1. Introduction. In this paper we investigate the asymptotic behaviour of global solutions for the following reaction-diffusion system:

S. B A D R A O U I (Guelma)

BEHAVIOUR OF GLOBAL SOLUTIONS FOR A SYSTEM OF REACTION-DIFFUSION EQUATIONS FROM COMBUSTION THEORY

Abstract. We are concerned with the boundedness and large time be- haviour of the solution for a system of reaction-diffusion equations mod- elling complex consecutive reactions on a bounded domain under homoge- neous Neumann boundary conditions. Using the techniques of E. Conway, D. Hoff and J. Smoller [3] we also show that the bounded solution converges to a constant function as t → ∞. Finally, we investigate the rate of this convergence.

1. Introduction. In this paper we investigate the asymptotic behaviour of global solutions for the following reaction-diffusion system:

(1.1) ∂Y

∂t = d

∆Y

− d

Y

Y

f

(T ), x ∈ Ω, t > 0, (1.2) ∂Y

∂t = d

∆Y

+ d

Y

Y

f

(T )

− d

Y

f

(T ) − d

Y

− d

Y

, x ∈ Ω, t > 0,

(1.3) ∂T

∂t = d

∆T + d

Y

Y

f

(T )

+ d

Y

f

(T ) + d

Y

+ d

Y

, x ∈ Ω, t > 0, (1.4) ∂Y

∂ν = ∂Y

∂ν = ∂T

∂ν = 0, x ∈ ∂Ω, t > 0, (1.5) (Y

, Y

, T )(x, 0) = (Y

, Y

, T

)(x), x ∈ Ω,

1991 Mathematics Subject Classification: 35K57, 35B40, 35B45.

Key words and phrases: reaction-diffusion equations, boundedness, global existence, large time behaviour.

[133]

where Ω is a bounded domain in R

with boundary ∂Ω, such that ∂Ω is a C

hypersurface separating Ω from R

/Ω (m ≥ 1), d

(j = 0, 1, . . . , 11) are positive constants, f

(i = 1, 2) are given by the Arrhenius law

f

(T ) = B

exp(−E

/T ),

where B

, E

are constants, and E

denotes the activation energy.

This system of reaction-diffusion equations arises as a model of chain branching and chain breaking kinetics of reactions with complex chemistry.

Here Y

is the concentration of fuel, Y

is the concentration of radicals, and T is the dimensionless temperature. Y

, Y

and T depend on x and t where (x, t) ∈ Ω × R

.

Under suitable conditions (see (CD) in Section 3), it is expected that (1.1)–(1.5) has a unique global solution (Y

, Y

, T ) and this solution tends to an equilibrium state uniformly in x as t → ∞.

We will show that (Y

(t), Y

(t), T (t)) approaches an equilibrium state (0, 0, T

) in C

(Ω)

as t → ∞ for every µ ∈ [0, 2), where T

is a con- stant, and we will consider the rate of this convergence, by means of in- tegral equations, fractional powers of operators, Poincar´e’s inequality and some imbedding theorems.

2. Preliminary results. We state some results needed in the sequel.

Lemma 2.1. Let (E, k · k

) and (F, k · k

) be two real Banach spaces with continuous inclusion E ⊂ F . Let A be a linear operator generating a strongly continuous semigroup G(t) in E such that:

(i) G(t)E ⊂ F for all t > 0,

(ii) there exists θ ∈ [0, 1) such that kG(t)ϕk

≤ ct

kϕk

for all t > 0.

Moreover , let p > 1/(1 − θ), f ∈ L

(R

, E) and sup

kf k

< ∞. Let u be a mild solution on R

of du

dt = Au(t) + f (t).

If u ∈ L

(0, ∞; E), then u(t) ∈ F for all t > 0 and u ∈ C

(δ, ∞; F ) for all δ > 0, where C

(δ, ∞; F ) is the space of all continuous functions u : (δ, ∞) → F such that sup{ku(t)k

: t ≥ δ} < ∞.

For the proof, see [4].

Lemma 2.2. Let G(t) be the semigroup generated by the operator d∆

in L

(Ω). Then for all 1 ≤ p < q ≤ ∞ and all ϕ ∈ L

(Ω) we have G(t)ϕ ∈ L

(Ω) and

kG(t)ϕk

≤ c(p, q)t

kϕk

.

