Georgian Mathematical Journal Volume 9 (2002), Number 1, 57–70

ESTIMATES OF A STABILIZATION RATE AS t → ∞ OF SOLUTIONS OF A NONLINEAR INTEGRO-DIFFERENTIAL

EQUATION

T. JANGVELADZE^{1} AND Z. KIGURADZE

Abstract. The asymptotic behavior as t → ∞ of solutions of a nonlinear integro-differential equation is studied. The equation arises as a model describing the penetration of the electromagnetic field in to a substance.

2000 Mathematics Subject Classification: 35K55, 45K05.

Key words and phrases: Nonlinear integro-differential parabolic equation, asymptotic behavior.

1. Introduction. The main result. This paper is devoted to the study of the stabilization of solutions of the first boundary value problem in a cylindrical domain Q = (0, 1) × {t > 0} for the system of nonlinear integro-differential parabolic equations

\frac{∂U}{∂t} = \frac{∂}{∂x}[a(S)\frac{∂U}{∂x} ], \frac{∂V}{∂t} = \frac{∂}{∂x}[a(S)\frac{∂V}{∂x} ], (x, t) ∈ Q,  (1)
U(0, t) = V (0, t) = 0, U(1, t) = ψ_{1}, V (1, t) = ψ_{2}, t ≥ 0,  (2)
U(x, 0) = U_{0}(x), V (x, 0) = V_{0}(x), x ∈ (0, 1),  (3)

where

S(x, t) = 1 + ∫_{0}^{t} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}]dτ,  (4)

or

S(t) = 1 + ∫_{0}^{t} ∫_{0}^{1} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}]dx dτ.  (5)

Here ψ_{1} = Const, ψ_{2} = Const, a(S), U_{0}(x) and V_{0}(x) are given functions.

The characteristic feature of equations (1), (4) and (1), (5) is the appearance of nonlinear members depending on the integral of searched functions in the coefficients of higher derivatives.

System (1), (4) arises as a model describing the penetration of the electromagnetic field into a substance [1].

1 In the literature this author is frequently referred to as T. A. Dzhangveladze.

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

A lot of scientific works are devoted to the investigation of the problem given in [1] and to some of its generalizations. These questions are considered in [2]–[9] and in a number of other works as well.

The study of equations of type (1), (4) began started in [1]. In this work, in particular, theorems of the existence of a generalized solution of the first boundary value problem for a(S) = S and the uniqueness for more general cases are proved. The case a(S) = S^{p}, 0 < p ≤ 1, is studied in [2], where a theorem of the existence and uniqueness of a solution of problem (1)–(4) is proved. Investigations for multidimensional space cases are carried out in [3] for the first time.

In [4], [5] an operational scheme with the so-called conditionally closed operators is proposed. This scheme is applied for the solution of problems of (1)–(4) type [4], [5].

Note that investigations of equations of (1), (4) type are also carried out in [6], [7].

In the work [5] some generalization of equations of type (1), (4) is proposed. In particular, assuming the temperature of the considered body to be constant throughout the material, i.e., depending on time, but independent of the space coordinates, the process of penetration of the magnetic field into the material is modelled by averaged integro-differential equations of type (1), (5).

The purpose of this note is to continue the study of the asymptotic behavior of solutions of the equations (1), (4) which began in [8], [9]. In the present paper estimates of stabilization rate as t → ∞ of solutions of problems (1)– (4) and (1)–(3), (5) are obtained for the case a(S) = S^{p}, 0 < p ≤ 1. We will use the scheme of [10] in which the adiabatic shearing of incompressible fluids with temperature-dependent viscosity is studied. We should note that boundary conditions (2) are used here taking into account the physical problem considered in [11].

We assume that (U, V) = (U(x, t), V (x, t)) is a solution of (1)–(4) on [0, 1] ×

[0, ∞) such that U, V, \frac{∂U}{∂x} , \frac{∂V}{∂x} , \frac{∂U}{∂t} , \frac{∂V}{∂t} , \frac{∂^{2}U}{∂x^{2}} , \frac{∂^{2}V}{∂x^{2}} are all in C^{0}([0, ∞); L_{2}(0, 1)),

while \frac{∂^{2}U}{∂t ∂x}, \frac{∂^{2}V}{∂t ∂x} are in C^{0}((0, ∞); L_{2}(0, 1)) and \frac{∂^{2}U}{∂t^{2}} , \frac{∂^{2}V}{∂t^{2}} are in L_{2,loc}((0, ∞);

L_{2}(0, 1)) (see [1], [2], [4], [5], [10]).

The main purpose of this work is to prove the following statement.

Theorem. Assume

a(S) = S^{p}, 0 < p ≤ 1,
U_{0}(0) = V_{0}(0) = 0, U_{0}(1) = ψ_{1}, V_{0}(1) = ψ_{2},
ψ_{1}^{2} + ψ_{2}^{2} ≠ 0, U_{0}, V_{0} ∈ W_{2}^{2}(0, 1).

