, p)|^{q_{2}}) .

Due to the boundedness of (ω_{k}) in L^{∞}(Q_{T}) and the convergence of (u_{k}) in L^{p_{1}}(Q_{T}) the left hand side of the above inequality is equi-integrable (see [10]), in addition, it a.e. converges to 0, therefore by Vitali’s theorem the left hand side converges in L^{1}(Q_{T}) to the zero function. Thus (because of the boundedness of (p_{k})) the right hand side of (22) tends to 0. Hence all terms on the right hand side of equation (21) converges to 0 thus (20) implies p_{k+1} → p in X_{2}.

Now by using the same arguments as in [6] one obtains that B̃(ω_{k}, u_{k}, p_{k+1}; ω, u, p) → B̃(ω, u, p; ω, u, p) = B(ω, u, p) weakly in X_{2}^{∗}. Further, by condition (B5) it is not difficult to see that B̃(ω_{k}, u_{k}, p_{k+1}; ω, u, p) − B̃(ω_{k}, u_{k}, p_{k+1}; ω_{k}, u_{k}, p_{k+1}) → 0 strongly in X_{2}^{∗} thus B̃(ω_{k}, u_{k}, p_{k+1}; ω_{k}, u_{k}, p_{k+1}) → B(ω, u, p). Then from recurrence (18) we conclude B(ω, u, p) = H, i.e., ω, u, p are solutions of problem (15).

Finally, A(ω, u, p) = G can be shown by similar arguments as above. The proof of the theorem is complete.

3 Examples

We show some examples for functions satisfying conditions (A1)–(A5), (B1)–(B5). Let functions a_{i}, b_{i} have the form

a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) = [π(w)](t, x)[φ(v_{1})](t, x)[ψ(v_{2})](t, x)P (ξ)Q(η_{0}, η)ζ_{i}|ζ|^{p_{1}−2}+

+ [π̃(w)](t, x)[φ̃(v_{1})](t, x)P̃(ξ)ζ_{i}|ζ|^{r_{1}−1}, if i ≠ 0, (23)

a_{0}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) = [π(w)](t, x)[φ(v_{1})](t, x)[ψ(v_{2})](t, x)P (ξ)Q(η_{0}, η)ζ_{0}|ζ_{0}|^{p_{1}−2}+

+ [π̃_{0}(w)](t, x)[φ̃_{0}(v_{1})](t, x)P̃_{0}(ξ)ζ_{0}|ζ_{0}|^{r_{1}−1}, (24)

b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2}) = [κ(w)](t, x)[λ(v_{1})](t, x)[ϑ(v_{2})](t, x)R(ξ)S(ζ_{0})η_{i}|(η_{0}, η)|^{p_{2}−2}+

+ [κ̃(w)](t, x)[ϑ̃(v_{2})](t, x)R̃(ξ)η_{i}|(η_{0}, η)|^{r_{2}−1}, i = 0, …, n, (25)

where 1 ≤ r_{i} < p_{i} − 1 (i = 1, 2) and the following hold.

E1. a) Operators π: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), φ: L^{p_{1}}(Q_{T}) → L^{∞}(Q_{T}), ψ: X_{2} → L^{∞}(Q_{T}) are bounded, φ and ψ are continuous, further, if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then π(ω_{k}) → π(ω) in L^{∞}(Q_{T}). In addition, P ∈ C(ℝ), Q ∈ C(ℝ^{n+1}) ∩ L^{∞}(ℝ^{n+1}), and there exists a positive lower bound for the values of π, φ, ψ, P, Q.

b) Operators π̃, π̃_{0} : L^{∞}(Q_{T}) → L^{∞}(Q_{T}), φ̃, φ̃_{0} : L^{p_{1}}(Q_{T}) → L^{\frac{p_{1}−1}{p_{1}−r_{1}−1}} (Q_{T}) are bounded, φ̃ and φ̃_{0} are continuous, further, if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then π̃(ω_{k}) → π̃(ω) and π̃_{0}(ω_{k}) → π̃_{0}(ω) in L^{∞}(Q_{T}). In addition, P̃, P̃_{0} ∈ C(ℝ), operators π̃, φ̃ and function P̃ are nonnegative and

lim_{∥v_{1}∥_{X_{1}} →+∞} \frac{∫_{Q_{T}} |φ̃_{0}(v_{1})|^{\frac{p_{1}−1}{p_{1}−r_{1}−1}}}{∥v_{1}∥_{X_{1}}^{p_{1}}} = 0.