For the proof, see [2].

3. Global existence and positivity. Throughout this paper, the following assumptions are in force:

(CD) (i) d

(j = 0, 1, . . . , 11) are positive constants,

(ii) Y

, Y

and T

are nonnegative measurable functions such that 0 ≤ Y

(x), Y

(x), T

(x) ≤ M

for almost every x ∈ Ω, for some positive constant M

.

Theorem 3.1. Assume (CD). Then there exists a unique nonnegative global solution (Y

, Y

, T ) for (1.1)–(1.5) which is smooth in Ω × (0, ∞).

P r o o f. For each 1 < p < ∞ and j ∈ {1, 2, 3} define the linear operator A

on L

(Ω) by

(3.1) D(A

) = {u ∈ W

(Ω) : (∂u/∂ν)|∂Ω = 0}, A

u = d

∆u, A

u = d

∆u, A

u = d

∆u,

where W

(Ω) is the usual Sobolev space. It is well known that A

gen- erates a compact, analytic contraction semigroup G

(t), t ≥ 0, of bounded linear operators on L

(Ω) (see, e.g., Amann [2]).

For the local existence we write (1.1)–(1.3) as a system of integral equa- tions via the variation of constants formula. For simplicity we set

F

(Y

, Y

, T )(t)(·) = − d

Y

(t)Y

(t)f

(T (t))(·),

F

(Y

, Y

, T )(t)(·) = (d

Y

(t)Y

(t)f

(T (t)) − d

Y

(t)f

(T (t))

− d

Y

(t) − d

Y

(t))(·),

F

(Y

, Y

, T )(t)(·) = (d

Y

(t)Y

(t)f

(T (t)) + d

Y

(t)f

(T (t)) + d

Y

(y) + d

Y

(t))(·),

for x ∈ Ω, t > 0; we then have Y

(t) = G

(t)Y

+

t

\

0

G

(t − τ )F

(Y

(τ ), Y

(τ ), T (τ )) dτ, (3.2)

Y

(t) = G

(t)Y

+

t

\

0

G

(t − τ )F

(Y

(τ ), Y

(τ ), T (τ )) dτ, (3.3)

T (t) = G

(t)T

+

t

\

0

G

(t − τ )F

(Y

(τ ), Y

(τ ), T (τ )) dτ.

(3.4)

For each α > 0 define the operator B

= I − A

. Then the fractional

powers B

= (I −A

)

exist and are injective, bounded linear operators

on L

(Ω) (see Pazy [8]). Let B

= (B

)

and X

= D(B

), the

domain of B

. Then X

is a Banach space with the graph norm kuk

=

kB

wk

, and for α > β ≥ 0, X

is a dense subspace of X

with the

inclusion X

⊂ X

compact (we use the convention X

= L

(Ω)). Also if 0 ≤ α < 1 we have

(3.5) X

⊂ C

(Ω) for every 0 ≤ µ < mα − n/p.

Note that this inclusion is valid even if p = 1 (see Henry [5], p. 39).

In addition, G

and B

have the properties summarised in the follow- ing lemma.

Lemma 3.2. The operators G

and B

satisfy (i) G

(t) : L

(Ω) → X

for all t > 0,

(ii) G

(t)B

u = B

G

(t)u for every u ∈ X

,

(iii) kG

(t)uk

≤ C

(α)t

e

kuk

for every t > 0 and u ∈ L

(Ω), (iv) k(G

(t) − I)uk

≤ C

(α)t

kuk

for 0 < α ≤ 1 and u ∈ X

. The proof can be found in Pazy [8].

Select 0 < α < 1 and p > 1 so that (3.5) holds, and use the tech- niques of Pazy [8] to show that there exists a unique noncontinuable solution (Y

, Y

, T ) to (3.2)–(3.4) for Y

∈ X

, Y

∈ X

and T

∈ X

. The solution satisfies

Y

∈ C([0, δ]; X

) ∩ C

((0, δ); L

(Ω)), Y

∈ C([0, δ]; X

) ∩ C

((0, δ); L

(Ω)), T ∈ C([0, δ]; X

) ∩ C

((0, δ); L

(Ω)),

for some δ > 0; and we have kY

(t)k

+ kY

(t)k

+ kT (t)k

→ ∞ as t → t

if t

< ∞.

