On a balance condition for a class of PDEs (PDF)

On a ”balance” condition for a class of PDEs
including porous medium and chemotaxis effect:
non-autonomous case
Messoud Efendiev
Helmholtz Center Munich, Institute of Biomathematics and Biometry,
85764 Neuherberg Germany
e-mail: messoud.efendiyev@helmholtz-muenchen.de

Anna Zhigun
Helmholtz Center Munich, Institute of Biomathematics and Biometry,
85764 Neuherberg Germany
e-mail: anna.zhigun@helmholtz-muenchen.de

Abstract
In this article we deal with a class of degenerate parabolic systems
that encompasses two different effects: porous medium and chemotaxis.
Such classes of equations arise in the mesoscale level modeling of biomass
spreading mechanisms via chemotaxis. We prove well-posedness under
certain ”balance conditions” on the order of the porous medium degeneracy and the growth of the chemotactic functions.
Keywords: porous-medium, chemotaxis, well-posedness, biofilms, nonautonomous equation
2000 Mathematics Subject Classification:
35A05, 35K15, 35K55, 35K55, 35K57, 35K65, 37L05

1

Introduction

This study aims to consider the following model
M˙ = ∇ · (|M |α ∇M ) − ∇ · (|M |γ ∇ρ) + f (t, M, ρ)
ρ˙ = ∆ρ − g(t, M, ρ)
M = 0, ρ = 1
M (·, 0) = M0 , ρ(·, 0) = ρ0

in
in
in
in

(0, ∞) × Ω,
(0, ∞) × Ω,
(0, ∞) × ∂Ω,
Ω.

(1.1)
(1.2)
(1.3)
(1.4)

where α and γ are given constants satisfying α2 + 1 ≤ γ ≤ α (we call these
conditions the balance conditions). Moreover, Ω ⊂ RN is a smooth bounded
domain (N = 1, 2, 3) and M0 ∈ L∞ (Ω), ρ0 ∈ W 1,∞ (Ω) with ρ0 , M0 ≥ 0. We
assume the functions f and g satisfy the following assumptions:
for all M, ρ ∈ R, t ∈ R+
0 let
1

1

|f (t, M, ρ)| ≤ f1 (t)(1 + |M |ξ ) 2 , f1 ∈ L2b (R+
0 ), 0 ≤ ξ < α − γ + 2,
2

f (t, M, ρ)M ≤ −F2 M + f3 (t)|M |, f3 ∈

Lκb (R+
0 ),

κ > 1,

(1.6)

1

g(t, M, ρ) = g1 (t)ρ + g2 (t, ρ)M, g1 ∈ C [0, +∞), g˙1 (t) ≤ 0,
|g2 (t, ρ)| ≤ g3 (t), g3 ∈

Lηb (R+
0 ),

(1.5)

η > 4,

(1.7)
(1.8)

and, in order to ensure the uniqueness and the non-negativity of solution,


2
2
∂ fe
+
+
+
fe(t, M, ρ) := f t, M 2+α , ρ − F4 M 2+α ,
∈ L∞
(1.9)
loc (R0 × R0 × R0 ),
∂M
∂g2
∂f ∂f
+
+
,
∈ L∞
∈ L∞
loc (R0 × R × R),
loc (R0 × R), f (t, 0, ρ) = 0, g2 (t, 0) = 0,
∂M ∂ρ
∂ρ
(1.10)
where f1 , f3 , g1 , g3 are non-negative functions and F2 , F4 are some constants, F2
is strict positive and for p ∈ [1, ∞], Q ⊂ Rm
Lploc (Q) = {u : Q → R : u ∈ Lp (K) for all compact sets K ⊂ Q} ,


Lpb (Q) = u ∈ Lploc (Q) : ||u||Lpb (Q) := sup ||u||Lp (Q∩B 1 ) < ∞
x0 ∈Rm

x0

where Bx10 is a ball of unit radius centered at x0 . The following example of
functions f and g satisfies the conditions (1.5)- (1.10):
Example 1.1.
2+α

f (t, M, ρ) = −M +

M+ 2

sin(t),
2+α
M+ 2 + 1
1
ρ+
g(t, M, ρ) =
ρ+M
cos(t),
1+t
ρ+ + 1

where M+ = max {M, 0}.
In the present paper, we treat weak solutions of the system (1.1)-(1.4). The
definition is as follows:
Definition 1.1. For T > 0, α > 1 and γ > 1, a pair of functions (M, ρ)
defined in Ω×[0, T ) is said to be a weak solution of (1.1)-(1.4) for M0 ∈ L∞ (Ω),
ρ0 ∈ W 1,∞ (Ω), if




α
(i) M ∈ L∞ ((0, T ) × Ω), |M | 2 M ∈ L2 0, T ; H01 (Ω) , M˙ ∈ L2 0, T ; H −1 (Ω) ,
(ii) ρ − 1 ∈ C([0, T ]; H01 (Ω)),
(iii) (M, ρ) satisfies the equation in the following sense:
Z T
(M, v) ϕ˙ − (|M |α ∇M − |M |γ ∇ρ, ∇v) ϕ + (f (τ, M, ρ), v) ϕ dτ = 0
0

for any v ∈ H01 (Ω), ϕ ∈ C0∞ (0, T ),
Z
(ρ(x, t) − 1) =
G(x, y, t)(ρ0 (y) − 1)dy
Ω
Z tZ
−
G(x, y, t − s)g(s, M (y, s), ρ(y, s))dyds
0

Ω

2

for a.e. (x, t) ∈ Ω × (0, T ), where G is a heat kernel in Ω with the homogeneous
Dirichlet boundary condition and the initial conditions hold: ρ(0) = ρ0 and, in
Cw ([0, T ]; L2 (Ω))-sense, M (0) = M0 .