E2. a) Operators κ: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), λ: L^{p_{1}}(Q_{T}) → L^{∞}(Q_{T}), ϑ: L^{p_{2}}(Q_{T}) → L^{∞}(Q_{T}) are bounded, λ and ϑ are continuous, further, if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then κ(ω_{k}) → κ(ω) in L^{∞}(Q_{T}). In addition, R ∈ C(ℝ), S ∈ C(ℝ) ∩ L^{∞}(ℝ), and there exists a positive lower bound for the values of κ, λ, ϑ, R, S.

b) Operators κ̃: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), ϑ̃: L^{p_{2}}(Q_{T}) → L^{\frac{p_{2}−1}{p_{2}−r_{2}−1}} (Q_{T}) are bounded, ϑ̃ is continuous, function R̃ ∈ C(ℝ), further, if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then κ̃(ω_{k}) → κ̃(ω) in L^{∞}(Q_{T}). In addition, operators κ̃, ϑ̃ and function R̃ ∈ C(ℝ) are nonnegative and

lim_{∥v_{2}∥_{X_{2}} →+∞} \frac{∫_{Q_{T}} |ϑ̃(v_{2})|^{\frac{p_{2}−1}{p_{2}−r_{2}−1}}}{∥v_{2}∥_{X_{2}}^{p_{2}}} = 0.

Proposition 6. Assume that E1-E2 hold, then functions (23)–(25) fulfil conditions (A1)–(A5), (B1)–(B5).

By using Young’s and Hölder’s inequalities it is not difficult to prove the above statement, a detailed proof can be found in [5].

Operators π, π̃, π̃_{0}, κ, κ̃ may have the form [π(w)](t, x) = ∫_{Q_{t}} |w|^{β}, where 1 ≤ β. Further, operators φ, λ may

have one of the forms

[φ(v)](t, x) = Φ (∫_{Q_{t}} |v|^{β}) or Φ (∫_{Q_{t}} dv) ,

where 1 ≤ β ≤ p_{1}, d ∈ L^{q_{1}}(Q_{T}), Φ ∈ C(ℝ) and Φ ≥ const > 0. Similarly, ψ may have in the form

[ψ(v)](t, x) = Ψ (∫_{Q_{t}} |v|^{β} + |Dv|^{β}) or Ψ (∫_{Q_{t}} d_{1}v + d_{2}|Dv|) ,

where 1 ≤ β ≤ p_{2}, d_{1}, d_{2} ∈ L^{q_{2}}(Q_{T}), Ψ ∈ C(ℝ) and Ψ ≥ const > 0 For φ̃ consider, e.g.,

[φ̃(v)](t, x) = Φ̃ (∫_{0}^{t} d(s, x)v(s, x) ds) , Φ̃ (∫_{Ω} d(t, x)v(t, x) dx) or Φ̃ ([∫_{0}^{t} |v(s, x)|^{β} ds]^{\frac{1}{β}} ) ,

where d ∈ L^{∞}(Q_{T}), 1 ≤ β ≤ p_{1}, Φ̃ ∈ C(ℝ), Φ̃ ≥ 0 and |Φ̃(τ)| ≤ const · |τ|^{p_{1}−r_{1}−1}. In the case of φ̃_{0} one has similar examples as for φ̃ above, except Φ̃ does not have to be nonnegative.

For operators ϑ, ϑ̃ we may consider similar examples as for φ, φ̃ above, by replacing exponents p_{1} with p_{2} and r_{1} with r_{2}.

It is not difficult to show that the above operators fulfil conditions E1–E2, for similar arguments see, e.g., [5].

As an example for function f consider, e.g., f(t, x, ξ, ζ_{0}; v) = −[φ(v)](t, x)f_{1}(t, x)f_{2}(ζ_{0})(ξ − ω^{∗}(x)), where φ: L^{p_{1}}(Q_{T}) → L^{∞}(Q_{T}) is bounded and nonnegative, further, f_{1} ∈ L^{∞}(Q_{T}), f_{2} : ℝ → ℝ is nonnegative, Lipschitz continuous and |f_{2}(ζ_{0})| ≤ const · |ζ_{0}|^{\frac{p_{1}}{q_{2}}} .

4 Solutions in (0, ∞)

In the previous section we have proved existence of solutions for all finite time interval (0, T). In what follows we shall show existence of weak solutions in (0, ∞). Denote by X_{i}^{∞} = L_{loc}^{p_{i}}(0, ∞; V_{i}) the space of measurable functions u: (0, ∞) → V_{i} such that u|_{(0,T)} ∈ L^{p_{i}}(0, T; V_{i}) for every 0 < T < ∞, further, let L_{loc}^{∞}(Q_{∞}) be the space of functions ω: Q_{∞} → ℝ such that ω|_{Q_{T}} ∈ L^{∞}(Q_{T}) for every 0 < T < ∞. In the following we suppose

