EXISTENCE OF A SOLUTION IN AN AGE
DEPENDENT TRANSPORT-DIFFUSION
PDE: A MODEL OF SETTLER
J.Ph. Chancelier, M. Cohen de Lara and F.Pacard
Cergrene. ENPC, La Courtine,
93167 Noisy le Grand Cédex, France.
e-mail: [email protected]
Abstract
The modeling of sludge particles settling in the final stage of a
waste water treatment plant may include both transport and diffusion.
When the residence time of sludge particles in the settler is considered,
this leads to a nonlinear age dependent transport-diffusion partial differential equation (PDE) with nonlocal condition. We investigate the
question of existence of a solution.
1
Introduction and Main Assumptions
The following PDE with non local condition

∂u ∂ 2 u
∂



 ∂a − ∂x2 + ∂x (V (a, x, c(x))u) = 0 with u(0, x) = u0 (x)
Z +∞




u(a, x)da = c(x)
0
1
(1)
is a possible model for the steady states of a settler (in a waste water treatment plant). This kind of equations is introduced in [1] as an extension
of traditional settling models (see [3] and the references in [1]) to take the
residence time in the settler into account.
2
Remark. The variable c(x) represents the sludge concentration at
point x and u(a, x) the density of sludge particles having spent time a in
the settler. The settling mechanism is modeled in the term V , or sludge
velocity. It includes the water velocity (depending on the geometry of the
settler and the output and recycle flows) and the settling velocity due to
gravity (depending on the concentration c(x) and on the “age” a of the sludge
particles). We assume that, outside the settler, the particles’ transport is only
2
due to water flows and thus constant. The ∂∂xu2 term accounts for diffusion
in the settler.
In the spirit of previous works on age dependent population models such
as in [5, 2], we shall prove that the previous problem has at least one solution.
We shall not address the question of uniqueness (or non uniqueness) of the
solution, which remains an open problem.
Remark. The age dependent population model (1) differs from those
in [2, 5] in that the population growth term is replaced here by a transport
term. It also differs by the boundary conditions.
The following assumptions are natural for our settling problem.
(H1) V is a smooth bounded function, Lipschitz with respect to c :
∀ a ∈ R+ , ∀ x ∈ R,
|V (a, x, c) − V (a, x, c′ )| ≤ α0 |c − c′ |.
(H2) There exists M > 0 such that