Suppose now that (Y

, Y

, T

) ∈ L

(Ω)

and let {Y

}

be a se- quence in X

, {Y

}

a sequence in X

and {T

}

a sequence in X

such that Y

, Y

, T

≥ 0 and kY

− Y

k

→ 0, kY

− Y

k

→ 0 and kT

− T

k

→ 0 as t → ∞. Using the equation (3.2) and the properties of A

stated in Lemma 3.2, it follows for α ≤ β < 1 that

kY

k

≤ C

t

kY

k

+

t

\

0

C

(t − τ )

kF

(Y

(τ ), Y

(τ ), Y

(τ ))k

dτ for all t ∈ [0, t

), where t

is the maximal time of existence for the system (1.1)–(1.5) with initial conditions 0 ≤ (Y

, Y

, T

) ∈ X

× X

× X

. From these estimates one can deduce the existence of a C

such that

max{kY

(t)k

, kY

(t)k

, kT

(t)k

} ≤ C

t

for all t ∈ [0, δ], k ≥ 1; thus {(Y

(t), Y

(t), T

(t))}

is contained in

a bounded subset of X

× X

× X

for each t ∈ (0, δ]. So by the

compact imbedding of X

in X

(j = 1, 2, 3) for α < β < 1, we see that

for each t ∈ (0, δ] the sequences {Y

(t)}

, {Y

(t)}

and {T

(t)}

contain convergent subsequences {Y

(t)}

, {Y

(t)}

and {T

(t)}

in X

, X

and X

respectively.

Now define Y

(t) = lim

i→∞

Y

(t), Y

(t) = lim

i→∞

Y

(t), T (t) = lim

i→∞

T

(t) for each t ∈ [0, δ]. Then (Y

(t), Y

(t), T (t)) satisfies (3.2)–(3.4) for each t ∈ [0, δ]. Replacing [0, t

) with [δ, t

) and (Y

, Y

, T

) by (Y

(δ), Y

(δ), T (δ)) and using the results already established when (Y

, Y

, T

) ∈ X

× X

× X

, we find that there is a unique, classical noncontinuable solution (Y

(t), Y

(t), T (t)) on Ω × [0, t

), for every (Y

, Y

, T

) ∈ (L

(Ω))

.

Since F

(0, Y

, T ) ≥ 0, F

(Y

, 0, T ) ≥ 0 and F

(Y

, Y

, 0) ≥ 0 it follows that Y

(t), Y

(t) and T (t) have nonnegative values on Ω (see [10]), and by the maximum principle we have

(3.6) kY

(t)k

≤ kY

k

for all t ∈ [0, t

).

Multiplying (1.2) by Y

and integrating the result over Ω × (0, t) we obtain

1 n

d dt

\

Ω

Y

dx ≤ c

\

Ω

Y

dx, where c = d

kY

k

kf

(T (t))k

, hence

\

Ω

Y

dx ≤ |Ω| kY

k

e

for all t < t

. We can then deduce

(3.7) kY

(t)k

≤ e

kY

k

for all t < t

.

From the expression of F

(Y

, Y

, T ) and (3.7) we can find two positive numbers c

and c

such that

(3.8) kF

(Y

(T ), Y

(T ), T (t))k

≤ e

(c

+ c

e

) for all t < t

, where c

= B

d

kY

k

+ d

B

+ d

and c

= d

kY

k

.

From (3.4) and (3.8) we obtain kT (t)k

≤ kT

k

+

t

\

0

e

(c

+ c

e

) dτ, from which we have

(3.9) kT (t)k

≤ kT

k

+ c

c (e

− 1) + c

2c (e

− 1) for all t < t

.

Inequalities (3.6), (3.7) and (3.9) contradict the fact that t

< ∞, hence

t

= ∞.

4. Boundedness of the solution. In fact, the solution obtained in Theorem 3.1 is uniformly bounded over Ω × (0, ∞).

Theorem 4.1. Assume (CD). Then there exists a positive number M such that

(4.1) (4.2)

0 ≤ Y

(x, t) ≤ kY

k

for x ∈ Ω, t ≥ 0, 0 ≤ Y

(x, t), T (x, t) ≤ M for x ∈ Ω, t ≥ 0.

P r o o f. The function Y

is uniformly bounded by kY

k

by the maxi- mum principle.