(Vol) Functions a_{i}: Q_{∞} × ℝ × ℝ^{n+1} × ℝ^{n+1} × L^{∞}(Q_{∞}) × X_{1}^{∞} × X_{2}^{∞} → ℝ, b_{i}: Q_{∞} × ℝ × ℝ × ℝ^{n+1} × L^{∞}(Q_{∞}) × X_{1}^{∞} × X_{2}^{∞} → ℝ (i = 0, …, n), f: Q_{∞} × ℝ^{2} × L_{loc}^{∞}(Q_{∞}) × X_{1}^{∞} → ℝ have the Volterra property, i.e., a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2})|_{Q_{T}}, b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2})|_{Q_{T}}, f(t, x, ξ, ζ_{0}; w)|_{Q_{T}} depend only on (w|_{Q_{T}} , v_{1}|_{Q_{T}} , v_{2}|_{Q_{T}}) for every 0 < T < ∞.

Now we may define the weak form of (5)–(7) in Q_{∞}. For fixed 0 < T < ∞ introduce operators A_{T} : L^{∞}(Q_{T})× L^{p_{1}}(0, T; V_{1}) × L^{p_{2}}(0, T; V_{2}) → L^{q_{1}}(0, T; V_{1}^{∗}), B_{T} : L^{∞}(Q_{T}) × L^{p_{1}}(0, T; V_{1}) × L^{p_{2}}(0, T; V_{2}) → L^{q_{2}}(0, T; V_{2}^{∗}), L_{T} : D(L_{T}) → L_{loc}^{q_{i}}(0, T; V_{i}^{∗}) by formulae (10)–(12). We say that ω ∈ L_{loc}^{∞}(Q_{∞}), u ∈ X_{1}^{∞}, p ∈ X_{2}^{∞} is a solution of (5)–(7) in (0, ∞) if for all 0 < T < ∞ (for the restrictions of the functions to Q_{T})

ω(t, x) = ω_{0}(x) + ∫_{0}^{t} f(s, x, ω(s, x), u(s, x); u) ds (t, x) ∈ Q_{T}  (26)
L_{T} u + A_{T} (ω, u, p) = G_{T}  (27)
B_{T} (ω, u, p) = H_{T}  (28)

where G_{T} = G|_{(0,T)}, H_{T} = H|_{(0,T)} with G ∈ L_{loc}^{q_{1}}(0, ∞; V_{1}^{∗}), H ∈ L_{loc}^{q_{2}}(0, ∞; V_{2}^{∗}). Observe that the Volterra property ensures that if ω, u, p is a solution in (0, T) for some T then these functions are solutions in (0, T̃) for all T̃ < T.

Theorem 7. Suppose that (Vol), (A1)–(A5), (B1)–(B5), (F1)–(F4) hold (in the sense that they are satisfied by the restrictions of functions a_{i}, b_{i}, f to Q_{T} for all 0 < T < ∞). Then there exists ω ∈ L^{∞}(Q_{∞}), u ∈ X_{1}^{∞}, p ∈ X_{2}^{∞} such that ω|_{Q_{T}} , u|_{Q_{T}} , p|_{Q_{T}} is a solution of problem (13)–(15) for all 0 < T < ∞ .

Idea of the proof. One may apply the arguments of the proof of Theorem 1 in [7] word for word. The idea is the following. Due to Theorem 1 we have solutions in (0, T_{k}) where T_{k} → ∞. By showing the boundedness of these solutions and using a diagonal process one may choose weakly convergent sequences of solutions. After taking k → ∞ we obtain a solution in (0, ∞).

Remark 8. Examples (23)–(25) fulfil the conditions of the above theorem if π, π̃, π̃_{0}, κ, κ̃: L_{loc}^{∞}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}),

φ, λ: L_{loc}^{p_{1}}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}), ψ, ϑ: L_{loc}^{p_{2}}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}), φ̃, φ̃_{0} : L_{loc}^{p_{1}}(Q_{∞}) → Lloc^{\frac{p_{1}−1}{p_{1}−r_{1}−1}} (Q_{∞}), ϑ̃: L_{loc}^{p_{2}}(Q_{∞}) →

Lloc^{\frac{p_{2}−1}{p_{2}−r_{2}−1}} (Q_{∞}) are of Volterra type and conditions E1–E2 are satisfied for all finite T > 0. E.g., the operators given after Proposition 6 serve as an example for the above.