Then for the solution of problem (1)–(4) the following estimates are true as t → ∞:

\frac{∂U(x, t)}{∂x} = ψ_{1} + O(t^{−1−p}), \frac{∂V (x, t)}{∂x} = ψ_{2} + O(t^{−1−p}),  (6)

\frac{∂U(x, t)}{∂t} = O(t^{−1}), \frac{∂V (x, t)}{∂t} = O(t^{−1}),  (7)

uniformly in x on [0, 1].

The proof of the theorem is based on a priori estimates which are obtained with the help of a number of identities derived below.

2. Proof of the theorem. Now let us proved to obtaining a priori estimates.

Lemma 1. For solving problem (1) − (4) the following estimations are true:

∫_{0}^{t} ∫_{0}^{1} (\frac{∂U}{∂τ} )^{2}dx dτ ≤ C, ∫_{0}^{t} ∫_{0}^{1} (\frac{∂V}{∂τ} )^{2}dx dτ ≤ C, t ≥ 0.  (8)

Proof. Let us differentiate the first equation of system (1) with respect to t:

\frac{∂^{2}U}{∂t^{2}} = \frac{∂}{∂x}[S^{p} \frac{∂^{2}U}{∂t ∂x} + pS^{p−1}([\frac{∂U}{∂x} ]^{3} + \frac{∂U}{∂x} [\frac{∂V}{∂x} ]^{2})].  (9)

Multiplying equation (9) by ∂U/∂t and integrating with respect to x on the interval (0, 1), we have

\frac{1}{2} \frac{d}{dt} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + ∫_{0}^{1} S^{p}(\frac{∂^{2}U}{∂t ∂x})^{2}dx + p ∫_{0}^{1} S^{p−1}(\frac{∂U}{∂x} )^{3} \frac{∂^{2}U}{∂t ∂x}dx +

+p ∫_{0}^{1} S^{p−1} \frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂^{2}U}{∂t ∂x}dx = 0.  (10)

Integration from 0 to t gives the identity

\frac{1}{2} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + ∫_{0}^{t} ∫_{0}^{1} S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ + \frac{p}{4} ∫_{0}^{1} S^{p−1}(\frac{∂U}{∂x} )^{4}dx −

−\frac{p(p − 1)}{4} ∫_{0}^{t} ∫_{0}^{1} S^{p−2}(\frac{∂U}{∂x} )^{4}[(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}]dx dτ +

+\frac{p}{2} ∫_{0}^{t} ∫_{0}^{1} S^{p−1}(\frac{∂V}{∂x} )^{2} \frac{∂}{∂τ} (\frac{∂U}{∂x} )^{2}dx dτ = \frac{1}{2} ∫_{0}^{1} (\frac{∂U(x, 0)}{∂t})^{2}dx + \frac{p}{4} ∫_{0}^{1} (\frac{∂U_{0}}{∂x} )^{4}dx.

It follows that

∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + 2 ∫_{0}^{t} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ + p ∫_{0}^{t} ∫_{0}^{1} S^{p−1}(\frac{∂V}{∂x} )^{2} \frac{∂}{∂τ} (\frac{∂U}{∂x} )^{2}dx dτ ≤ C.

Here and below c, C and C_{i} denote the positive constants dependent only on ψ_{i} = Const, i = 1, 2, U_{0}(x), V_{0}(x) and consequently independent of t.

60 T. JANGVELADZE AND Z. KIGURADZE

Similarly, using the second equation of system (1), we get

∫_{0}^{1} (\frac{∂V}{∂t} )^{2}dx + 2 ∫_{0}^{t} ∫_{0}^{1} (\frac{∂^{2}V}{∂τ ∂x})^{2}dx dτ + p ∫_{0}^{t} ∫_{0}^{1} S^{p−1}(\frac{∂U}{∂x} )^{2} \frac{∂}{∂τ} (\frac{∂V}{∂x} )^{2}dx dτ ≤ C

and, therefore

∫_{0}^{1} [(\frac{∂U}{∂t} )^{2} + (\frac{∂V}{∂t} )^{2}]dx + 2 ∫_{0}^{t} ∫_{0}^{1} [(\frac{∂^{2}U}{∂τ ∂x})^{2} + (\frac{∂^{2}V}{∂τ ∂x})^{2}]dx dτ

+p ∫_{0}^{t} ∫_{0}^{1} S^{p−1} \frac{∂}{∂τ} [(\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{2}]dx dτ ≤ C.