Let B(x, t) = d

Y

(x, t)f

(T (x, t))− d

f

(T (x, t))− d

− d

Y

(x, t). Then we can write

∂Y

∂t = d

∆Y

+ B(x, t)Y

with B(x, t) ≤ a (for example a = d

kY

kB

) and B(x, t) is locally Lipschitz in (x, t). Moreover, Y

∈ L

(R

, L

(Ω)). In fact, integrating (1.1) over Ω × (0, t) we obtain

(4.3)

\

Ω

Y

(x, t) dx =

\

Ω

Y

(x) dx − d

\

0

\

Ω

Y

(x, τ )Y

(x, τ )f

(T (x, τ )) dx dτ, which implies

(4.4)

t

\

0

\

Ω

(Y

Y

f

(T ))(x, τ ) dx dτ ≤ |Ω|

d

kY

k

for all t ≥ 0, where |Ω| is the Lebesgue measure of Ω. Similarly, we get (4.5)

\

Ω

Y

(x, t) dx

≤

\

Ω

Y

(x) dx + d

\

0

\

Ω

(Y

Y

f

(T ))(x, τ ) dx dτ for all t ≥ 0.

From (4.4) and (4.5) we obtain (4.6) kY

(t)k

≤ |Ω|



kY

k

+ d

d

kY

k



for all t ≥ 0.

An application of the result of Alikakos ([1], §3) shows that Y

(t) is uniformly bounded over Ω × (0, ∞):

(4.7) kY

(t)k

≤ K for all t ≥ 0,

for some K > 0.

Now, integrating (1.3) over Ω × (0, t) we obtain

\

Ω

T (x, t) dx =

\

Ω

T

(x) dx + d

\

0

\

Ω

(Y

Y

f

(T ))(x, τ ) dx dτ (4.8)

+ d

\

0

\

Ω

(Y

f

(T ))(x, τ ) dx dτ

+ d

\

0

\

Ω

Y

(x, τ ) dx dτ + d

\

0

\

Ω

Y

(x, τ ) dx dτ.

Integrating (1.2) over Ω × (0, t) we obtain (4.9)

\

Ω

Y

(x, t) dx + d

t

\

0

\

Ω

(Y

f

)(x, τ ) dx dτ + d

t

\

0

\

Ω

Y

(x, τ ) dx dτ

+ d

\

0

\

Ω

Y

(x, τ ) dx dτ = d

\

0

\

Ω

(Y

Y

)(x, τ ) dx dτ +

\

Ω

Y

(x) dx, from which we deduce that

(4.10)

∞

\

0

\

Ω

(Y

f

)(x, τ ) dx dτ < ∞ and

∞

\

0

\

Ω

Y

(x, τ ) dx dτ < ∞.

From (4.4)–(4.7) and (4.10) in (4.8) we obtain (4.11)

\

Ω

T (x, t) dx ≤ C for all t ≥ 0, i.e., T ∈ L

(R

, L

(Ω)).

To prove that T ∈ L

(R

, L

(Ω)) we distinguish two cases. We define S

(t) ≡ G

(t).

Case 1: n = 1, i.e., Ω = (a, b) ⊂ R. In this case we take E := L

(Ω) and F = C(Ω). Then Lemma 2.2 shows that

(4.12) kS

(t)ϕk

≤ ct

kϕk

for all ϕ ∈ L

(Ω).

Take α = 3/4; from Lemma 2.2 and (3.5) we have S

(t)L

(Ω) ⊂ C(Ω).

Applying Lemma 2.1, we conclude that T ∈ C

(δ, ∞; C(Ω)) for all δ > 0, hence from the result concerning the local existence we obtain

kT (t)k

≤ C for all t ≥ 0.

Case 2: n ≥ 2. Let q

= 1, q

= n/(n − r) and E = L

(Ω), F =

L

(Ω) for r ∈ {1, . . . , n−1}. We have T ∈ C

(R

, L

(Ω)), S

(t)L

(Ω) ⊂

L

(Ω) and kS

(t)ϕk

≤ ct

kϕk

. Application of Lemma 2.1 gives

T ∈ C

(R

, L

(Ω)). Next we take E = L

(Ω) and F = L

(Ω) to ob-

tain T ∈ C

(R

, L

(Ω)). Continuing this process we finally have T ∈

C

(R

, L

(Ω)). In the last iteration we take E = L

(Ω) and F = C(Ω).