4.1 Boundedness

Now we show that under some further assumptions, the solutions, formulated in the previous theorem are bounded in appropriate norms in the time interval (0, ∞). First suppose

A4^{∗}. There exist a constant c_{2} > 0, a continuous function γ: ℝ → ℝ and Volterra operators Γ: L_{loc}^{∞}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}), k_{2} : X_{1}^{∞} → L_{loc}^{1}(Q_{∞}) such that

∑_{i=0}^{n} a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2})ζ_{i} ≥ c_{2} (|ζ_{0}|^{p_{2}} + |ζ|^{p_{2}}) − γ(ξ)[Γ(w)](t, x)[k_{2}(v_{1})](t, x)

for a.a. (t, x) ∈ Q_{∞}, every (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1} and (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{∞}) × X_{1}^{∞} × X_{2}^{∞}. Further,

(8) holds and for all v ∈ X_{1}^{∞} and

∫_{Ω} |[k_{2}(v_{1})](t, x)| dx ≤ α_{1} [ sup_{τ∈[0,t]} ∥v_{1}(τ)∥_{L^{2}(Ω)}^{ρ_{1}} + χ_{1}(t) sup_{τ∈[0,t]} ∥v_{1}(τ)∥_{L^{2}(Ω)}^{p_{1}} + 1]

with some α_{1} > 0, ρ_{1} < p_{1} and χ_{1} : ℝ → ℝ such that lim_{t→∞} χ_{1}(t) = 0.

B4^{∗}. There exist a constant ĉ_{2} > 0, a continuous function γ̂: ℝ → ℝ and Volterra operators Γ̂: L^{∞}(Q_{∞}) → L^{∞}(Q_{∞}), k_{2} : X_{2}^{∞} → L_{loc}^{1}(Q_{∞}) such that

∑_{i=0}^{n} b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2})η_{i} ≥ ĉ_{2} (|η_{0}|^{p_{2}} + |η|^{p_{2}}) − γ̂(ξ)[Γ̂(w)](t, x) (|ζ_{0}|^{2} + [k̂_{2}(v_{2})](t, x))

for a.a. (t, x) ∈ Q_{∞}, every (ξ, ζ_{0}, η_{0}, η) ∈ ℝ × ℝ × ℝ^{n+1} and (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{∞}) × X_{1}^{∞} × X_{2}^{∞}. Further,

(9) holds and for all v ∈ X_{2}^{∞} and

∫_{Ω} |[k̂_{2}(v_{2})](t, x)| dx ≤ α_{2} [ess sup_{τ∈[0,t]} ∥v_{2}(τ)∥_{V_{2}}^{ρ_{2}} + χ_{2}(t) ess sup_{τ∈[0,t]} ∥v_{2}(τ)∥_{V_{2}}^{p_{2}} + 1]

with some α_{2} > 0, ρ_{2} < p_{2} and χ_{2} : ℝ → ℝ such that lim_{t→∞} χ_{2}(t) = 0.

Theorem 9. Suppose that p_{1}, p_{2} > 2 and conditions (Vol), (A1)–A3, (A4^{∗}), (A5), (B1)–(B3), (B4^{∗}), (B5), (F1)–(F4) are satisfied, further, ∥G(·)∥_{V_{1}^{∗}}, ∥H(·)∥_{V_{2}^{∗}} ∈ L^{∞}(0, ∞). Then for the solutions ω, u, p formulated in Theorem 9, ω ∈ L^{∞}(Q_{∞}), u ∈ L^{∞}(0, ∞; L^{2}(Ω)), p ∈ L^{∞}(0, ∞; V_{2}).

Idea of the proof. We may apply the arguments of the proof of Theorem 2 in [7]. Introduce the notation y(t) = ∥u(t, ·)∥_{L^{2}(Ω)}^{2} (then y is continuous see, e.g., [27]). By choosing arbitrary 0 < T_{1} < T_{2} < ∞, (13), conditions (A4^{∗}), Young’s inequality and the continuous embedding V_{1} ↪ L^{2}(Ω) imply

\frac{1}{2} (y(T_{1}) − y(T_{2})) + \frac{1}{2}c_{2} ∫_{T_{1}}^{T_{2}} y(t)^{\frac{p_{1}}{2}} dt ≤ const · ∫_{T_{1}}^{T_{2}} ( sup_{τ∈[0,t]} y(τ)^{\frac{ϱ_{1}}{2}} + χ_{1}(t) sup_{τ∈[0,t]} y(τ)^{\frac{p_{1}}{2}} + 1) dt.

It is not difficult to see that the above inequality implies the boundedness of y in (0, ∞), a detailed argument can be found in [24]. The boundedness of p follows from the boundedness of y similarly as above, by using conditions (B4^{∗}), see [7].

