Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 20 1-13; http://www.math.u-szeged.hu/ejqtde/

On qualitative properties of a system containing a singular parabolic functional equation

László Simon ^{∗}

L. Eötvös University of Budapest, Institute of Mathematics

Abstract

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation containing functional dependence on the unknown functions. The existence and some properties of solutions in (0, ∞) will be proved.

Introduction

In this work we consider initial-boundary value problems for the system

D_{t}u − ∑_{i=1}^{n} D_{i}[a_{i}(t, x, u(t, x), Du(t, x); u, w)]+  (0.1)

a_{0}(t, x, u(t, x), Du(t, x); u, w) = f(t, x),
D_{t}w = F (t, x; u, w) in Q_{T} = (0, T) × Ω, T ∈ (0, ∞)  (0.2)

where the functions

a_{i} : Q_{T} × ℝ^{n+1} × L^{p}(0, T; V_{1}) × L^{2}(Q_{T}) → ℝ

(with a closed linear subspace V_{1} of the Sobolev space W^{1,p}(Ω), 2 ≤ p < ∞) satisfy conditions which are generalizations of the usual conditions for quasilinear parabolic differential equations, considered by using the theory of monotone type operators (see, e.g., [2], [7], [13]) but the equation (0.1) is not uniformly parabolic in a sense, analogous to the linear case. Further,

F : Q_{T} × L^{p}(0, T; V_{1}) × L^{2}(Q_{T}) → ℝ

satisfies a Lipschitz condition.

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

In [12] the existence of weak solutions in Q_{T} was proved. In this present paper this result will be extended to Q_{∞} = (0, ∞) × Ω and some properties (boundedness, asymptotic property as t → ∞) of the solutions will be shown.

Such problems arise, e.g., when considering diffusion and transport in porous media with variable porosity, see [5], [8]. In [8] J.D. Logan, M.R. Petersen, T.S. Shores considered and numerically studied a nonlinear system, consisting of a parabolic, an elliptic and an ODE which describes reaction-mineralogy-porosity changes in porous media. System (0.1), (0.2) was motivated by that system. In [3], [4] Á. Besenyei considered a more general system of a parabolic PDE, an elliptic PDE and an ODE.

1 Existence of solutions

Let Ω ⊂ ℝ^{n} be a bounded domain having the uniform C^{1} regularity property (see [1]) and p ≥ 2 be a real number. Denote by W^{1,p}(Ω) the usual Sobolev space of real valued functions with the norm

∥ u ∥= [∫_{Ω}(|Du|^{p} + |u|^{p})]^{1/p} .

Let V_{1} ⊂ W^{1,p}(Ω) be a closed linear subspace containing W_{0}^{1,p}(Ω) (the closure of C_{0}^{∞}(Ω) in W^{1,p}(Ω)). Denote by L^{p}(0, T; V_{1}) the Banach space of the set of measurable functions u : (0, T) → V_{1} such that ∥ u ∥_{V_{1}}^{p} is integrable and define the norm by

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

The dual space of L^{p}(0, T; V_{1}) is L^{q}(0, T; V_{1}^{⋆}) where 1/p + 1/q = 1 and V_{1}^{⋆} is the dual space of V_{1} (see, e.g., [7], [13]).

On functions a_{i} we assume:

(A_{1}). The functions a_{i} : Q_{T} × ℝ^{n+1} × L^{p}(0, T; V_{1}) × L^{2}(Ω) → ℝ satisfy the Carathéodory conditions for arbitrary fixed (u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Ω) (i = 0, 1, ..., n).

(A_{2}). There exist bounded (nonlinear) operators g_{1} : L^{p}(0, T; V_{1})×L^{2}(Ω) → ℝ^{+} and k_{1} : L^{p}(0, T; V_{1}) × L^{2}(Ω) → L^{q}(Ω) such that

|a_{i}(t, x, ζ_{0}, ζ; u, w)| ≤ g_{1}(u, w)[|ζ_{0}|^{p−1} + |ζ|^{p−1}] + [k_{1}(u, w)](x), i = 0, 1, ..., n

for a.e. (t, x) ∈ Q_{T}, each (ζ_{0}, ζ) ∈ ℝ^{n+1} and (u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Ω). (A_{3}). ∑_{i=1}^{n}[a_{i}(t, x, ζ_{0}, ζ; u, w) − a_{i}(t, x, ζ_{0}, ζ^{⋆}; u, w)](ζ_{i} − ζ_{i}^{⋆}) > 0 if ζ ≠ ζ^{⋆}. (A_{4}). There exist bounded operators g_{2} : L^{p}(0, T; V_{1}) × L^{2}(Ω) → C[0, T],

k_{2} : L^{p}(0, T; V_{1}) × L^{2}(Ω) → L^{1}(Q_{T}) such that

∑_{i=0}^{n} a_{i}(t, x, ζ_{0}, ζ; u, w)ζ_{i} ≥ [g_{2}(u, w)](t)[|ζ_{0}|^{p} + |ζ|^{p}] − [k_{2}(u, w)](t, x)

for a.e. (t, x) ∈ Q_{T}, all (ζ_{0}, ζ) ∈ ℝ^{n+1}, (u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Ω) and (with some positive constants)