As S

(t)L

(Ω) ⊂ X

and kS

(t)ϕk

≤ ct

kϕk

for all ϕ ∈ L

(Ω) and T ∈ C

(R

, L

(Ω)), from Lemma 2.1 we conclude that T ∈ C

(R

; C(Ω)).

5. Asymptotic behaviour. First, let us establish a preparatory lemma. Consider the problem

(P)  ∂u/∂t + Au = ϕ(t),

u(0) = u

,

where −A generates an analytic semigroup G(t) in a Banach space (X, k · k) with Re σ(A) > a > 0. We have the following lemma.

Lemma 5.1. Let X be a Banach space. If ϕ ∈ L

(R

, X) and the problem (P) has a bounded global solution u ∈ L

(R

, X) then for all 0 <

α < 1 we have

(A) sup

kA

u(t)k ≤ C(α, δ) for any δ > 0, and

(B) the function t 7→ A

u(t) is H¨ older continuous from [δ, ∞) to X for any δ > 0.

P r o o f. The solution u of (P) satisfies the integral equation u(t) = G(t)u

+

t

\

0

G(t − τ )ϕ(τ ) dτ, t > 0.

Applying A

to both sides yields kA

u(t)k ≤ kA

G(t)u

k +

t

\

0

kA

G(t − τ )ϕ(τ )k dτ.

From this and Lemma 3.2, we obtain kA

u(t)k

≤ C

(α)t

e

ku

k +

t

\

0

C

(α)(t − τ )

e

kϕ(τ )k dτ

≤ C

(α)ku

k + C

(α)M Γ (1 − α)a

.

Here Γ is the gamma function of Euler. Hence kA

u(t)k is uniformly bounded on [δ, ∞) for any δ > 0.

To prove (B), we have

kA

u(t + h) − A

u(t)k ≤ k(G(h) − I)A

G(t)u

k +

t+h

\

t

kA

G(t + h − τ )ϕ(τ )k dτ

+

t

\

0

k(G(h) − I)A

G(t − τ )ϕ(τ )k dτ.

Set

I

= k(G(h) − I)A

G(t)u

k, I

=

t+h

\

t

kA

G(t + h − τ )ϕ(τ )k dτ,

I

=

t

\

0

k(G(h) − I)A

G(t − τ )ϕ(τ )k dτ.

From the inequalities of Lemma 3.2, there exist two constants C

(α), C

(α) such that

I

≤ C

(α + β)C

(α)t

e

ku

kh

, I

≤ M C

(α)h

,

I

≤ M C

(α + β)C

(β)Γ (1 − α − β)a

h

,

where M = sup

kϕ(t)k

for every 0 < β < 1. Taking β < 1 − α, we then have for all t ≥ δ,

kA

u(t + h) − A

u(t)k ≤ C(α, ku

k) max{h

, h

}.

Remark . As a consequence of this lemma, the function t 7→ A

u(t) is uniformly continuous.

The following proposition is also useful in the sequel.

Proposition 5.2. For any δ > 0, the family {Y

(t) : t ≥ δ} is relatively compact in C(Ω).

P r o o f. We have ∂Y

/∂t = d

∆Y

+ F

(Y

, Y

, T ) where F

(Y

, Y

, T ) =

−d

Y

Y

f

(T ). There is a positive constant N such that kF

(Y

, Y

, T )k

≤ N for all t ≥ 0. Let 0 < ε < 1 and t > ε. Then we can write Y

(t) = G

(ε)Y

(t−ε)+[Y

(t)−G

(ε)Y

(t−ε)], where G

(t) is the semigroup generated by d

∆ with homogeneous Neumann boundary conditions in the Banach space C(Ω). We set

Y

(t) = G

(ε)Y

(t − ε) and Y

(t) = Y

(t) − G

(ε)Y

(t − ε).

Then {Y

(t) : t ≥ δ} is relatively compact in C(Ω) since {Y

(t − ε) : t ≥ δ}

is bounded and G

(δ) is a compact operator. Also, kY

(t)k

=

t

\

t−ε

G

(t − s)F

(Y

, Y

, T )(s) ds

≤ εN,

therefore {Y

(t) : t ≥ 1} is totally bounded, hence {Y

(t) : t ≥ 1} is relatively

compact in C(Ω). As {Y

(t) : δ ≤ t ≤ 1} is compact in C(Ω), it follows

that {Y

(t) : t ≥ δ} is relatively compact in C(Ω). The same holds true for

{Y

(t) : t ≥ δ} and {T (t) : t ≥ δ}.