Remark 10. Example (23)–(25) fulfil the conditions of Theorem 9 if assumptions of Remark 8 are satisfied, in addition

∫_{Ω} |[φ̃_{0}(v_{1})](t, x)|^{\frac{p_{1}−1}{p_{1}−r_{1}−1}} dx ≤ α_{1} [ sup_{τ∈[0,t]} ∥v_{1}(τ)∥_{L^{2}(Ω)}^{ρ_{1}} + χ_{1}(t) sup_{τ∈[0,t]} ∥v_{1}(τ)∥_{L^{2}(Ω)}^{p_{1}} + 1]

for all v_{1} ∈ L_{loc}^{p_{1}}(Q_{∞}) with some constants α_{1} > 0, ϱ_{1} < p_{1} and function χ_{1} : ℝ → ℝ such that lim_{t→∞} χ_{1}(t) = 0,

further, similar condition holds for ϑ̃ (by changing the indeces from 1 to 2, and L^{2}(Ω) to V_{2}). For example, operator φ̃_{0} may have the form

[φ̃(v)](t, x) = Φ̃ (∫_{Ω} d(t, x)v(t, x) dx) , Φ̃ (∫_{Ω} |d(t, x)||v(t, x)|^{β} dx) or χ_{1}(t)Φ̃_{0} ([∫_{Ω} |d(t, x)||v(t, x)|^{2} dx]^{\frac{1}{β}} ) ,

where d ∈ L^{∞}(Q_{∞}), 1 ≤ β ≤ 2, Φ̃, Φ̃_{0}, χ_{1} ∈ C(ℝ) and |Φ̃(τ)| ≤ const · |τ|^{p_{1}−ϱ_{1}−1}, |Φ̃_{0}(τ)| ≤ const · |τ|^{p_{1}−r_{1}−1},

lim_{τ→∞} χ_{1}(τ) = 0.

4.2 Stabilization

In this section we consider a special case of problem (26)–(28), namely, let p_{1} = p_{2} = p (thus q_{1} = q_{2} = q, V_{1} = V_{2} = V and X_{1} = X_{2} = X). In what follows, we prove stabilization of the solutions of the system, that is, we show the convergence (in some sense) of solutions as t → ∞ to the solutions of a stationary system. We need some further assumptions:

A6. There exist Carathéodory functions a_{i,∞}: Ω × ℝ × ℝ^{n+1} × ℝ^{n+1} → ℝ (i = 0, …, n) such that for a.a. x ∈ Ω and every (ζ_{0}, ζ, η_{0}, η) ∈ ℝ × ℝ^{n} × ℝ × ℝ^{n}, ξ^{∗} ∈ ℝ, w ∈ L^{∞}(Q_{∞}), v_{1} ∈ X^{∞} ∩ L^{∞}(0, ∞; L^{2}(Ω)), v_{2} ∈ X^{∞} ∩ L^{∞}(0, ∞; V),

lim_{ξ→ξ^{∗}}_{t→∞} a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) = a_{i,∞}(x, ξ^{∗}, ζ_{0}, ζ, η_{0}, η).

B6. There exist Carathéodory functions b_{i,∞}: Ω×ℝ×ℝ×ℝ^{n+1} → ℝ (i = 0, …, n) such that for a.a. x ∈ Ω and every (ζ_{0}, η_{0}, η) ∈ ℝ × ℝ^{n+1}, ξ^{∗} ∈ ℝ, w ∈ L^{∞}(Q_{∞}), v_{1} ∈ X^{∞} ∩ L^{∞}(0, ∞; L^{2}(Ω)), v_{2} ∈ X^{∞} ∩ L^{∞}(0, ∞; V),

lim_{ξ→ξ^{∗}}_{t→∞} b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2}) = b_{i,∞}(x, ξ^{∗}, ζ_{0}, η_{0}, η).

AB There exists a positive constant 𝒞 such that for a.a. (t, x) ∈ Q_{∞} and every ξ ∈ ℝ, (ζ_{0}, ζ, η_{0}, η), (ζ̃_{0}, ζ̃, η̃_{0}, η̃) ∈ ℝ × ℝ^{n} × ℝ × ℝ^{n}, (w, v_{1}, v_{2}) ∈ L^{∞}(Q_{∞}) × X_{1}^{∞} × X_{2}^{∞},

∑_{i=0}^{n} (a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) − a_{i}(t, x, ξ, ζ̃_{0}, ζ̃, η̃_{0}, η̃; w, v_{1}, v_{2})) (ζ_{i} − ζ̃_{i})+

+∑_{i=0}^{n} (b_{i}(t, x, ξ, ζ_{0}, η_{0}, η) − b_{i}(t, x, ξ, ζ̃_{0}, η̃_{0}, η̃)) (η_{i}−η̃_{i}) ≥ 𝒞·(|ζ_{0} − ζ̃_{0}|^{p} + |ζ − ζ̃|^{p} + |η_{0} − η̃_{0}|^{p} + |η − η̃|^{p}) .

F5. For every fixed v ∈ X^{∞} ∩L^{∞}(0, ∞; L^{2}(Ω)) there is a constant m > 0 such that (ξ −ω^{∗}(x))f(t, x, ξ, ζ_{0}; v) ≤ −m(ξ − ω^{∗}(x))^{2} for a.a. (t, x) ∈ Q_{∞} and every (ξ, ζ_{0}) ∈ ℝ^{2}.