[g_{2}(u, w)]t) ≥ const(1+ ∥ u ∥_{L^{p}(0,t;V_{1})})^{−σ^{⋆}}(1+ ∥ w ∥_{L^{2}(Q_{t})})^{−β^{⋆}}  (1.3)
∥ k_{2}(u, w) ∥_{L^{1}(Q_{t})}≤ const(1+ ∥ u ∥_{L^{p}(0,t;V_{1})})^{σ}(1+ ∥ w ∥_{L^{2}(Q_{t})})^{β}  (1.4)

where

0 < σ^{⋆} < p − 1, 0 ≤ σ < p − σ^{⋆}

and β, β^{⋆} ≥ 0 satisfy

β^{⋆} + σ^{⋆} < p − 1, β^{⋆} + σ^{⋆} + β + σ < p.

(A_{5}). There exists δ > 0 such that if (u_{k}) → u weakly in L^{p}(0, T; V_{1}), strongly in L^{p}(0, T; W^{1−δ,p}(Ω)) and (w_{k}) → w in L^{2}(Ω) then for i = 0, 1, ..., n

a_{i}(t, x, u_{k}(t, x), Du_{k}(t, x); u_{k}, w_{k}) − a_{i}(t, x, u_{k}(t, x), Du_{k}(t, x); u, w) → 0

in L^{q}(Q_{T}).

Definition Assuming (A_{1})-(A_{5}) we define operator A : L^{p}(0, T; V_{1})×L^{2}(Q_{T}) → L^{q}(0, T; V_{1}^{⋆}) by

[A(u, w), v] = ∫_{0}^{T} ⟨A(u, w)(t), v(t)⟩dt =

∫_{Q_{T}} {∑_{i=1}^{n} a_{i}(t, x, u(t, x), Du(t, x); u, w)D_{i}v + a_{0}(t, x, u(t, x), Du(t, x); u, w)v} dtdx,

(u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Q_{T}), v ∈ L^{p}(0, T; V_{1})

where the brackets ⟨·, ·⟩, [·, ·] mean the dualities in spaces V_{1}^{⋆}, V_{1} and L^{q}(0, T; V_{1}^{⋆}), L^{p}(0, T; V_{1}), respectively.

On function F : Q_{T} × L^{p}(0, T; V_{1}) × L^{2}(Q_{T}) → ℝ assume

(F_{1}). For each fixed (u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Q_{T}), F (·; u, w) ∈ L^{2}(Q_{T}).

(F_{2}). F satisfies the following (global) Lipschitz condition: for each t ∈ (0, T], (u, w), (u^{⋆}, w^{⋆}) ∈ X we have

∥ F (·; u, w) − F (·; u^{⋆}, w^{⋆}) ∥_{L^{2}(Q_{t})}^{2}≤
K [∥ u − u^{⋆} ∥_{L^{p}(0,t;W^{1−δ,p}(Ω))}^{2} + ∥ w − w^{⋆} ∥_{L^{2}(Q_{t})}^{2}] .

In [12] the following theorem was proved.

Theorem 1.1 Assume (A_{1}) - (A_{5}) and (F_{1}), (F_{2}). Then for any f ∈ L^{q}(0, T; V_{1}^{⋆}) and w_{0} ∈ L^{2}(Q_{T}) there exists u ∈ L^{p}(0, T; V_{1}), w ∈ L^{2}(Q_{T}) such that D_{t}u ∈ L^{q}(0, T; V_{1}^{⋆}), D_{t}w ∈ L^{2}(Q_{T}) and

D_{t}u + A(u, w) = f, u(0) = 0, .  (1.5)
D_{t}w = F (t, x; u, w) for a.e. (t, x) ∈ Q_{T} , w(0) = w_{0}.  (1.6)

Now assume

(F′_{1}) F has the form F (t, x; u, w) = F̂(t, x, w(t, x); u, w) and F̂ : Q_{T} × ℝ × X → R satisfies: for each fixed (u, w) ∈ L^{p}(0, T; V_{1}) × L^{2}(Q_{T}), ξ ∈ ℝ, F̂(·, ξ; u, w) ∈ L^{2}(Q_{T}).