Theorem 5.3. Under the assumptions (CD) we have

(5.1) lim

t→∞

kY

(t)k

= 0, lim

t→∞

kY

(t)k

= 0 and there exists a positive constant T

such that

(5.2) lim

t→∞

kT (t) − T

k

= 0.

P r o o f. From (1.1) we have

(5.3) d

dt

\

Ω

Y

(x, t) dx = −d

\

Ω

(Y

(t)Y

(t)f

(T (t)))(x) dx ≤ 0, hence the function t 7→

T

Ω

Y

(x, t) dx is nonincreasing. Let Y

be a constant such that

(5.4) lim

t→∞

\

Ω

Y

(x, t) dx = Y

. From (1.2) we have

(5.5) d dt

\

Ω

Y

(x, t) dx =

\

Ω

(d

Y

Y

f

(T )−d

Y

f

(T )−d

Y

−d

Y

)(x, t) dx.

From (5.3) and (5.5) we deduce (5.6) d

dt

\

Ω

 1 d

Y

+ 1 d

Y



(x, t) dx

= −

\

Ω

 d

d

Y

f

(T ) + d

d

Y

+ d

d

Y



(x, t) dx ≤ 0, from which we infer that there is a constant K such that

(5.7) 1

d

\

Ω

Y

(x, t) dx + 1 d

\

Ω

Y

(x, t) dx → K as t → ∞.

Combining (5.1) and (5.7) we conclude that there is a positive constant Y

such that

(5.8) lim

t→∞

\

Ω

Y

(x, t) dx = Y

.

Integrating (5.6) over (0, ∞) we conclude that there is a constant C such that

(5.9)

∞

\

0

\

Ω

Y

(x, τ ) dx dτ ≤ C.

Combining (5.8) and (5.9) we find that Y

= 0, whence

(5.10) lim

t→∞

\

Ω

Y

(x, t) dx = 0.

As Y

(x, t) ≥ 0, the invariance principle of La Salle [5] and (5.10) imply lim

kY

(t)k

= 0.

Multiplying (1.1) by Y

and integrating over Ω and using Poincar´e’s inequality we obtain

d dt

\

Ω

Y

(x, t) dx ≤ −c

\

Ω

Y

(x, t) dx for some positive constant c > 0, from which we deduce (5.11) kY

(t)k

≤ e

kY

k

.

Also, as a consequence of the maximum principle we have (5.12) kY

(t)k

≤ kY

(s)k

for t ≥ s > 0.

According to Proposition 5.2, {Y

(t) : t ≥ δ} is relatively compact in C(Ω) for all δ > 0; so from this, (5.11) and (5.12) we have

(5.13) lim

t→∞

kY

(t)k

= 0.

Multiplying (1.2) by Y

and integrating over Ω × (0, t) we have (5.14) kY

(t)k

+ 2d

t

\

0

k∇Y

(τ )k

dτ + 2d

\

0

\

Ω

Y

f

(T ) dx dτ

+ 2d

\

0

kY

(τ )k

dτ + 2d

\

0

\

Ω

Y

dx dτ

= kY

k

+ 2d

\

0

\

Ω

Y

Y

f

(T ) dx dτ.

Similarly for (1.3), (5.15) kT (t)k

+ 2d

t

\

0

k∇T (τ )k

dτ

= kT

k

+ 2d

\

0

\

Ω

Y

Y

T f

(T ) dx dτ

+ 2d

\

0

\

Ω

Y

T f

(T ) dx dτ

+ 2d

\

0

\

Ω

Y

T dx dτ + 2d

\

0

\

Ω

Y

T dx dτ.

By (4.4) and as Y

, Y

and T are uniformly bounded, it follows from (5.14)

and (5.15) that ∇Y

, ∇T ∈ L

(R, L

(Ω)), i.e.

(5.16)

∞

\

0

k∇Y

(τ )k

dτ < ∞,

∞

\

0

k∇Y

(τ )k

dτ < ∞,

∞

\

0

k∇T (τ )k

dτ < ∞.