Now introduce operators A_{∞}: L^{∞}(Ω) × V × V → V^{∗}, B_{∞}: L^{∞}(Ω) × V × V → V^{∗} by

⟨A_{∞}(ω, u, p), v⟩ := ∫_{Ω} ∑_{i=1}^{n} a_{i,∞}(x, ω(x), u(x), Du(x), p(x), Dp(x))D_{i}v(x) dx+

+ ∫_{Ω} a_{0,∞}(x, ω(x), u(x), Du(x), p(x), Dp(x))v(x) dx,

⟨B_{∞}(ω, u, p), v⟩ := ∫_{Ω} ∑_{i=1}^{n} b_{i,∞}(x, ω(x), u(x), p(x), Dp(x))D_{i}v(x) dx+

+ ∫_{Ω} b_{0,∞}(x, ω(x), u(x), p(x), Dp(x))v(x) dx,

Theorem 11. Assume conditions (A1)–(A3), (A4^{∗}), (A5)–(A6), (B1)–(B3), (B4^{∗}), (B5)–(B6), (AB), (F1)– (F5) are satisfied (with p = p_{1} = p_{2}), further, there exist F_{∞}, G_{∞} ∈ V^{∗} such that

lim_{t→∞} ∥F (t) − F_{∞}∥_{V^{∗}} = 0, lim_{t→∞} ∥G(t) − G_{∞}∥_{V^{∗}} = 0.

Then there exist u_{∞} ∈ V, p_{∞} ∈ V such that for the solutions ω, u, p of problem (26)–(28), ω(t, ·) → ω^{∗} in

L^{∞}(Ω), u(t) → u_{∞} in L^{2}(Ω), ∫_{t−1}^{t+1} ∥u(s) − u_{∞}∥_{V}^{p} ds → 0 ,∫_{t−1}^{t+1} ∥p(s) − p_{∞}∥_{V}^{p} ds → 0, further,

A_{∞}(ω^{∗}, u_{∞}, p_{∞}) = G_{∞}  (29)
B_{∞}(ω^{∗}, u_{∞}, p_{∞}) = H_{∞}.  (30)

Sketch of the proof. We follow the proof of Theorem 3 in [7]. Let ω, u, p be solutions of (26)–(28) then by Theorem 9, ω ∈ L^{∞}(Q_{∞}), u ∈ L^{∞}(0, ∞; L^{2}(Ω)), p ∈ L^{∞}(0, ∞; V_{2}). By using the same arguments as in the above mentioned paper, conditions (F3), (F5) imply estimate ∥ω(t, ·) − ω^{∗}(·)∥_{L^{∞}(Ω)} ≤ ∥ω_{0}∥_{L^{∞}(Ω)}e^{−mt} which yields the convergence ω(t, ·) → ω^{∗}(·) in L^{∞}(Ω).

Now by the using the idea of Proposition 5 and condition AB it is easy to see that for fixed ω^{∗} there exist a unique solution u_{∞} ∈ V, p_{∞} ∈ V of problem (29)–(30) see, e.g., [27].

In order to show the desired convergences we prove a differential inequality for u and p. From equations (26)–(28) and (29)–(30) we obtain

⟨D_{t}(u(t) − u_{∞}), u(t) − u_{∞}⟩ + ⟨[A(ω, u, p)](t) − A_{∞}(ω^{∗}, u_{∞}, p_{∞}), u(t) − u_{∞}⟩+ + ⟨[B(ω, u, p)](t) − B_{∞}(ω^{∗}, u_{∞}, p_{∞}), p(t) − p_{∞}⟩ = ⟨G(t) − G_{∞}, u(t) − u_{∞}⟩ + ⟨F (t) − F_{∞}, p(t) − p_{∞}⟩. (31)

Observe that the first term equals to \frac{1}{2}y′(t) where y(t) = ∫_{Ω}(u(t) − u_{∞})^{2}. Further, for the second and third

terms of the above equation we have by condition AB and Young’s inequality

⟨[A(ω, u, p)](t) − A_{∞}(ω^{∗}, u_{∞}, p_{∞}), u(t) − u_{∞}⟩ + ⟨[B(ω, u, p)](t) − B_{∞}(ω^{∗}, u_{∞}, p_{∞}), p(t) − p_{∞}⟩ ≥

≥ 𝒞 · (∥u(t) − u_{∞}∥_{V}^{p} + ∥p(t) − p_{∞}∥_{V}^{p}) − \frac{ε^{p}}{p} ∥u(t) − u_{∞}∥_{V}^{p} − \frac{ε^{p}}{p} ∥p(t) − p_{∞}∥_{V}^{p} −