 V (a, x, c) = V− < 0 for x ≤ −M

V (a, x, c) = V+ > 0 for x ≥ M
(H3) There exists a function H(x, c) such that
φ(A) =
sup
|H(x, c) − W (a, x, c)| →A→+∞ 0
x∈R,c≥0,a≥A
(H4) The source function u0 (x) is non negative and smooth with compact
support.
3
The first and fourth assumptions are technical, while the second one
means that the particles right above (resp. under) the settler have an upward
(resp. a downward) fixed velocity and the third one that the age, when it is
large enough, has no influence on the velocity.
Our main result concerns the existence of solutions of (1) under the above
assumptions.
Theorem 1.1 Under the above assumptions (H1)-(H4), there exists a solution of (1). What is more, u is non negative, u ∈ C 0 (R+ × R) and c ∈ C 1 (R).
2
Preliminary Estimates
When looking at (1), it is worthwile to note that the velocity term V (a, x, c)
is not necessarily smooth because c may not be regular. This is why, in this
preliminary section, we state some existence results and estimations concerning the linear heat equation with bounded coefficients. If the existence
result is indeed well known in a bounded domain [4], it seems to the authors
knowledge that the uniform estimate depending only on the L∞ norm of the
coefficients does not appear in classical references.
Proposition 2.1 For any function W (a, x) ∈ C 0 (R+ × R), there exists a
unique weak solution u(a, x) ∈ C 0 (R+ × R) to the problem
∂u ∂ 2 u ∂(W u)
−
+
=0
∂a ∂x2
∂x
(2)
with the initial condition given by u(0, x) = u0 (x). Moreover, u is non
negative and there exists a constant α1 > 0, depending only on ||W ||L∞ ,
such that the following estimate is satisfied for all a0 > 0:
sup ||u||L∞ (R) ≤ α1a0 +1 ||u0 ||L∞ (R)
(3)
0≤a≤a0
Proof. For the time being, we assume that W is a sufficiently regular
function (at least C 1,α ), which allows us to state the existence and uniqueness
of a (non negative) solution to (2) (see [4] for instance). In addition, we can
write
Z a Z +∞
Z +∞
∂
Γ(a−τ, x−y) (W (τ, y)u(τ, y))dydτ
Γ(a, x−y)u0 (y)dy−
u(a, x) =
∂y
−∞
0
−∞
4
where we have defined
Γ(τ, y) ≡
1
2(πτ )
1
2
e−
|y|2
4τ
.
By an integration by parts, we find the following representation of the solution u
Z a Z +∞
Z +∞
x−y
Γ(a, x−y)u0 (y)dy+
u(a, x) =
Γ(a−τ, x−y)W (τ, y)u(τ, y)dydτ
−∞ 2(a − τ )
0
−∞
and, after a change of variables, we get:
Z aZ
Z +∞
Γ(a, x−y)u0 (y)dy+
u(a, x) =
−∞
0
+∞
−∞
ze−z
2
(πτ )
1
2
1
[W u](a−τ, x−2τ 2 z)dzdτ
(4)
When the function W is only continuous, we notice that a solution of (4)
is a solution of (2) in the weak sense. Thus, it is enough to prove the existence
of a solution of (4) for small ages and then show that this latter can be defined
for all age.
The existence of a solution of (4) for small ages is obtained in the space
Ea0 = {u(a, x) ∈ C 0 ([0, a0 ] × R) |
sup ||u||L∞ (R) < +∞}.
0≤a≤a0
By (4), it is appropriate to introduce the operator Π on Ea0 defined by
Z +∞
Z a Z +∞ −z2
1
ze
2
Π(u) ≡
Γ(a, x − y)u0 (y)dy +
1 [W u](a − τ, x − 2τ z)dzdτ
−∞
0
−∞ (πτ ) 2
(5)
for all u ∈ Ea0 . It is clear that this operator maps Ea0 into itself for all a0 > 0.
Moreover, we have the estimates