(F′_{2}) There exist constants K, K_{1}(c_{1}) such that if |ξ|, |ξ^{⋆}| ≤ c_{1} then for each t ∈ (0, T], (u, w), (u^{⋆}, w^{⋆}) ∈ L^{p}(0, T; V_{1}) × L^{2}(Q_{T} with the property |w|, |w^{⋆}| ≤ c_{1} in Q_{T}

|F̂(t, x, ξ; u, w) − F̂(t, x, ξ^{⋆}; u^{⋆}, w^{⋆})|^{2} ≤

K ∥ u − u^{⋆} ∥_{L^{p}(0,t;W^{1−δ,p}(Ω))}^{2} +K_{1}(c_{1}) [∥ w − w^{⋆} ∥_{L^{2}(Q_{t})}^{2} +|ξ − ξ^{⋆}|^{2}] .

(F′_{3}) There exists a constant c_{0} > 0 such that for a.e. (t, x) and all u, w

F̂(t, x, ξ; u, w)ξ ≤ 0 if |ξ| ≥ c_{0}.

Theorem 1.2 Assume (A_{1}) - (A_{5}) and (F′_{1}) - (F′_{3}) such that operators g_{2}, k_{2} in (A_{4}) satisfy the following modified inequalities instead of (1.3) and (1.4):

[g_{2}(u, w)](t) ≥ const(1+ ∥ u ∥_{L^{p}(0,t;V_{1})})^{−σ^{⋆}}(1 + g_{3}(∥ w ∥_{L^{2}(Q_{t})})^{−1},
∥ k_{2}(u, w) ∥_{L^{1}(Q_{t})}≤ const(1+ ∥ u ∥_{L^{p}(0,t;V_{1})})^{σ}(1+ ∥ w ∥_{L^{2}(Q_{t})})

where g_{3}, g_{4} are monotone nondecreasing positive functions, 0 < σ^{⋆} < p − 1, 0 ≤ σ < p − σ^{⋆}.

Then for any f ∈ L^{q}(0, T; V_{1}^{⋆}) and w_{0} ∈ L^{2}(Q_{T}) there exists u ∈ L^{p}(0, T; V_{1}), w ∈ L^{2}(Q_{T}) such that D_{t}u ∈ L^{q}(0, T; V_{1}^{⋆}), D_{t}w ∈ L^{2}(Q_{T}) and (1.5), (1.6) hold.

This theorem is a consequence of Theorem 1.1 (see also [12]): set

c_{0}^{⋆} = max{c_{0}, ∥ w_{0} ∥_{L^{∞}(Ω)}}

and let χ ∈ C_{0}^{∞}(ℝ) be such that χ(ξ) = ξ for |ξ| ≤ c_{0}^{⋆} and define functions F̃, ã_{i} by

F̃(t, x; u, w) = F̂(t, x, χ(w(t, x)); u, χ(w)),
ã_{i}(t, x, ζ_{0}, ζ; u, w) = a_{i}(t, x, ζ_{0}, ζ; u, χ(w)),

Then by Theorem 1.1 there exists a solution (u, w) of (1.5), (1.6) with F̃, ã_{i} (instead of F, a_{i}, respectively). It is not difficult to show that for a.e. x ∈ Ω, all t ∈ [0, T], |w(t, x)| ≤ c_{0}^{⋆} by (F′_{3}) and so (u, w) satisfies the original problem, too.

Now we formulate existence theorems in (0, ∞). Denote by L_{loc}^{p}(0, ∞; V_{1}) the set of functions v : (0, ∞) → V_{1} such that for each fixed finite T > 0, v|_{(0,T)} ∈ L^{p}(0, T; V_{1}) and let Q_{∞} = (0, ∞) × Ω, L_{loc}^{α}(Q_{∞}) the set of functions v : Q_{∞} → R such that v|_{Q_{T}} ∈ L^{α}(Q_{T}) for any finite T > 0.

Theorem 1.3 Assume that the functions

a_{i} : Q_{∞} × ℝ^{n+1} × L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞} → ℝ

satisfy the assumptions (A_{1}) - (A_{5}) for any finite T and that a_{i}(t, x, ζ_{0}, ζ; u, w)|_{Q_{T}} depend only on u |_{(0,T)} and w |_{Q_{T}} (Volterra property). Further, the function

F : Q_{∞} × L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞} → ℝ

satisfies (F_{1}), (F_{2}) for arbitrary fixed T and has the Volterra property.

Then for each f ∈ L_{loc}^{q}(0, ∞; V_{1}^{⋆}), w_{0} ∈ L^{2}(Ω) there exist u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{2}(Q_{∞}) which satisfy (1.5), (1.6) for any finite T.