For the equation (1.1) for example, we define the operator B

as follows:

D(B

) = {u ∈ W

(Ω) : (∂u/∂ν)|∂Ω = 0}, B

u = (−d

∆ + a)u, with a fixed positive real number a > 0. It is well known that −B

generates an analytic semigroup and Re σ(B

) > a > 0. Also, if we set ϕ(t) = aY

(t)+

F

(Y

, Y

, T )(t), then ϕ ∈ L

(R

, L

(Ω)). Application of Lemma 5.1 then implies

(5.17) sup

t≥δ

kB

Y

(t)k

≤ C(p, α, δ) for any δ > 0, and

(5.18) t 7→ B

Y

(t) is uniformly continuous from [δ, ∞) to L

(Ω) for any δ > 0.

The same holds for Y

and T .

By (5.18) we find that t 7→ k∇Y

(t)k

, t 7→ k∇Y

(t)k

and t 7→ k∇T (t)k

are uniformly continuous on [δ, ∞) by choosing p = 2 and suitable α ∈ (0, 1) and m. From this and (5.16), Lemma 5.1 gives

(5.19) lim

t→∞

k∇Y

(t)k

= 0, lim

t→∞

k∇Y

(t)k

= 0, lim

t→∞

k∇T (t)k

= 0.

The interested reader can see [7] for details.

Since {T (t) : t ≥ δ} is compact in C(Ω) it follows that there is a sequence {t

} such that

tk

lim

T (t

) = T

in C(Ω),

where T

is a constant. Owing to the Poincar´e inequality (see [11]) we have λ

T (t) − |Ω |

−1

\

Ω

T (x, t) dx

2

2

≤ k∇T (t)k

.

Here λ is the smallest positive eigenvalue of −∆ with homogeneous Neumann boundary conditions on ∂Ω. Since the limit T

is uniquely determined we have

t→∞

lim T (t) = T

in C(Ω).

6. Rates of convergence. In this section we study the rates of con- vergence obtained in Theorem 5.3.

Theorem 6.1. Assume (CD). Then for given µ ∈ [0, 2), there exist

K

(µ), K

(µ), K(µ) > 0 and ̺, σ, ω > 0 such that

kY

(t)k

≤ K

(µ)e

, kY

(t)k

≤ K

(µ)e

, kT (t)k

≤ K(µ)e

,

for some t

> 0, as t → ∞, where 0 < σ < d

, ̺ = min{σ, d

λ}, ω = min{σ, d

λ} and λ is the smallest positive eigenvalue of −∆ with homoge- neous Neumann boundary condition on ∂Ω.

Let us recall the following two lemmas.

Lemma 6.2. For 1 < p < ∞ and d > 0, let L

be the operator defined by D(L

) = {u ∈ W

(Ω) : (∂u/∂ν)|∂Ω = 0}, L

u = −d∆u, and let the operators Q

, Q

: L

(Ω) → L

(Ω) be defined by

Q

u = 1

|Ω|

\

Ω

u(x) dx, Q

u = u − Q

u.

Define the operator L

as L

≡ L

|Q

L

(Ω), the restriction of L

to Q

L

(Ω). Then there exists a constant C

(α) > 0 such that for u ∈ L

(Ω) and t > 0,

kL

e

Q

uk

≤ C

(α)q(t)

e

kQ

uk

,

where q(t) = min{t, 1} and λ is the smallest positive eigenvalue of −∆ with homogeneous Neumann boundary conditions on ∂Ω.

Lemma 6.2 is proved by Rothe [9].

Lemma 6.3. For α ∈ [0, 1) and β > 0, there exists a constant C(α, β) > 0 such that

t

\

0

q(ξ)

e

dξ ≤ C(α, β)e

. For the proof, see [6].

Proof of Theorem 6.1. For 1 < p < ∞ we take the operators D(A

) = D(B

) = D(R

) = {u ∈ W

(Ω) : (∂u/∂ν)|∂Ω = 0}, A

u = −d

∆u, B

= −(d

∆ − d

)u, R

= −d

∆u.

By Theorem 5.3 we already know that Y

(t) → 0 and Y

(t) → 0 in C(Ω) as t → ∞, hence for any ε > 0 there exists a constant t

> 0 such that (6.1) d

Y

f

(T ) < ε for all t ≥ t

.