1
sup ||Π(u)||L∞ (R) ≤ ||u0 ||L∞ (R) + 2(a0 /π) 2 ||W ||L∞ sup ||u||L∞ (R)
0≤a≤a0
0≤a≤a0
and also
1
sup ||Π(u1 ) − Π(u2 )||L∞ (R) ≤ 2(a0 /π) 2 ||W ||L∞ sup ||u1 − u2 ||L∞ (R)
0≤a≤a0
0≤a≤a0
1
by an easy upper estimate in (5). Thus, for 2(a0 /π) 2 ||W ||L∞ < 1, the operator Π is a contraction from Ea0 into itself and this ensures existence and
uniqueness of a solution of (2) for small ages.
5
What is more, since it can be noticed that the interval of existence of the
solution does not depend on the norm of the initial data, the solution can be
extended in a unique way to all ages and satisfies, for all a0 > 0,
1
sup ||u||L∞ (R) ≤ ||u0 ||L∞ (R) + 2(a0 /π) 2 ||W ||L∞ sup ||u||L∞ (R)
0≤a≤a0
0≤a≤a0
1
For a0 > 0 such that 2(a0 /π) 2 ||W ||L∞ < 1/2, we find that sup0≤a≤a0 ||u||L∞ (R) ≤
2||u0 ||L∞ (R) . By recursively using this latter estimate as initial condition
in (2), it is easy to get (3).
The following result will be useful in the sequel.
Lemma 2.2 Let W1 , W2 ∈ C 0 (R+ × R) and u1 , u2 the corresponding solutions of (2). For any β1 > 0, there exists a constant α2 > 0 such that, if
||Wi ||L∞ < β1 for i = 1, 2, then
sup ||u2 − u1 ||L∞ (R) ≤ α22a0 +1 ||W2 − W1 ||L∞ ||u0 ||L∞ (R)
(6)
0≤a≤a0
Proof. The function w ≡ u1 − u2 is weak solution of
∂w ∂ 2 w ∂(W1 u1 − W2 u2 )
−
= 0,
+
∂a
∂x2
∂x
with initial data w(0, x) = 0. As seen above, we get the following estimates
for a1 < a2
supa1 ≤a≤a2 ||w||L∞ (R)
1
1 2
) supa1 ≤a≤a2 ||W1 u1 − W2 u2 ||L∞ (R)
≤ ||w||L∞ (R) | + 2( a2 −a
π
a1
1
1 2
sup
≤ ||w||L∞ (R) | + 2( a2 −a
||(W
−
W
)u
||
+
||W
w||
)
1
2 2 L∞ (R)
1
a1 ≤a≤a2
L∞ (R)
π
a1
By (3), u1 and u2 are bounded in C 0 ([0, a0 ] × R) by α1a0 +1 ||u0 ||L∞ (R) , so that
for a2 − a1 small enough, a1 , a2 ≤ a0 , we have:
a0
sup ||w||L∞ (R) ≤ α3 ||w||L∞ (R) | + ||W2 − W1 ||L∞ α1 ||u0 ||L∞ (R) .
a1
a1 ≤a≤a2
By recursively using this latter estimate (starting from a non zero initial
condition at “age” a1 ) as in the previous proposition, we can see that there
exists a constant α2 > 0, depending only upon β1 , such that
sup ||w||L∞ (R) ≤ α22a0 +1 ||W2 − W1 ||L∞ ||u0 ||L∞ (R)
0≤a≤a0
and the lemma is proved.
6
3
A Priori Estimates
The following proposition is a step towards the proof of Theorem 1.1.
Proposition 3.1 Let W (a, x) ∈ C 0 (R+ × R) be a bounded function such
that
• W (a, x) does not depend on (a, x) for |x| ≥ M ,
• W (a, x) = W+ > 0 for x ≥ M and W (a, x) = W− < 0 for x ≤ −M .
Let u(a, x) ∈ C 0 (R+ × R) be the solution of (2) with the initial condition
given by u(0, x) = u0 (x) having support in [−M, +M ]. Then
1. the following convergence is uniform in C 0 -norm on every compact of
R
Z A
Z +∞
u(a, x)da →A→+∞
u(a, x)da
0
0
and the limit function
c(x) =
and its derivative
dc
dx
Z
+∞
u(a, x)da
(7)
0
are bounded by a constant C(M, ||u0 ||L1 (R) , ||W ||L∞ , W+ , W− ),
2. the function c(x) is of class C 1 , is constant outside [−M, M ] and satisfies
Z +∞
d2 c
d
− 2 (x) +
W (a, x)u(a, x)da = u0 (x)
(8)
dx
dx
0
in the weak sense.
Proof. Let A > 0 and
cA (x) =
Z
A
u(a, x)da
0
By (3), it is clear that cA (x) is bounded independently of x. To prove point 1,