− \frac{1}{qε^{q}} ∥[Ã(ω, u_{∞}, p_{∞}; ω, u, p)](t) − A_{∞}(ω^{∗}, u_{∞}, p_{∞})∥_{V^{∗}}^{q}−

− \frac{1}{qε^{q}} ∥[B̃(ω, u_{∞}, p_{∞}; ω, u, p)](t) − B(ω^{∗}, u_{∞}, p_{∞})∥_{V^{∗}}^{q} (32)

with some ε > 0 (and operator Ã is defined similarly as B̃, see (19)). We show that last two terms on the right hand side of the above inequality converges to 0 as t → ∞. Clearly,

∥[Ã(ω, u_{∞}, p_{∞}; ω, u, p)](t) − A_{∞}(ω^{∗}, u_{∞}, p_{∞})∥_{V^{∗}}^{q} ≤

≤ ∑_{i=0}^{n} ∫_{Ω} |a_{i}(t, ·, ω(t, ·), u_{∞}, Du_{∞}, p_{∞}, Dp_{∞}; ω, u, p) − a_{i,∞}(ω^{∗}, u_{∞}, Du_{∞}, p_{∞}, Dp_{∞})|^{q}.

The integrand on the right hand side is a.e. convergent in Ω as t → ∞ by condition A6 and since ω(t, x) → ω^{∗}(x) for a.a. x ∈ Ω. Further, it is integrable in Ω by conditions (A2), (A6) and estimate

|a_{i}(t, ·, ω(t, ·), u_{∞}, Du_{∞}, p_{∞}, Dp_{∞}) − a_{i,∞}(ω^{∗}, u_{∞}, Du_{∞}, p_{∞}, Dp_{∞})|^{q} ≤

≤ const · (∥c_{1}(ω)∥_{L^{∞}(Q_{∞})} + ∥c_{1}(ω^{∗})∥_{L^{∞}(Q_{∞})}) (|u_{∞}|^{p} + |Du_{∞}|^{p} + |p_{∞}|^{p} + |Dp_{∞}|^{p} + ∥k_{1}∥_{L^{q}(Ω)})

thus by Lebesgue’s theorem we obtain ∥[Ã(ω, u_{∞}, p_{∞}; ω, u, p)](t) − A_{∞}(ω, u_{∞}, p_{∞})∥_{V^{∗}}^{q} → 0 as t → ∞. The convergence of the last term in (32) can be proved similarly.

On the right hand side of (31) by Young’s inequality we obtain

|⟨G(t) − G_{∞}, u(t) − u_{∞}⟩ + ⟨F (t) − F_{∞}, p(t) − p_{∞}⟩| ≤

\frac{ε^{p}}{p} ∥u(t) − u_{∞}∥_{V}^{p} + \frac{ε^{p}}{p} ∥p(t) − p_{∞}∥_{V}^{p} + \frac{1}{qε^{q}} ∥G(t) − G_{∞}∥_{V^{∗}}^{q} + \frac{1}{qε^{q}} ∥F (t) − F_{∞}∥_{V^{∗}}^{q} (33)

where the last two terms tend to 0 as t → ∞.

Now, by choosing sufficiently small ε in (31) and by using (32), (33), the above convergences and the continuous embedding L^{p}(Ω) ↪ L^{2}(Ω) we obtain

y′(t) + const · y(t)^{\frac{p}{2}} + const · ∥p(t) − p_{∞}∥_{V}^{p} ≤ φ(t)

where φ(t) → 0 as t → ∞ and the constants are positive. It is not difficult to show that this inequality implies lim_{t→∞} y(t) = 0 (see the proof of Theorem 2 in [24]), furthermore, by integrating (31) over (t − 1, t + 1) one can

deduce the convergences ∫_{t−1}^{t+1} ∥u(s)−u_{∞}∥_{V}^{p} ds → 0, ∫_{t−1}^{t+1} ∥p(s)−p_{∞}∥_{V}^{p} ds → 0, too. The proof of stabilization

is complete.

Consider the following functions for i = 0, …, n

a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) = [π(w)](t, x)[φ(v_{1})](t, x)[ψ(v_{2})](t, x)P (ξ)ζ_{i}|(ζ_{0}, ζ, η_{0}, η)|^{p−2},  (34)
b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2}) = [κ(w)](t, x)[λ(v_{1})](t, x)[ϑ(v_{2})](t, x)R(ξ)η_{i}|(ζ_{0}, η_{0}, η)|^{p−2}.  (35)

Suppose E3. a) Operators π: L_{loc}^{∞}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}), φ, ψ: L_{loc}^{p}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}) are of Volterra type, further, for

every 0 < T < ∞, π: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), φ, ψ: L^{p}(Q_{T}) → L^{∞}(Q_{T}) are bounded, φ and ψ are continuous, and if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then π(ω_{k}) → π(ω) in L^{∞}(Q_{T}). In addition, P ∈ C(ℝ), and there exists a positive lower bound for the values of π, φ, ψ, P.