Note that

p ∫_{0}^{t} ∫_{0}^{1} S^{p−1} \frac{∂}{∂τ} [(\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{2}]dx dτ = p ∫_{0}^{1} S^{p−1}(\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{2}dx

−p ∫_{0}^{1} (\frac{∂U_{0}}{∂x} )^{2}(\frac{∂V_{0}}{∂x} )^{2}dx − p(p − 1) ∫_{0}^{t} ∫_{0}^{1} S^{p−2}[(\frac{∂U}{∂x} )^{2}

+(\frac{∂V}{∂x} )^{2}](\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{2}dx dτ.

We have

∫_{0}^{1} [(\frac{∂U}{∂t} )^{2} + (\frac{∂V}{∂t} )^{2}]dx + 2 ∫_{0}^{t} ∫_{0}^{1} [(\frac{∂^{2}U}{∂τ ∂x})^{2} + (\frac{∂^{2}V}{∂τ ∂x})^{2}]dx dτ ≤ C. (11)

From this, taking into consideration the relations

∫_{0}^{t} ∫_{0}^{1} (\frac{∂U}{∂τ} )^{2}dx dτ ≤ ∫_{0}^{t} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ,

∫_{0}^{t} ∫_{0}^{1} (\frac{∂V}{∂τ} )^{2}dx dτ ≤ ∫_{0}^{t} ∫_{0}^{1} (\frac{∂^{2}V}{∂τ ∂x})^{2}dx dτ,

we get a priori estimates (8).

Lemma 2. The following estimations are true:

cφ^{\frac{1}{1+2p}} (t) ≤ S(x, t) ≤ Cφ^{\frac{1}{1+2p}} (t), 0 ≤ x ≤ 1, t ≥ 0,  (12)

where

φ(t) = 1 + ∫_{0}^{t} ∫_{0}^{1} S^{2p}[(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}]dx dτ.

STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION 61

Proof. From (4) it follows that

\frac{∂S}{∂t} = (\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}, S(x, 0) = 1.

Let us multiply this equation by S^{2p}:

\frac{1}{1 + 2p} \frac{∂S^{1+2p}}{∂t} = (\frac{∂U}{∂x} )^{2}S^{2p} + (\frac{∂V}{∂x} )^{2}S^{2p}.

Now let us introduce the notations:

σ_{1} = S^{p} \frac{∂U}{∂x} , σ_{2} = S^{p} \frac{∂V}{∂x} ,

then (1) can be rewritten as

\frac{∂U}{∂t} = \frac{∂σ_{1}}{∂x} , \frac{∂V}{∂t} = \frac{∂σ_{2}}{∂x} .

We have

\frac{1}{1 + 2p} \frac{∂S^{1+2p}}{∂t} = σ_{1}^{2} + σ_{2}^{2},  (13)

σ_{1}^{2}(x, t) = ∫_{0}^{1} σ_{1}^{2}(y, t)dy + 2 ∫_{0}^{1} ∫_{y}^{x} σ_{1}(ξ, t)\frac{∂U(ξ, t)}{∂t}dξ dy,

σ_{2}^{2}(x, t) = ∫_{0}^{1} σ_{2}^{2}(y, t)dy + 2 ∫_{0}^{1} ∫_{y}^{x} σ_{2}(ξ, t)\frac{∂V (ξ, t)}{∂t}dξ dy,

φ(t) = 1 + ∫_{0}^{t} ∫_{0}^{1}(σ_{1}^{2} + σ_{2}^{2})dx dτ.

From (8) and (13) we get

\frac{1}{1 + 2p}S^{1+2p} = ∫_{0}^{t}(σ_{1}^{2} + σ_{2}^{2})dτ + \frac{1}{1 + 2p}

= ∫_{0}^{t} ∫_{0}^{1} (σ_{1}^{2}(y, τ) + σ_{2}^{2}(y, τ))dy dτ + 2 ∫_{0}^{t} ∫_{0}^{1} ∫_{y}^{x} σ_{1}(ξ, τ)\frac{∂U(ξ, τ)}{∂τ}dξ dy dτ

+2 ∫_{0}^{t} ∫_{0}^{1} ∫_{y}^{x} σ_{2}(ξ, τ)\frac{∂V (ξ, τ)}{∂τ}dξ dy dτ + \frac{1}{1 + 2p}

≤ 2 ∫_{0}^{t} ∫_{0}^{1} (σ_{1}^{2}(y, τ) + σ_{2}^{2}(y, τ))dy dτ + C + \frac{1}{1 + 2p} ≤ (2 + C_{1})φ(t),

i.e.,

S(x, t) ≤ Cφ^{\frac{1}{1+2p}} (t).  (14)

62 T. JANGVELADZE AND Z. KIGURADZE

Analogously,

\frac{1}{1 + 2p}S^{1+2p} = ∫_{0}^{t} ∫_{0}^{1} (σ_{1}^{2}(y, τ) + σ_{2}^{2}(y, τ))(y, τ)dy dτ