we shall show that cA (x) and dcdxA (x) are bounded independently of A. The
basic relation upon which we shall dwell is obtained by integrating (2) between 0 and x and 0 and A, that is
Z A
dcA
−
(x) +
W (a, x)u(a, x)da = FA (x) + kA
(9)
dx
0
7
where kA is a constant and
FA (x) =
Z
x
(u(0, y) − u(A, y)) dy
0
By a well known property of solutions of (2), we have
Z +∞
Z +∞
u0 (y)dy = ||u0 ||L1 (R) (10)
u(A, y)dy ≤ FA (y) ≤
−||u0 ||L1 (R) = −
−∞
−∞
We first study the case |x| ≥ M .
Let x ≥ M . By (9) and the assumption on W , we see that cA (x) satisfies
−
dcA
(x) + W+ cA (x) = FA (x) + kA
dx
The solution to this latter ODE is given by
Z x
−W+ x
−W+ M
e−W+ y (FA (y) + kA )dy
e
cA (x) = e
cA (M ) −
M
Now, since cA (x) is bounded independently of x, we have e−W+ x cA (x) → 0
when x → +∞, so that
Z +∞
W+ x
cA (x) = e
e−W+ y) (FA (y) + kA )dy
(11)
x
Since cA (x) ≥ 0 in (11), we get
R +∞ −W y
−e + FA (y)dy
kA ≥ sup x R +∞
≥ −||u0 ||L1 (R)
−W+ y dy
x≥M
e
x
by (10)
For x ≤ −M , we get
cA (x) = −e
W− x
Z
x
e−W− y (FA (y) + kA )dy
−∞
and, as well,
kA ≤ inf
x≤−M
Rx
−e−W− y FA (y)dy
Rx
≤ ||u0 ||L1 (R)
−W− y dy
e
−∞
−∞
8
(12)
so that both estimates on kA give
|kA | ≤ ||u0 ||L1 (R)
(13)
Thus, by (11), (12), (10) and (13), we get
sup cA (x) ≤
|x|≥M
2||u0 ||L1 (R)
inf(W+ , −W− )
What is more, (9) yields
sup |
|x|≥M
dcA
(x)| ≤ 2||u0 ||L1 (R) + ||W ||L∞ sup cA (x)
dx
|x|≥M
We now study the case |x| ≤ M .
Let x ∈ [−M, +M ]. By (9) and the previous estimates, we get
|
dcA
(x)| ≤ ||W ||L∞ cA (x) + 2||u0 ||L1 (R)
dx
so that |cA (x)| is bounded by a constant depending upon M , ||W ||L∞ , ||u0 ||L1 (R) ,
cA (−M ) and cA (M ) and the same property holds for | dcdxA (x)|.
Thus, combining results for |x| ≥ M and |x| ≤ M , we find that
||cA ||L∞ (R) + ||
dcA
||
≤ C(M, ||u0 ||L1 (R) , ||W ||L∞ , W+ , W− )
dx L∞ (R)
(14)
This proves that the sequence of functions (cA )A≥0 is bounded in C 1 (R).
Thus, by the Ascoli theorem as well as a diagonal argument, there exists a
continuous function c(x) which is the limit of (cAn )n≥0 in C 0 norm on every
compact of R (where An goes to +∞ with n). Taking advantage from the
fact that A → cA is increasing, we get that (cA )A≥0 converges towards c in
C 0 norm on every compact of R. We now prove that the function c(x) thus
defined satisfies (8) (in the weak sense).
For this, we integrate (2) between 0 and A and estimate it against a test
function φ(x) to get
R
R R A
2
dφ
− cA (x) ddxφ2 (x)dx −
W
(a,
x)u(a,
x)da
(x)dx
dx
0
(15)
R
=
(u0 (x) − u(A, x))φ(x)dx
9
When A → +∞, the first term converges to
Z
d2 φ
c(x) 2 (x)dx
dx
since cA → c in C 0 norm on every compact of R. By a similar argument, the
second term converges to
Z Z +∞
dφ
W (a, x)u(a, x)da
(x)dx
dx
0
since
Z
A
Z
+∞
W (a, x)u(a, x)da|
W (a, x)u(a, x)da −
0
0
Z +∞
≤ ||W ||L∞
u(a, x)da = ||W ||L∞ (c(x) − cA (x))
|
(16)
A
Now, we prove that there exists a sequence (An ) such that
Z
u(An , x)φ(x)dx →n→+∞ 0
RA
R
R
da dx u(a, x)φ(x) is bounded by ||c||L∞ (R) |φ(x)|dx, there exists a
R
R A′
sequence (A′n ) such that ( 0 n da dx u(a, x)φ(x))n is a convergent sequence.
In particular, it is necessary that
Z
lim
inf