The idea of the proof. The Volterra property implies that if u, w are solutions in Q_{T} then for arbitrary T̃ < T, their restriction to QT̃ are solutions in QT̃. Therefore, if lim(T_{j}) = +∞, T_{1} < T_{2} < ... < T_{j} < ... and u_{j}, w_{j} are solutions in Q_{T_{j}} then, by using a ’diagonal process’, we can select a subsequence of (u_{j}, w_{j}) which converges in Q_{T} for arbitrary finite T to (u, w), a solution of (1.5), (1.6) in Q_{∞}. (For more details see, e.g., [10].) Similarly can be proved

Theorem 1.4 Assume that the functions

a_{i} : Q_{∞} × ℝ^{n+1} × L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞} → ℝ

satisfy the assumptions of Theorem 1.2 for any finite T and they have the Volterra property; the function

F̂ : Q_{∞} × ℝ × L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞} → ℝ

satisfies (F′_{1}) - (F′_{3}) for arbitrary fixed T and has the Volterra property.

Then for each f ∈ L_{loc}^{q}(0, ∞; V_{1}^{⋆}), w_{0} ∈ L^{2}(Ω) there exist u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{2}(Q_{∞}) which satisfy (1.5), (1.6) for any finite T.

2 Boundedness and stabilization

Theorem 2.1 Assume that the functions a_{i}, F̂ satisfy the conditions of Theorem 1.4 such that for all u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{∞}(Q_{∞}), t ∈ (0, ∞) the operators g_{2}, k_{2} in (A_{4}) satisfy

[g_{2}(u, w)](t) ≥ const [1 + sup_{τ∈[0,T]} y(τ)]^{−σ^{⋆}/2} · [1 + g_{3}( sup_{τ∈[0,T]} z(τ))]^{−1} (2.7)

∫_{Ω}[k_{2}(u, w)](t, x)dx ≤  (2.8)

const [1 + sup_{τ∈[0,T]} y(τ)^{σ/2} + φ(t) sup_{τ∈[0,T]} y(τ)^{(p−σ^{⋆})/2}] · [1 + g_{4}( sup_{τ∈[0,T]} z(τ))]

where 0 < σ^{⋆} < p − 1, 0 < σ < p − σ^{⋆}, lim_{∞} φ = 0, g_{3}, g_{4} are monotone nondecreasing positive functions,

y(τ) = ∫_{Ω} u(τ, x)^{2}dx, z(τ) =∥ w(τ, ·) ∥_{L^{∞}(Ω)} .

Further, the constant c_{0} in (F′_{3}) is independent of T, ∥ f(t) ∥_{V_{1}^{⋆}} is bounded for t ∈ (0, ∞).

Then for a solution u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{∞}(Q_{∞}) of (1.5), (1.6) with w_{0} ∈ L^{∞}(Ω) and arbitrary initial condition on u, y and z are bounded in (0, ∞).

Proof Since the constant c_{0} in (F′_{3}) is independent of T, it is easy to show that

|w(t, x)| ≤ max {c_{0}, ∥ w_{0} ∥_{L^{∞}(Ω)}}

for a.e. x ∈ Ω, all t > 0 (see the idea of the proof of Theorem 1.2, i.e. z is bounded.

Further, applying (1.5) to u(t) ∈ V_{1}, by (A_{4}) we obtain

\frac{1}{2}y′(t) + [g_{2}(u, w)](t) ∥ u(t) ∥_{V_{1}}^{p} − ∫_{Ω}[k_{2}(u, w)](t, x)dx ≤  (2.9)
⟨f(t), u(t)⟩ ≤∥ f(t) ∥_{V_{1}^{⋆}}∥ u(t) ∥_{V_{1}}≤ const ∥ u(t) ∥_{V_{1}}

since ∥ f(t) ∥_{V_{1}^{⋆}} is bounded. Young’s inequality implies

∥ u(t) ∥_{V_{1}}≤ ε[g_{2}(u, w)](t)^{1/p} ∥ u(t) ∥_{V_{1}} ·\frac{1}{ε[g_{2}(u, w)](t)^{1/p}} ≤  (2.10)

\frac{ε^{p}}{p} [g_{2}(u, w)](t) ∥ u(t) ∥_{V_{1}}^{p} +\frac{1}{qε^{q}[g_{2}(u, w)](t)^{q/p}} .

Choosing sufficiently small ε < 0, one obtains from (2.9), (2.10)

\frac{1}{2}y′(t) + \frac{1}{2}[g_{2}(u, w)](t) ∥ u(t) ∥_{V_{1}}^{p}≤ ∫_{Ω}[k_{2}(u, w)](t, x)dx + const[g_{2}(u, w)](t)^{−q/p}

(2.11)

Since by Hölder’s inequality , p ≥ 2,

∥ u(t) ∥_{V_{1}}^{p}≥ consty(t)^{p/2},

(2.7), (2.8), (2.11) and the boundedness of z imply (with some positive constant c^{⋆})

y′(t) + c^{⋆}y(t)^{p/2} [1 + sup_{τ∈[0,T]} y(τ)]^{−σ^{⋆}/2} ≤  (2.12)

const [1 + sup_{τ∈[0,T]} y(τ)^{σ/2} + φ(t) sup_{τ∈[0,T]} y(τ)^{(p−σ^{⋆})/2} + sup_{τ∈[0,T]} y(τ)^{(q/p)(σ^{⋆}/2)}] .