We take 0 < ε < d

.

(I) The decay rate of kY

(t)k

. From (1.2) and (6.1) we get

(6.2) ∂Y

∂t ≤ d

∆Y

− (d

− ε)Y

, t > t

.

Multiplying both sides by Y

for p ∈ [1, ∞), integrating over Ω and using Green’s formula, we obtain

(6.3) d

dt kY

(t)k

≤ −p(d

− ε)kY

(t)k

for t > t

, which leads to

kY

(t)k

≤ kY

(t

)k

e

for t > t

. The H¨older inequality then yields

(6.4) kY

(t)k

≤ M |Ω|

e

for t > t

, where M is the positive number appearing in (4.2).

(II) The decay rate of kY

(t)k

. To investigate the decay rate of kY

(t)k

, we treat the following integral equation which is equivalent to (1.2) with (1.4) for t > t

:

∂Y

∂t = d

∆Y

− d

Y

+ F (Y

, Y

, T ),

where F (Y

, Y

, T ) = d

Y

Y

f

(T ) − d

Y

f

(T ) − d

Y

. Let G

(t) be the semigroup generated by −B

. Then

(6.5) Y

(t) = G

(t − t

)Y

(t

) +

t

\

t^∗

G

(t − τ )F (τ ) dτ, t > t

. From Lemma 3.2(iii), we obtain

(6.6) kB

Y

(t)k

≤ C

(α)q(t − t

)

e

kY

(t

)k

+ d

M B

J

(t), where

J

(t) =

t

\

t^∗

kG

(t − τ )k

kY

(τ )k

dτ.

It is sufficient to estimate J

(t). By (6.4) and Lemmas 3.2 and 6.3 we have

J

(t) ≤ C

(α)M |Ω|

t−t\^∗

0

q(t − t

− τ )

e

dτ (6.7)

≤ C

(α)M |Ω|

e

for t ≥ t

.

Consequently, the imbedding D(B

) ⊂ C

(Ω) ensures the existence of a

constant K

(µ) > 0 such that for every 0 < σ < d

there is t

> 0 such that

(6.8) kY

(t)k

≤ K

(µ)e

for t > t

.

(III) The decay rate of kY

(t)k

. First we write Y

(t) = Q

Y

(t) + Q

Y

(t), where

Q

Y

(t) = 1

|Ω|

\

Ω

Y

(x, t) dx, Q

Y

(t) = Y

(t) − Q

Y

(t).

We see from (4.3) and the fact that Y

→ 0 as t → ∞ in C(Ω) that Q

Y

(t) ≡ 1

|Ω|

\

Ω

Y

(x, t) dx (6.9)

= 1

|Ω|

\

Ω

Y

(x, t) dx − d

|Ω|

t

\

0

\

Ω

(Y

Y

f

)(x, τ ) dx dτ

= d

|Ω|

∞

\

t

\

Ω

(Y

Y

f

)(x, τ ) dx dτ

≤ d

|Ω| B

M

∞

\

t

\

Ω

Y

(x, τ ) dx dτ

≤ d

B

M

∞

\

t

e

dτ

≤ 1

̺ d

B

M

e

for t > t

.

Next, we study the decay of Q

Y

(t). We consider the integral equation associated with (1.1) and apply A

Q

to get

A

Q

Y

(t) = G

(t − t

)Q

Y

(t

) − d

\

t^∗

G

(t − τ )(Q

Y

Y

f

)(τ ) dτ for t > t

. By Lemma 6.2, we get

kA

Q

Y

(t)k

≤ C

(α)q(t − t

)

e

kQ

Y

(t

)k

(6.10)

+ M B

d

C

(α)kQ

kJ

(t) for t > t

, where J

(t) =

Tt

t^∗

q(t−τ )

e

kY

(τ )k

dτ for t > t

and kQ

k is the norm of the linear operator Q

: L

(Ω) → L

(Ω). Here Q

Y

(t

) ∈ D(A

) because t > t

and by the smoothness of Y

(t

). By Lemma 6.3,

J

(t) =

t

\

t^∗

q(t − τ )

e

kY

(τ )k

dτ (6.11)

≤ M |Ω|

t−t^∗

\

0

q(ξ)

e

dξ

≤ C(α, d

λ)e

for t > t

.