u(A′n , x)φ(x)dx = 0
′
Since
0
An →+∞
R
and thus there exists a subsequence (An ) such that u(An , x)φ(x)dx → 0.
Applying these convergence results to (15), we see that c(x) satisfies (8) in
the weak sense.
Now, from (8), we find that
Z x
Z +∞
dc
dc
u0 (z)dz
W (a, x)u(a, x)da −
(x) =
(0) +
dx
dx
0
0
Thus, to prove that c(x) is of class C 1 , it is enough to show that
Z +∞
W (a, x)u(a, x)da
x 7→
0
10
RA
is continuous. This is clear since x 7→ 0 W (a, x)u(a, x)da is continuous
for
R +∞every A > 0 and since we have just seen by (16) that the remainder
W (a, x)u(a, x)da converges uniformly towards zero on every compact of
A
R when A → +∞.
By (14), we have
||c||L∞ (R) + ||
dc
||
≤ C(M, ||u0 ||L1 (R) , ||W ||L∞ , W+ , W− )
dx L∞ (R)
(17)
To end up, we prove that c(x) is constant outside [−M, M ]. By the
assumptions on W and (8), we have
∀x≥M,
−
d2 c
dc
= 0 and ∀ x ≤ −M ,
+ W+
2
dx
dx
−
d2 c
dc
=0
+ W−
2
dx
dx
Since c(x) is bounded and the only bounded solutions of these latter ODE’s
are constant functions, the result follows.
4
The Existence Result
In this section, we prove the existence of a solution of (1) by Schauder’s fixed
point theorem on the bounded closed convex set
ΣK = {c ∈ C 0 (R) | ||c||L∞ (R) ≤ K}
For all c ∈ ΣK , we can define thanks to Proposition 2.1 the solution uc ≡ S(c)
of
∂u ∂ 2 u ∂(V (a, x, c(x))u)
−
+
= 0 , u(0, x) = u0 (x)
(18)
∂a ∂x2
∂x
We can also define the operator I(·) by
Z +∞
uc (a, x)da
I(u)(x) ≡
0
The proof of Theorem 1.1 is a consequence of Schauder’s fixed point
theorem and of the following proposition.
11
Proposition 4.1 There exists a constant K0 > 0 such that, for all K > K0 ,
the operator I ◦ S maps ΣK into itself and is continuous and compact. A
solution c(x) of c = I ◦ S(c) given by Schauder’s fixed point theorem is
necessarily of class C 1 and constant outside [−M, M ].
The proof consists of the two following lemmas.
Lemma 4.2 The operator I ◦ S is continuous on ΣK .
Proof. By Proposition 3.1 with W (a, x) ≡ V (a, x, c(x)), we know that
c̃ = I ◦ S(c) is a continuous function, constant outside [−M, M ]. Thus, we
just need to prove that the following mapping is continuous:
c ∈ ΣK 7→ I ◦ S(c)|[−M,M ]
Let c1 ∈ ΣK and c2 ∈ ΣK be given. We note u1 = S(c1 ), u2 = S(c2 ) and
c̃1 = I(u1 ), c̃2 = I(u2 ). We also note
Wi (a, x) = V (a, x, ci (x)) for i = 1, 2
First, we estimate ∆c̃(x) = c̃2 (x) − c̃1 (x). Using (8), we see that there
exists a constant α such that :
Z +∞
d∆c̃
−
(x) +
(W2 (a, x)u2 (a, x) − W1 (a, x)u1 (a, x))da = α
(19)
dx
0
With H2 (x) = H(x, c2 (x)), this equation leads to
−
d∆c̃
(x) + H2 (x)∆c̃(x) = Γ(x) + α
dx
(20)
where
Γ(x) =
+
Z
Z
+∞
((H2 (x) − W2 (a, x))(u2 (a, x) − u1 (a, x))) da
0
(21)
+∞
((W1 (a, x) − W2 (a, x))u1 (a, x)) da
0
Since the function ∆c̃ is of class C 1 and is constant outside [−M, M ], it must
(±M ) = 0, and α is a constant which must be fixed to fulfill these
satisfy d∆c̃
dx
conditions. We shall prove that
||∆c̃||L∞ ([−M,M ]) ≤ C(M, ||V ||L∞ , V + , V − )||Γ||L∞ ([−M,M ])
12
(22)
Integrating (20), we easily find that there exists a constant β such that
∆c̃(x) = −αt1 (x) + βt2 (x) + t3 (x)
with
(23)