+2 ∫_{0}^{t} ∫_{0}^{1} ∫_{y}^{x} σ_{1}(ξ, τ)\frac{∂U(ξ, τ)}{∂τ}dξ dy dτ + 2 ∫_{0}^{t} ∫_{0}^{1} ∫_{y}^{x} σ_{2}(ξ, τ)\frac{∂V (ξ, τ)}{∂τ}dξ dy dτ

+\frac{1}{1 + 2p} ≥ \frac{1}{2} ∫_{0}^{t} ∫_{0}^{1} (σ_{1}^{2}(y, τ) + σ_{2}^{2}(y, τ))dy dτ − C_{2} ≥ \frac{1}{2}φ(t) − C_{3}. (15)

From (4) it follows that S(x, t) ≥ 1. So

C_{3}S^{1+2p} ≥ C_{3}.  (16)

Taking into account (15) and (16) we easily get

(\frac{1}{1 + 2p} + C_{3})S^{1+2p} ≥ \frac{1}{2}φ(t),

or

S(x, t) ≥ cφ^{\frac{1}{1+2p}} (t).  (17)

Finally, from (14) and (17) we obtain (12). Lemma 3. The following esimates are true:

cφ^{\frac{2p}{1+2p}} (t) ≤ ∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx ≤ Cφ^{\frac{2p}{1+2p}} (t), t ≥ 0.  (18)

Proof. Taking into account (12), we get

∫_{0}^{1}(σ_{1}^{2} + σ_{2}^{2})dx = ∫_{0}^{1} S^{2p}[(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}]dx ≥ cφ^{\frac{2p}{1+2p}} (t) ∫_{0}^{1} [(\frac{∂U}{∂x} )^{2}

+(\frac{∂V}{∂x} )^{2}]dx ≥ cφ^{\frac{2p}{1+2p}} (t)\{[ ∫_{0}^{1} \frac{∂U}{∂x} dx]^{2} + [ ∫_{0}^{1} \frac{∂V}{∂x} dx]^{2}\} = (ψ_{1}^{2} + ψ_{2}^{2})cφ^{\frac{2p}{1+2p}} (t),

or

∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx ≥ cφ^{\frac{2p}{1+2p}} (t).  (19)

At the same time, from (11) we derive

∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx ≤ C, ∫_{0}^{1} (\frac{∂V}{∂t} )^{2}dx ≤ C, t ≥ 0.  (20)

STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION 63

Let us multiply the first and second equations of system (1) by U(x, t) and V (x, t), respectively. Using the boundary conditions (2), we have

∫_{0}^{1} U \frac{∂U}{∂t} dx + ∫_{0}^{1} S^{p}(\frac{∂U}{∂x} )^{2}dx = ψ_{1}σ_{1}(1, t),

∫_{0}^{1} V \frac{∂V}{∂t} dx + ∫_{0}^{1} S^{p}(\frac{∂V}{∂x} )^{2}dx = ψ_{2}σ_{2}(1, t).

Using these equalities, (12), (19), (20) and the maximum principle

|U(x, t)| ≤ max_{0≤y≤1} |U_{0}(y)|, |V (x, t)| ≤ max_{0≤y≤1} |V_{0}(y)|, 0 ≤ x ≤ 1, t ≥ 0,

we have

{ ∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx}^{2} ≤ 2{ ∫_{0}^{1} σ_{1}^{2}(x, t)dx}^{2} + 2{ ∫_{0}^{1} σ_{2}^{2}(x, t)dx}^{2}

≤ 2C_{1}φ^{\frac{2p}{1+2p}} (t)[\{ ∫_{0}^{1} S^{p}(\frac{∂U}{∂x} )^{2}dx\}^{2} + \{ ∫_{0}^{1} S^{p}(\frac{∂V}{∂x} )^{2}dx\}^{2}]

≤ 4C_{1}φ^{\frac{2p}{1+2p}} (t)[(ψ_{1}σ_{1}(1, t))^{2} + ( ∫_{0}^{1} U \frac{∂U}{∂t} dx)^{2} + (ψ_{2}σ_{2}(1, t))^{2}

+( ∫_{0}^{1} V \frac{∂V}{∂t} dx)^{2}] ≤ 4C_{1}φ^{\frac{2p}{1+2p}} (t)[(ψ_{1}^{2} + ψ_{2}^{2})(σ_{1}^{2}(1, t) + σ_{2}^{2}(1, t))

+C_{2}{( max_{0≤y≤1} |U_{0}(y)|)^{2} + ( max_{0≤y≤1} |V_{0}(y)|)^{2}}]

≤ 8C_{1}φ^{\frac{2p}{1+2p}} (t)[(ψ_{1}^{2} + ψ_{2}^{2})\{ ∫_{0}^{1} σ_{1}^{2}dx + ∫_{0}^{1} (\frac{∂σ_{1}}{∂x} )^{2}dx + ∫_{0}^{1} σ_{2}^{2}dx