Since 0 ≤ σ < p − σ^{⋆} < p, (q/p)σ^{⋆} < p − σ^{⋆}, lim_{∞} φ = 0, it is not difficult to show that (2.12) implies the boundedness of y(t) (see [11]).

Now we formulate an attractivity result.

Theorem 2.2 Assume that the functions a_{i}, F̂ satisfy the conditions of Theorem 2.1 such that for all u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{∞}(Q_{∞}), t ∈ (0, ∞)

∫_{Ω}[k_{2}(u, w)](t, x)dx ≤ φ(t) [ sup_{τ∈[0,T]} y(τ)^{(p−σ^{⋆})/2}] · [1 + g_{4}( sup_{τ∈[0,T]} z(τ))] .

(2.13)

Further,

lim_{t→∞} ∥ f(t) ∥_{V_{1}^{⋆}}= 0,  (2.14)
ξF̂(t, x, ξ; u, w) ≤ −g(ξ)ξ  (2.15)

with a strictly monotone increasing continuous function g satisfying g(0) = 0.

Then for a solution u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{∞}(Q_{∞}) of (1.5), (1.6) with w_{0} ∈ L^{∞}(Ω) and arbitrary initial condition on u, for the functions defined in Theorem 2.1 we have

lim_{∞} y = 0,  (2.16)
lim_{∞} z = 0.  (2.17)

Proof By (1.6) and (2.15) for a.e. x ∈ Ω, t ↦ w(t, x) is continuous and monotone decreasing and for a.e. (t, x)

D_{t}w(t, x) ≤ −g(w(t, x)) if w(t, x) > 0

thus for a.e. x ∈ Ω satisfying w_{0}(x) > 0,

w(t, x) ≤ w_{0}(x) − tg(w(t, x)) for a.e. x ∈ Ω until w(t, x) > 0

(g is monotone increasing, t ↦ w(t, x) is monotone decreasing). Consequently,

tg(w(t, x)) ≤ w_{0}(x), thus w(t, x) ≤ g^{−1} (\frac{∥ w_{0} ∥_{L^{∞}(Ω)}}{t} )

for a.e. x ∈ Ω with w_{0}(x) > 0 until w(t, x) > 0. In the case w_{0}(x) < 0 we obtain

w(t, x) ≥ −g^{−1} (\frac{∥ w_{0} ∥_{L^{∞}(Ω)}}{t} )

for a.e. x ∈ Ω until w(t, x) < 0. If for some t_{1}, w(t_{1}, x) = 0 then w(t, x) = 0 for t > t_{1}. Hence we obtain (2.17).

In order to prove (2.16), we use (2.13) and so we obtain (similarly to (2.12))

y′(t) + c^{⋆}y(t)^{p/2} [1 + sup_{τ∈[0,T]} y(τ)]^{−σ^{⋆}/2} ≤  (2.18)

const φ(t) [1 + sup_{τ∈[0,T]} y(τ)^{(p−σ^{⋆})/2}] + const ∥ f(t) ∥_{V_{1}^{⋆}} sup_{τ∈[0,T]} y(τ)^{(q/p)(σ^{⋆}/2)}.

Since y is bounded and lim φ_{∞} = 0, by using (2.14) one can derive from (2.18) the equality (2.16) (see, e.g., [9]).

Remark In the case g(ξ) = −α_{1}ξ (where α_{1} is a positive constant)

|w(t, x)| ≤ |w_{0}(x)| exp(−α_{1}t) for a.e. (t, x)

and the inequality

−α_{2}ξ^{2} ≤ ξF̂(t, x, ξ; u, w) ≤ 0 (α_{2} > 0)

implies

|w(t, x)| ≥ |w_{0}(x)| exp(−α_{2}t) for a.e. (t, x).

Now we formulate a stabilization result.

Theorem 2.3 Assume that conditions of Theorem 2.1 are satisfied such that (A_{2}), (A_{4}) hold for all T > 0 with operators

g_{1}, g_{2} : L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞}) → ℝ^{+},  (2.19)
k_{1} : L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞}) → L^{q}(Ω);  (2.20)

for arbitrary fixed u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{2}(Q_{∞}) such that

∫_{Ω} u(t, x)^{2}dx, ∥ w(t, ·) ∥_{L^{∞}}, t ∈ (0, ∞) are bounded  (2.21)

and for every (ζ_{0}, ζ) ∈ ℝ, a.a. x ∈ Ω

lim_{t→∞} a_{i}(t, x, ζ_{0}, ζ; u, w) = a_{i,∞}(x, ζ_{0}, ζ), i = 0, 1, ..., n  (2.22)

exist and are finite where a_{i,∞} satisfy the Carathéodory conditions.