Z x
Z x


H2 (z)dz)dy
exp(
t1 (x) =



y
−M


Z x

H2 (z)dz)
t2 (x) = exp(

−M


Z x
Z x




H2 (z)dz)Γ(y)dy
exp(
 t3 (x) = −
−M
By (20), the boundary conditions
(24)
y
d∆c̃
(±M )
dx
∆c̃(±M ) =
= 0 may be replaced by
Γ(±M ) + α
H2 (±M )
since H2 (±M ) 6= 0 (for H2 (M ) = H(M, c2 (M )) = lima→+∞ V (a, M, c2 (M )) =
V+ > 0 and H2 (−M ) = V− < 0). Thus, α and β are determined by the
relations
∆c̃(±M ) = −αt1 (±M ) + βt2 (±M ) + t3 (±M ) =
Γ(±M ) + α
H2 (±M )
and, since t1 (−M ) = 0 and t2 (−M ) = 1 (see (24)), by the linear system
Γ(M ) − V+ t3 (M )
α
−(1 + V+ t1 (M )) t2 (M )V+
=
Γ(−M ) − V− t3 (−M )
β
−1
V−
It is easily seen by the expressions of t1 (x) and t2 (x) that each element of the
above matrix is bounded by a constant C(M, ||V ||L∞ ), while the determinant
(−V− ) + t1 (M )V+ × (−V− ) + t2 (M )V+
is bounded below by a positive constant C(M, ||V ||L∞ , V+ , V− ), since −V− , V+
are positive and t1 (M ) and t2 (M ) are bounded below by a positive constant
C(M, ||V ||L∞ ). On the other hand, since by (24)
||t3 ||L∞ ([−M,M ]) ≤ C(M, ||V ||L∞ )||Γ||L∞ ([−M,M ])
we can easily estimate the right hand side of the above matrix equation to
get
|α| + |β| ≤ C(M, ||V ||L∞ , V+ , V− )||Γ||L∞ ([−M,M ])
13
Combined with (23), this gives (22).
Let ∆c(x) = c2 (x) − c1 (x). Now, to prove the continuity, it suffices to
show that ||Γ||L∞ ([−M,M ]) goes to zero with ||∆c||L∞ . For this, we split Γ(x)
defined in (21) in three parts.
First, by assumption (H1) and (6), we get
Z A
(H2 (x)−W2 (a, x))(u2 (a, x)−u1 (a, x))| ≤ 2A||V ||L∞ α22A+1 ||u0 ||L1 (R) α0 ||∆c||L∞
|
0
where α2 depends only on ||V ||L∞ .
Second, by assumption (H3) and (17), we get
Z +∞
(H2 (x)−W2 (a, x))(u2 (a, x)−u1 (a, x))da ≤ (||c1 ||L∞ +||c2 ||L∞ )φ(A) ≤ 2Kφ(A)
|
A
where φ(A) depends only on A and goes to zero as A tends to +∞.
Third, by assumption (H1) and (17), we get:
Z +∞
(W1 (a, x) − W2 (a, x))u1 (a, x)da| ≤ α0 ||∆c||L∞ ||c1 ||L∞ ≤ Kα0 ||∆c||L∞
|
0
Thus, by (21), we get:
||Γ(x)||L∞ ≤ C(K, ||u0 ||L1 (R) , ||V ||L∞ , α0 , A)||∆c||L∞ + 2Kφ(A)
By first fixing A large enough, then ||∆c||L∞ small enough, we can easily
show that ||Γ||L∞ ([−M,M ]) goes to zero with ||∆c||L∞ .
Lemma 4.3 The operator I ◦ S maps ΣK into ΣK0 (for K0 > 0 depending
upon M , ||u0 ||L1 (R) , ||V ||L∞ , V+ , V− ) and is compact.
Proof. From the estimate (17), we see that c̃ = I ◦ S(c) is such that
||c̃||L∞ (R) ≤ K0 = C(M, ||u0 ||L1 (R) , ||V ||L∞ , V+ , V− )
since V (a, x, c) is bounded by assumption (H1).
Let (cn )n be a sequence in ΣK . We note un = S(cn ) and c̃n = I(un ) ∈
1
C (R). Since c̃n (x) = c̃n (M ) for x ≥ M and c̃n (x) = c̃n (−M ) for x ≤ −M ,
n
it suffices to show that the sequence ( dc̃
) is bounded on [−M, M ] to prove
dx n
the compactness of I ◦ S. This follows from the estimate (17).
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5
Conclusion
Following our study of a PDE model of settler in [1], we were asked by
practitioners to see if we could evaluate the residence time of sludge particles
in the settler. The task is challenging since the introduction of the new
variable “age” a considerably changes the mathematical methods of study.
Our result in this paper is a first attempt in this direction. Analysis of
numerical schemes are currently investigated and may lead to new discussions
with waste water treatment plant engineers. The question of uniqueness,
clearly of both practical and theoretical interest, is still open.
References
[1] J. Ph. Chancelier, M. Cohen de Lara, and F. Pacard. Analysis of a
conservation pde with discontinuous flux: a model of settler. SIAM J. of
Applied Math., 54(4):954–995, 1994.
[2] M. Kubo and M. Langlais. Solutions périodiques pour un problème de
dynamique des populations. C. R. Acad. Sci. Paris Sér. I Math., 313:387–
390, 1991.
[3] G.J. Kynch. A theory of sedimentation. Transactions of the Faraday
Society, 48:166, July 1952.
[4] O. Ladyzhenskaya, V. Solonnikov, and N. Uraltseva. Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, Rhode Island, 1968.
[5] G. Webb. Theory of nonlinear age dependant population dynamics. Pure
and Applied Math. Series. 89. Marcel Dekker, New York, 1985.
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