+ ∫_{0}^{1} (\frac{∂σ_{2}}{∂x} )^{2}dx\} + C_{3}] ≤ 8C_{1}φ^{\frac{2p}{1+2p}} (t)[(ψ_{1}^{2} + ψ_{2}^{2}) ∫_{0}^{1}(σ_{1}^{2} + σ_{2}^{2})dx + C_{4}]

≤ 8C_{1}φ^{\frac{2p}{1+2p}} (t)[(ψ_{1}^{2} + ψ_{2}^{2}) ∫_{0}^{1}(σ_{1}^{2} + σ_{2}^{2})dx + \frac{C_{4}}{cφ^{\frac{2p}{1+2p}} (t)} ∫_{0}^{1}(σ_{1}^{2} + σ_{2}^{2})dx]

≤ Cφ^{\frac{2p}{1+2p}} (t) ∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx,

i.e.,

∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx ≤ Cφ^{\frac{2p}{1+2p}} (t).

Finally, using this estimate and (19) we obtain (18).

64 T. JANGVELADZE AND Z. KIGURADZE

Lemma 4. The following inequalities take place:

ct^{2p} ≤ ∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{2}^{2}(x, t))dx ≤ Ct^{2p}, t ≥ 1,  (21)
ct ≤ S(x, t) ≤ Ct, 0 ≤ x ≤ 1, t ≥ 1.  (22)

Proof. From (18) taking into account the relation

\frac{dφ(t)}{dt} = ∫_{0}^{1} (σ_{1}^{2}(x, t) + σ_{1}^{2}(x, t))dx

we get

cφ^{\frac{2p}{1+2p}} (t) ≤ \frac{dφ(t)}{dt} ≤ Cφ^{\frac{2p}{1+2p}} (t).

From this we have ct^{1+2p} ≤ φ(t) ≤ Ct^{1+2p}, t ≥ 1. Now taking into account (12) and (18) from the last inequality we obtain (21) and (22).

Lemma 5. ∂U/∂t and ∂V/∂t statisfy the inequality

∫_{0}^{1} [(\frac{∂U}{∂t} )^{2} + (\frac{∂V}{∂t} )^{2}]dx ≤ Ct^{−2}, t ≥ 1.  (23)

Proof. By Schwarz’s inequality, (10) yields

\frac{d}{dt} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + ∫_{0}^{1} S^{p}(\frac{∂^{2}U}{∂t ∂x})^{2}dx ≤ 2p^{2} ∫_{0}^{1} S^{p−2}(\frac{∂U}{∂x} )^{6}dx

+2p^{2} ∫_{0}^{1} S^{p−2}(\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{4}dx.  (24)

Now using (21), (22), the relations σ_{1} = S^{p} \frac{∂U}{∂x}, σ_{2} = S^{p} \frac{∂V}{∂x} and

∫_{0}^{1} (\frac{∂σ_{1}}{∂x} )^{2}dx = − ∫_{0}^{1} σ_{1} \frac{∂^{2}σ_{1}}{∂x^{2}} dx, ∫_{0}^{1} (\frac{∂σ_{2}}{∂x} )^{2}dx = − ∫_{0}^{1} σ_{2} \frac{∂^{2}σ_{2}}{∂x^{2}} dx,

from (24) we get

\frac{d}{dt} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + ct^{p} ∫_{0}^{1} (\frac{∂^{2}U}{∂t ∂x})^{2}dx ≤ C_{1} \frac{t^{p−2}}{t^{6p}} ∫_{0}^{1}(σ_{1}^{6} + σ_{1}^{2}σ_{2}^{4})dx

≤ C_{1}t^{−5p−2} ∫_{0}^{1} σ_{1}^{2}(x, t)dx([ max_{0≤x≤1} σ_{1}^{2}(x, t)]^{2} + [ max_{0≤x≤1} σ_{2}^{2}(x, t)]^{2})

≤ C_{2}t^{−3p−2}(\{ ∫_{0}^{1} σ_{1}^{2}dx + 2[ ∫_{0}^{1} σ_{1}^{2}dx]^{\frac{1}{2}} [ ∫_{0}^{1} (\frac{∂σ_{1}}{∂x} )^{2}dx]^{\frac{1}{2}} \}^{2}

STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION 65

+\{ ∫_{0}^{1} σ_{2}^{2}dx + 2[ ∫_{0}^{1} σ_{2}^{2}dx]^{\frac{1}{2}} [ ∫_{0}^{1} (\frac{∂σ_{2}}{∂x} )^{2}dx]^{\frac{1}{2}} \}^{2})