Further, for every fixed u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{2}(Q_{∞})

∑_{i=0}^{n}[a_{i}(t, x, ζ_{0}, ζ; u, w) − a_{i}(t, x, ζ_{0}^{⋆}, ζ^{⋆}; u, w)](ζ_{i} − ζ_{i}^{⋆}) ≥  (2.23)

[g_{2}(u, w)](t)[|ζ_{0} − ζ_{0}^{⋆}|^{p} + |ζ − ζ^{⋆}|^{p}] − [k_{3}(u, w)](t, x)

with some operator

k_{3} : L_{loc}^{p}(0, ∞; V_{1}) × L_{loc}^{2}(Q_{∞}) → L^{1}(Q_{∞})  (2.24)

satisfying

lim_{t→∞} ∫_{Ω}[k_{3}(u, w)](t, x)dx = 0  (2.25)

for all fixed u, w satisfying (2.21).

On F̂ assume

F̂(t, x, ξ; u, w)[ξ − w_{∞}(x)] ≤ −g(ξ − w_{∞}(x))[ξ − w_{∞}(x)]  (2.26)

with some w_{∞} ∈ L^{∞}(Ω) where g is a strictly monotone increasing function with g(0) = 0.

Finally, there exists f_{∞} ∈ V_{1}^{⋆} such that

lim_{t→∞} ∥ f(t) − f_{∞} ∥_{V_{1}^{⋆}}= 0.  (2.27)

Then for a solution u ∈ L_{loc}^{p}(0, ∞; V_{1}), w ∈ L_{loc}^{∞}(Q_{∞}) of (1.5), (1.6) in (0, ∞) with w_{0} ∈ L^{∞}(Ω), any initial condition on u we have

lim_{t→∞} ∥ u(t) − u_{∞} ∥_{L^{2}(Ω)}= 0,  (2.28)

lim_{T →∞} ∫_{T −a}^{T +a} ∥ u(t) − u_{∞} ∥_{V_{1}}^{p} dt = 0 for arbitrary fixed a > 0,  (2.29)
lim_{t→∞} ∥ w(t, ·) − w_{∞} ∥_{L^{∞}(Ω)}= 0,  (2.30)

where u_{∞} ∈ V_{1} is the unique solution to

A_{∞}(u_{∞}) = f_{∞}  (2.31)

and the operator A_{∞} : V_{1} → V_{1}^{⋆} is defined by

⟨A_{∞}(z), v⟩ = ∑_{i=1}^{n} ∫_{Ω} a_{i,∞}(x, z(x), Dz(x))D_{i}v(x)dx+

∫_{Ω} a_{0,∞}(x, z(x), Dz(x))v(x)dx, z, v ∈ V_{1}.

Proof Equality (2.30) follows from (2.26) similarly as it was proved in Theorem 2.2. By Theorem 2.1 (2.21) holds. Applying (A_{2}) (by using (2.19), (2.20)) to u(t) = ũ, w(t) = w̃ where ũ ∈ V_{1}, w̃ ∈ L^{2}(Ω), we obtain from (2.22)

|a_{i,∞}(x, ζ_{0}, ζ)| ≤ c_{1}(|ζ_{0}|^{p−1} + |ζ|^{p−1}) + k̃_{1}(x)

with some constant c_{1} and k̃_{1} ∈ L^{q}(Ω). Similarly, Vitali’s theorem, (2.19), (2.20), (2.22) - (2.25) imply

∑_{i=1}^{n} ∫_{Ω}[a_{i,∞}(x, z(x), Dz(x)) − a_{i,∞}(x, z^{⋆}(x), Dz^{⋆}(x))][D_{i}z(x) − D_{i}z^{⋆}(x)]dx+

∫_{Ω}[a_{0,∞}(x, z(x), Dz(x)) − a_{0,∞}(x, z^{⋆}(x), Dz^{⋆}(x))][z(x) − z^{⋆}(x)]dx ≥
c_{2} ∫_{Ω}[|z(x) − z^{⋆}(x)|^{p} + |Dz(x) − Dz^{⋆}(x)|^{p}]dx for any z, z^{⋆} ∈ V_{1}.

Consequently, A_{∞} : V_{1} → V_{1}^{⋆} is bounded, hemicontinuous, strictly monotone and coercive which implies the existence of a unique solution of (2.31) (see, e.g., [13]).

If u, w are solutions of (1.5), (1.6) in (0, ∞) then by (2.31) we obtain

⟨D_{t}[u(t) − u_{∞}], u(t) − u_{∞}⟩ + ⟨A(u, w)(t) − A_{∞}(u_{∞}), u(t) − u_{∞}⟩ = (2.32)
⟨f(t) − f_{∞}, u(t) − u_{∞}⟩.