b) There exist π_{∞}, φ_{∞}, ψ_{∞} ∈ L^{∞}(Ω) such that for every w ∈ L^{∞}(Q_{∞}), v_{1} ∈ X^{∞}∩L^{∞}(0, ∞; L^{2}(Ω)), v_{2} ∈ X^{∞} ∩ L^{∞}(0, ∞; V),

lim_{t→∞} (∥[π(w)](t, ·) − π_{∞}∥_{L^{∞}(Ω)} + ∥[φ(v_{1})](t, ·) − φ_{∞}∥_{L^{∞}(Ω)} + ∥[ψ(v_{2})](t, ·) − ψ_{∞}∥_{L^{∞}(Ω)}) = 0.

E4. a) Operators κ: L_{loc}^{∞}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}), λ, ϑ: L_{loc}^{p}(Q_{∞}) → L_{loc}^{∞}(Q_{∞}) are of Volterra type, further, for every 0 < T < ∞, κ: L^{∞}(Q_{T}) → L^{∞}(Q_{T}), λ, ϑ: L^{p}(Q_{T}) → L^{∞}(Q_{T}) are bounded, λ and ϑ are continuous, and if (ω_{k}) is bounded in L^{∞}(Q_{T}) and ω_{k} → ω a.e. in Q_{T} then κ(ω_{k}) → κ(ω) in L^{∞}(Q_{T}). In addition, R ∈ C(ℝ), and there exists a positive lower bound for the values of π, φ, ψ, R.

b) There exist κ_{∞}, λ_{∞}, ϑ_{∞} ∈ L^{∞}(Ω) such that for every w ∈ L^{∞}(Q_{∞}), v_{1} ∈ X^{∞}∩L^{∞}(0, ∞; L^{2}(Ω)), v_{2} ∈ X^{∞} ∩ L^{∞}(0, ∞; V)

lim_{t→∞} (∥[κ(w)](t, ·) − κ_{∞}∥_{L^{∞}(Ω)} + ∥[λ(v_{1})](t, ·) − λ_{∞}∥_{L^{∞}(Ω)} + ∥[ϑ(v_{2})](t, ·) − ϑ_{∞}∥_{L^{∞}(Ω)}) = 0.

It is not difficult to prove (for some arguments see, e.g., [9, 5, 22])

Proposition 12. Suppose 2 ≤ p ≤ 4 and E3–E4, then the above (34)–(35) functions satisfy conditions (A1)– (A3), (A4^{∗}), (A5)–(A6), (B1)–(B3), (B4^{∗}), (B5)–(B6), (AB) with p_{1} = p_{2} = p.

If we consider

a_{i}(t, x, ξ, ζ_{0}, ζ, η_{0}, η; w, v_{1}, v_{2}) = ζ_{i}|(ζ_{0}, ζ)|^{p−2} + [π(w)](t, x)[ϕ(v_{1})](t, x)P (ξ)ζ_{i}|(ζ_{0}, ζ, η_{0}, η)|^{r−2},
b_{i}(t, x, ξ, ζ_{0}, η_{0}, η; w, v_{1}, v_{2}) = ζ_{i}|(η_{0}, η)|^{p−2} + [κ(w)](t, x)(t, x)[ϑ(v_{2})](t, x)R(ξ)η_{i}|(ζ_{0}, η_{0}, η)|^{r−2}

where 1 ≤ r ≤ 4 and E3–E4 hold then it is easy to see that these functions satisfy conditions (A1)–(A3), (A4^{∗}), (A5)–(A6), (B1)–(B6), (AB) with p_{1} = p_{2} = p ≥ max{2, r}. E.g. operators π, φ may have the form

[π(w)](t, x) = χ(t) ∫_{Q_{t}} |w|^{β} + π_{∞}(x), [φ(v)](t, x) = χ̃(t) ∫_{Ω} |d(t, x)||v(t, x)|^{β}dx + φ_{∞}(x),

where lim_{t→∞} χ(t) = 0, lim_{t→∞} χ̃(t) = 0 and d, π_{∞}, φ_{∞} ∈ L^{∞}(Ω), 1 ≤ α, 1 ≤ β ≤ 2. The other operators may have similar form.

As an example for function f consider, e.g., f(t, x, ξ, ζ_{0}; v) = −(ξ − ω^{∗}(x)) ∫_{Ω} |v(t, x)|^{β} dx where 1 ≤ β ≤ 2.

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Ádám Besenyei Department of Applied Analysis Eötvös Loránd University Pázmány Péter sétány 1/C H-1117 Budapest Hungary

email: badam@cs.elte.hu

(Received August 28, 2007)