≤ C_{2}t^{−3p−2}(\{ ∫_{0}^{1} σ_{1}^{2}dx + 2[ ∫_{0}^{1} σ_{1}^{2}dx]^{\frac{3}{4}} [ ∫_{0}^{1} (\frac{∂^{2}σ_{1}}{∂x^{2}} )^{2}dx]^{\frac{1}{4}} \}^{2}

+\{ ∫_{0}^{1} σ_{2}^{2}dx + 2[ ∫_{0}^{1} σ_{2}^{2}dx]^{\frac{3}{4}} [ ∫_{0}^{1} (\frac{∂^{2}σ_{2}}{∂x^{2}} )^{2}dx]^{\frac{1}{4}} \}^{2})

≤ C_{3}t^{p−2} + C_{4}t^{−3p−2}t^{3p}([ ∫_{0}^{1} (\frac{∂^{2}U}{∂t ∂x})^{2}dx]^{\frac{1}{2}} + [ ∫_{0}^{1} (\frac{∂^{2}V}{∂t ∂x})^{2}dx]^{\frac{1}{2}} )

≤ C_{3}t^{p−2} + C_{5}t^{−p−4} + \frac{ct^{p}}{4} ( ∫_{0}^{1} (\frac{∂^{2}U}{∂t ∂x})^{2}dx + ∫_{0}^{1} (\frac{∂^{2}V}{∂t ∂x})^{2}dx).

Hence we have

\frac{d}{dt} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx + \frac{c}{4}t^{p} ∫_{0}^{1} [3(\frac{∂^{2}U}{∂t ∂x})^{2} − (\frac{∂^{2}V}{∂t ∂x})^{2}]dx ≤ Ct^{p−2}, t ≥ 1.

Alalogously, we can show that

\frac{d}{dt} ∫_{0}^{1} (\frac{∂V}{∂t} )^{2}dx + \frac{c}{4}t^{p} ∫_{0}^{1} [3(\frac{∂^{2}V}{∂t ∂x})^{2} − (\frac{∂^{2}U}{∂t ∂x})^{2}]dx ≤ Ct^{p−2}, t ≥ 1.

As a consequence we get the following estimate:

\frac{d}{dt} ∫_{0}^{1} [(\frac{∂U}{∂t} )^{2}+(\frac{∂V}{∂t} )^{2}]dx+ \frac{c}{2}t^{p} ∫_{0}^{1} [(\frac{∂^{2}U}{∂t ∂x})^{2}+(\frac{∂^{2}V}{∂t ∂x})^{2}]dx ≤ Ct^{p−2}. (25)

By the Poincarè inequality

∥\frac{∂U}{∂t} ∥ ≤ ∥\frac{∂^{2}U}{∂t ∂x}∥, ∥\frac{∂V}{∂t} ∥ ≤ ∥\frac{∂^{2}V}{∂t ∂x}∥,

(25) gives

\frac{d}{dt} ∫_{0}^{1} [(\frac{∂U}{∂t} )^{2} + (\frac{∂V}{∂t} )^{2}]dx + \frac{ct^{p}}{2} ∫_{0}^{1} [(\frac{∂U}{∂t} )^{2} + (\frac{∂V}{∂t} )^{2}]dx ≤ Ct^{p−2}.

From this we obtain (23). Let us now estimate ∂S/∂x in L_{1}(0, 1). Lemma 6. For ∂S/∂x the following estimate is true:

∫_{0}^{1} |\frac{∂S}{∂x} |dx ≤ Ct^{−p}, t ≥ 1.  (26)

66 T. JANGVELADZE AND Z. KIGURADZE

Proof. We have

\frac{∂}{∂t}[S^{2p} \frac{∂S}{∂x} ] = 2σ_{1} \frac{∂σ_{1}}{∂x} + 2σ_{2} \frac{∂σ_{2}}{∂x} = 2σ_{1} \frac{∂U}{∂t} + 2σ_{2} \frac{∂V}{∂t} .  (27)

From (21) and (23) it follows that

| ∫_{0}^{1} σ_{1} \frac{∂U}{∂t} dx| ≤ C_{1}t^{p}t^{−1} = Ct^{p−1}, | ∫_{0}^{1} σ_{2} \frac{∂V}{∂t} dx| ≤ C_{1}t^{p}t^{−1} = Ct^{p−1} (28)

and, applying (22), (27) and (28), we get

S^{2p} \frac{∂S}{∂x} = ∫_{0}^{t} (2σ_{1} \frac{∂U}{∂τ} + 2σ_{2} \frac{∂V}{∂τ} )dτ,

∫_{0}^{1} |\frac{∂S}{∂x} |dx ≤ \frac{1}{c}t^{−2p} ∫_{0}^{t} C_{1}τ^{p−1}dτ = Ct^{−p}.