It is well known ( see [13]) that

y(t) = ⟨u(t) − u_{∞}, u(t) − u_{∞}⟩ = ∫_{Ω}[u(t) − u_{∞}]^{2}dx

is absolutely continuous and the first term in (2.32) equals to 1/2y′(t) for a.e. t. Further, for the second term in (2.32) we have by (2.23) and Young’s inequality

⟨[A(u, w)](t) − A_{∞}(u_{∞}), u(t) − u_{∞}⟩ =  (2.33)
⟨[A(u, w)](t) − [A_{u,w}(u_{∞})](t), u(t) − u_{∞}⟩+
⟨[A_{u,w}(u_{∞})](t) − A_{∞}(u_{∞}), u(t) − u_{∞}⟩ ≥
g_{2}(u, w) ∥ u(t) − u_{∞} ∥_{V_{1}}^{p} − ∫_{Ω}[k_{3}(u, w)](t, x)dx−

\frac{ε^{p}}{p} ∥ u(t) − u_{∞} ∥_{V_{1}}^{p} −\frac{1}{qε^{q}} ∥ [A_{u,w}(u_{∞})](t) − A_{∞}(u_{∞}) ∥_{V_{1}^{⋆}}^{q}

with arbitrary ε > 0 where we used the notation

⟨[A_{u,w}(u_{∞})](t), z⟩ =

∫_{Ω} {∑_{i=1}^{n} a_{i}(t, x, u_{∞}(x), Du_{∞}(x); u, w)D_{i}z + a_{0}(t, x, u_{∞}(x), Du_{∞}(x); u, w)z} .

By Vitali’s theorem we obtain from (A_{2}), (2.19), (2.20), (2.22)

lim_{t→∞} ∥ [A_{u,w}(u_{∞})](t) − A_{∞}(u_{∞}) ∥_{V_{1}^{⋆}}= 0.  (2.34)

Finally, by Young’s inequality, for the right hand side of (2.32) we have

|⟨f(t) − f_{∞}, u(t) − u_{∞}⟩| ≤ \frac{ε^{p}}{p} ∥ u(t) − u_{∞} ∥_{V_{1}}^{p} +\frac{1}{qε^{q}} ∥ f(t) − f_{∞} ∥_{V_{1}^{⋆}}^{q} . (2.35)

Thus, choosing sufficiently small ε > 0, (2.21), (2.25), (2.27), (2.32) - (2.35) yield

y′(t) + c^{⋆} ∥ u(t) − u_{∞} ∥_{V_{1}}^{p}≤ ψ(t),  (2.36)

thus by Hölder’s inequality

y′(t) + c^{⋆⋆}y(t)^{p/2} ≤ ψ(t),  (2.37)

where lim_{t→∞} ψ(t) = 0 and c^{⋆}, c^{⋆⋆} are positive constants. Similarly to (2.16), one obtains (2.28) from (2.37). Combining (2.28) and (2.36) one obtains (2.29).

Examples Now we consider examples satisfying the conditions of Theorems 1.4 - 2.3. (Examples for Theorems 1.1, 1.3 see in [12].) Let a_{i}(t, x, ζ_{0}, ζ; u, w) have the form

a_{i}(t, x, ζ_{0}, ζ; u, w) = b_{1}([H_{1}(u)])b_{2}([H_{2}(w)])α_{i}(t, x, ζ_{0}, ζ), i = 1, ..., n, (2.38)
a_{0}(t, x, ζ_{0}, ζ; u, w) = b_{1}([H_{1}(u)])b_{2}([H_{2}(w)])α_{0}(t, x, ζ_{0}, ζ)+  (2.39)
b̂_{0}([F_{0}(u)](t, x))b̃_{0}(G_{0}(w))α̂_{0}(t, x, ζ_{0}, ζ)

where α_{i} satisfy the usual conditions: they are Carathéodory functions;

|α_{i}(t, x, ζ_{0}, ζ)| ≤ c_{1}(|ζ_{0}|^{p−1} + |ζ|^{p−1}) + k_{1}(x)

with some constant c_{1}, k_{1} ∈ L^{q}(Ω), i = 0, 1, ..., n;

∑_{i=1}^{n}[α_{i}(t, x, ζ_{0}, ζ) − α_{i}(t, x, ζ_{0}, ζ^{⋆})](ζ_{i} − ζ_{i}^{⋆}) > 0 if ζ ≠ ζ^{⋆};

∑_{i=0}^{n} α_{i}(t, x, ζ_{0}, ζ)ζ_{i} ≥ c_{2}(|ζ_{0}|^{p} + |ζ|^{p})

with some constant c_{2} > 0. E.g. functions

α_{i} = ζ_{i}|ζ|^{p−2}, i = 1, ..., n, α_{0} = ζ_{0}|ζ_{0}|^{p−2}

satisfy the above conditions. The function α̂_{0} satisfies the Carathéodory condition and

|α̂_{0}(t, x, ζ_{0}, ζ)| ≤ c_{1}(|ζ_{0}|ρ̂ + |ζ|ρ̂), 0 ≤ ρ̂ < p − 1  (2.40)