Now we are ready to prove the theorem. Let us estimate ∂^{2}U/∂x^{2} in L_{1}(0, 1). We have

\frac{∂U}{∂x} = σ_{1}S^{−p}, \frac{∂σ_{1}}{∂x} = \frac{∂U}{∂t} , \frac{∂^{2}U}{∂x^{2}} = \frac{∂U}{∂t} S^{−p} − pσ_{1}S^{−p−1} \frac{∂S}{∂x} ,

σ_{1}^{2}(x, t) ≤ ∫_{0}^{1} σ_{1}^{2}(y, t)dy + 2 ∫_{0}^{1} |σ_{1}(y, t)||\frac{∂U(y, t)}{∂t}|dy ≤ C_{1}t^{2p} + C_{2}t^{−2}.

From the latter we get

σ_{1}(x, t) ≤ Ct^{p}, t ≥ 1.  (29)

Applying now (22), (23), (26) and (29), we derive

∫_{0}^{1} |\frac{∂^{2}U(x, t)}{∂x^{2}} |dx ≤ ∫_{0}^{1} |\frac{∂U}{∂t} S^{−p}|dx + p ∫_{0}^{1} |σ_{1}S^{−p−1} \frac{∂S}{∂x} |dx

≤ [ ∫_{0}^{1} S^{−2p}dx]^{\frac{1}{2}} [ ∫_{0}^{1} |\frac{∂U}{∂t} |^{2}dx]^{\frac{1}{2}} + p ∫_{0}^{1} |σ_{1}S^{−p−1} \frac{∂S}{∂x} |dx ≤ Ct^{−1−p}.

Hence we have

∫_{0}^{1} |\frac{∂^{2}U(x, t)}{∂x^{2}} |dx ≤ Ct^{−1−p}, t ≥ 1.

From this estimate, taking into account the relation

\frac{∂U(x, t)}{∂x} = ∫_{0}^{1} \frac{∂U(y, t)}{∂y}dy + ∫_{0}^{1} ∫_{y}^{x} \frac{∂^{2}U(ξ, t)}{∂ξ^{2}}dξ dy,

STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION 67

it follows that

\frac{∂U(x, t)}{∂x} − ψ_{1} = ∫_{0}^{1} ∫_{y}^{x} \frac{∂^{2}U(ξ, t)}{∂ξ^{2}}dξ dy ≤ ∫_{0}^{1} |\frac{∂^{2}U(y, t)}{∂y^{2}} |dy ≤ Ct^{−1−p}.

Thus the following asymptotic formula takes place:

\frac{∂U(x, t)}{∂x} = ψ_{1} + O(t^{−1−p}).

The same estimate is valid for ∂V/∂x:

\frac{∂V (x, t)}{∂x} = ψ_{2} + O(t^{−1−p}).

Let us now establish the asymptotic behaviour of the derivatives ∂U/∂t and ∂V/∂t. For this multiply (10) by t^{2}. By integrating on (0, t) we have

\frac{t^{2}}{2} ∫_{0}^{1} (\frac{∂U}{∂t} )^{2}dx − ∫_{0}^{t} ∫_{0}^{1} τ (\frac{∂U}{∂τ} )^{2}dx dτ + ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+p ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−1}(\frac{∂U}{∂x} )^{3} \frac{∂^{2}U}{∂τ ∂x}dx dτ + p ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−1} \frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂^{2}U}{∂τ ∂x}dx dτ = 0

and, using Schwarz’s inequality, we conclude that

\frac{1}{2} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ ≤ ∫_{0}^{t} ∫_{0}^{1} τ (\frac{∂U}{∂τ} )^{2}dx dτ

+p^{2} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−2}(\frac{∂U}{∂x} )^{6}dx dτ + p^{2} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−2}(\frac{∂U}{∂x} )^{2}(\frac{∂V}{∂x} )^{4}dx dτ.

From this using (6), (22) and (23) we get

∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ ≤ Ct^{p+1}.  (30)

Hence

∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ ≤ Ct^{p+1}.  (31)

Analogously,

∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}V}{∂τ ∂x})^{2}dx dτ ≤ Ct^{p+1},  (32)
∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}V}{∂τ ∂x})^{2}dx dτ ≤ Ct^{p+1}.  (33)

Multiplying (9) by t^{3}∂^{2}U/∂t^{2}, applying the formula of integrating by parts and a priori estimates (6), (22), (30), (31), (32) and (33), we get

∫_{0}^{t} ∫_{0}^{1} τ^{3}(\frac{∂^{2}U}{∂τ^{2}} )^{2}dx dτ + \frac{1}{2} ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p} \frac{∂}{∂τ} [\frac{∂^{2}U}{∂τ ∂x}]^{2}dx dτ

+p ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1}(\frac{∂U}{∂x} )^{3} \frac{∂}{∂τ} [\frac{∂^{2}U}{∂τ ∂x}]dx dτ

+p ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1} \frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂}{∂τ} [\frac{∂^{2}U}{∂τ ∂x}]dx dτ = 0,