Further, b_{1}, b_{2}, b̂_{0}, b̃_{0} are continuous functions, satisfying (with some positive constants)

b_{1}(Θ) ≥ \frac{const}{1 + |Θ|^{σ^{⋆}}} , |b̂_{0}(Θ)| ≤ const|Θ|^{p−1−ρ^{⋆}}

with 0 < σ^{⋆} < p − 1, σ^{⋆} + ρ̂ < ρ^{⋆} < p − 1,

b_{2}(Θ) ≥ \frac{const}{1 + g_{3}(Θ)}, |b̃_{0}(Θ)| ≤ const[1 + g_{4}(Θ)].

(g_{3}, g_{4} are monotone nondecreasing positive functions.)

Finally,

H_{1} : L_{loc}^{p}(0, ∞; W^{1−δ,p}(Ω))) → C(\overline{Q_{∞}}), H_{2} : L_{loc}^{2}(Q_{∞}) → C(\overline{Q_{∞}}),

F_{0} : L_{loc}^{p}(0, ∞; W^{1−δ,p}(Ω))) → L_{loc}^{p}(Q_{∞}), G_{0} : L_{loc}^{2}(Q_{∞}) → L_{loc}^{2}(Q_{∞})

are linear operators of Volterra type such that for any fixed finite T > 0 their restrictions

H_{1} : L^{p}(0, T; W^{1−δ,p}(Ω))) → C(\overline{Q_{T}}), H_{2} : L^{2}(Q_{T}) → C(\overline{Q_{T}}),

F_{0} : L^{p}(0, T; W^{1−δ,p}(Ω))) → L^{p}(Q_{T}), G_{0} : L^{2}(Q_{T}) → L^{2}(Q_{T})

are uniformly bounded with respect to T ∈ (0, ∞). [H_{1}(u)](t, x) may have e.g. one of the forms:

∫_{Q_{t}} d(t, x, τ, ξ)u(τ, ξ)dτdξ where sup_{(t,x)∈Q_{T}} ∫_{Q_{T}} |d(t, x, τ, ξ)|^{q}dτdξ < ∞,

∫_{Γ_{t}} d(t, x, τ, ξ)v(τ, ξ)dτdσ_{ξ} where sup_{(t,x)∈Q_{T}} ∫_{Γ_{T}} |d(t, x, τ, ξ)|^{q}dτdσ_{ξ} < ∞.

Examples for F_{0}, G_{0} see in [11].

By using Young’s inequality, one can prove that the assumptions on a_{i} in Theorem 1.4 are fulfilled for the above example (see [11]). The assumptions on a_{i} in Theorem 2.1 are satisfied for the above example if

∥ H_{1}(u) ∥_{C(\overline{Q_{T}})}≤ const sup_{τ∈[0,T]} \{∫_{Ω} u(τ, x)^{2}dx\}^{1/2} ,  (2.41)

∥ F_{0}(u) ∥_{L^{p}(Q_{T})}≤ const sup_{τ∈[0,T]} {∫_{Ω} u(τ, x)^{2}dx}^{1/2}  (2.42)

with constants not depending on T. (2.41) is satisfied if e.g.

[H_{1}(u)](t, x) = ∫_{Q_{t}} d(t, x, τ, ξ)u(τ, ξ)dτdξ where

sup_{(t,x)∈Q_{∞}} ∫_{0}^{∞} [∫_{Ω} |d(t, x, τ, ξ)|^{2}dξ]^{1/2} dτ < ∞.

The assumptions on a_{i} in Theorem 2.2 are satisfied if (2.41), (2.42) hold and (instead of (2.40))

|α̂_{0}(t, x, ζ_{0}, ζ)| ≤ φ_{1}(t)(|ζ_{0}|ρ̂ + |ζ|ρ̂), 0 ≤ ρ̂ < p − 1  (2.43)

where lim_{∞} φ_{1} = 0.

Finally, the following modification of functions (2.38), (2.39) satisfy the conditions of Theorem 2.3: for simplicity e.g.

α_{i} = ζ_{i}|ζ|^{p−2} for i = 1, ..., n, α_{0} = ζ_{0}|ζ_{0}|^{p−2},

(2.43) is valid and instead of b_{1}(H_{1}(u)), b_{2}(H_{2}(w)) we have b_{1}(t, H_{1}(u)), b_{2}(t, H_{2}(w)), respectively, where (with some positive constants)

b_{1}(t, Θ) = \frac{const}{1 + ψ_{1}(t)|Θ|^{σ^{⋆}}} , b_{2}(t, Θ) = \frac{const}{1 + ψ_{2}(t)g_{3}(Θ)}, lim_{∞} ψ_{j} = 0.

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(Received September 1, 2007)