∫_{0}^{t} ∫_{0}^{1} τ^{3}(\frac{∂^{2}U}{∂τ^{2}} )^{2}dx dτ + \frac{1}{2} ∫_{0}^{1} t^{3}S^{p}(\frac{∂^{2}U}{∂t ∂x})^{2}dx = \frac{3}{2} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ∂x})^{2}dx dτ

+\frac{p}{2} ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1}[(\frac{∂U}{∂x} )^{2}+(\frac{∂V}{∂x} )^{2}](\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

−pt^{3} ∫_{0}^{1} S^{p−1}(\frac{∂U}{∂x} )^{3} \frac{∂^{2}U}{∂t ∂x}dx + 3p ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−1}(\frac{∂U}{∂x} )^{3} \frac{∂^{2}U}{∂τ ∂x}dx dτ

+p(p − 1) ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−2}[(\frac{∂U}{∂x} )^{5} + (\frac{∂U}{∂x} )^{3}(\frac{∂V}{∂x} )^{2}]\frac{∂^{2}U}{∂τ ∂x}dx dτ

+3p ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1}(\frac{∂U}{∂x} )^{2}(\frac{∂^{2}U}{∂ τ∂x})^{2}dx dτ −pt^{3} ∫_{0}^{1} S^{p−1} \frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂^{2}U}{∂t ∂x}dx

+3p ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−1} \frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂^{2}U}{∂τ ∂x}dx dτ + p(p − 1) ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−2}[(\frac{∂U}{∂x} )^{2}

+(\frac{∂V}{∂x} )^{2}]\frac{∂U}{∂x} (\frac{∂V}{∂x} )^{2} \frac{∂^{2}U}{∂x ∂τ} dx dτ +p ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1}(\frac{∂V}{∂x} )^{2}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+2p ∫_{0}^{t} ∫_{0}^{1} τ^{3}S^{p−1} \frac{∂U}{∂x} \frac{∂V}{∂x} \frac{∂^{2}U}{∂τ ∂x} \frac{∂^{2}V}{∂τ ∂x}dx dτ,

t^{3} ∫_{0}^{1} S^{p}(\frac{∂^{2}U}{∂t ∂x})^{2}dx≤3 ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ +C_{1} ∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+\frac{t^{3}}{2} ∫_{0}^{1} S^{p}(\frac{∂^{2}U}{∂t ∂x})^{2}dx + C_{2}t^{3} ∫_{0}^{1} S^{p−2}dx + C_{3} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION 69

+C_{4} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p−2}dx dτ + ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+C_{5} ∫_{0}^{t} ∫_{0}^{1} τ^{4}S^{p−4}dx dτ + C_{7}t^{p−1} + C_{6} ∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+C_{8} ∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ + C_{9} ∫_{0}^{t} ∫_{0}^{1} τ^{2}S^{p}(\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+C_{10} ∫_{0}^{t} ∫_{0}^{1} τ^{4}S^{p−4}dx dτ + C_{11} ∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ

+C_{12} ∫_{0}^{t} τ^{p+2} ∫_{0}^{1} (\frac{∂^{2}U}{∂τ ∂x})^{2}dx dτ ≤ C_{13}t^{p+1},

ct^{p+3} ∫_{0}^{1} (\frac{∂^{2}U}{∂t ∂x})^{2}dx ≤ Ct^{p+1}.

From this we have

∫_{0}^{1} (\frac{∂^{2}U}{∂t ∂x})^{2}dx ≤ Ct^{−2}.  (34)

Taking into account the relation

\frac{∂U(x, t)}{∂t} = ∫_{0}^{1} \frac{∂U(y, t)}{∂t}dy + ∫_{0}^{1} ∫_{y}^{x} \frac{∂^{2}U(ξ, t)}{∂t ∂ξ}dξ dy ≤ Ct^{−1}

+ ∫_{0}^{1} ∫_{y}^{x} \frac{∂^{2}U(ξ, t)}{∂t ∂ξ}dξ dy ≤ Ct^{−1} + [ ∫_{0}^{1} (\frac{∂^{2}U(y, t)}{∂t ∂y})^{2}dy]^{\frac{1}{2}} ,

from (34) we get

\frac{∂U(x, t)}{∂t} = O(t^{−1}).

Analogously, we can show that

\frac{∂V (x, t)}{∂t} = O(t^{−1}).

So the proof of the theorem is over. Finally we note that estimates similar to (6) and (7) are true for the averaged

integro-differential problem (1)–(3), (5).

References

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(Received 4.06.2001; revised 8.10.2001)

Author’s addresses:

T. Jangveladze I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia E-mail: djante.math@viam net.edu.ge

Z. Kiguradze Department of Mechanics and Mathematics I. Javakhishvili bilisi State University 1, University St., Tbilisi 380043 Georgia E-mail: zura2806@yahoo.com