Remark 1.1. From M ∈ L∞ (0, T ; L2 (Ω)) and M˙ ∈ L2 0, T ; H −1 (Ω) it follows that M ∈ Cw ([0, T ]; L2 (Ω)) (see [2]), therefore the initial condition for M
does make sense.
This system of partial differential equations, models, for example, a population
described in terms of its density M , which grows in dependence of a substrate
with concentration ρ. The substrate is degraded by the abiotic decay. The
spatial movement of the population is caused by two different effects. Firstly,
the model includes a density dependent diffusion term. This non-linear diffusion effect becomes stronger as the population grows larger locally, following a
power law as in the case of the porous medium equation. Secondly, the population moves towards regions with increased substrate availability, i.e. follows
the chemical signal ρ. This effect is also controlled by the population density
and its intensity increases as the local population density grows. Both effects
of population mobility increase/diminish with the population, each following a
power law. Thus, the model degenerates for M = 0. Finally, our model includes
a ”source term”: a non-linear reaction-interaction term f . As usual, it stays for
the sink/source density (net number of particles created per unit time and per
unit volume). At a high level of population density M the depth rate (caused by
the exterior forces such as predation or intoxication) is no less then F2 while it’s
order is governed by the square root of the ratio of density-dependent diffusion
coefficient to the intensity of response to the chemical signal. in particular this
effect also vanishes with the population.
The study of the model (1.1)-(1.4) expands the previous work of the first coauthor of this article (see [4], where the autonomous case was studied). In our
research, we generalize the model to include the non-autonomous case, in which
the source term may now be time dependent, additionally, its dependence of
population density is more general. We also relax the balance condition.
The main focus of the present study is on proving the well-posedness of (1.1)(1.4). We emphasize on the fact that the analysis of equations with a chemotaxistype term even in autonomous case and without degeneracy (α = 0) is quite
difficult, (see [8, 9, 10, 11, 12] and the references therein) and in proving the
well-posedness in our degenerate case, we face significant difficulties. In order
to overcome these difficulties we impose so-called ”balance conditions” between
the order of porous-medium degeneracy and the growth order of the chemotaxis
function: α2 + 1 ≤ γ ≤ α. We show that our model is a well-posed one and that
it exhibits no singular behavior. For each pair of starting values the solution is
uniformly bounded in time and space. Recall that this is not the case for the
models that contain the chemotaxis effect alone (the solution may blow up, see
[5]).
The condition α ≥ γ (an improvement over the condition α ≥ γ + 1 imposed in
[4]) reads: the density-dependent diffusion coefficient ”dominates” the intensity
of response to the chemical signal as the population density grows. This, as we
will see later, results in the uniform boundedness of M and ρ.
3

α
On the other hand, we show that the
R chemotactic effect is due to 2 + 1 ≤ γ
strong enough for the total biomass Ω M (t, x) dx to remain bounded from below for all time intervals [0, T ] by some positive constant and thus the population
will not vanish in finite time. In this paper we use the condition α2 + 1 ≤ γ to
establish the well-posedness for our model.

Our main result can be summarized as follows:
Theorem 1.1. Let the functions f and g satisfy the
(1.5)-(1.10)
 assumptions

and let the given constants α and γ satisfy γ ∈ α2 + 1, α . Then the initial
boundary-value problem (1.1)-(1.4) has at most one non-negative solution.
The paper is organized as follows: in Section 2 we obtain several a priori estimates for the solutions of (1.1)-(1.4), which in turn lead to L∞ - bounds for both
the biomass component and the substrate concentration; Section 3 is devoted
to the uniqueness of solutions. For the convenience of the reader, we present
some standard ideas which we used while proving the well-posedness of solutions
in the Appendix A and we show the positivity of total biomass in the Appendix B.
R
Notation: || · || stands for || · ||L2 (Ω) -norm and (u, v) for Ω u(x)v(x) dx or,
more generally (in the case of distributional derivatives for instance), for hu, vi.
Acknowledgments: The authors express their thanks to R. Czaja, H.J. Eberl
and D. Wrzosek for many stimulating discussions.

2

A priori estimates

In this section we derive several a priori estimates for the solutions of the problem (1.1)-(1.4), which in turn lead to L∞ -bounds for both M and ρ. The
existence of a solution for this problem can be then proved by means of a standard approximation procedure (see Appendix A).
We start with rewriting the equation (1.1) in the following way:



1
γ
(α−γ+1)−1
˙
M = ∇ · |M | ∇
M |M |
−ρ
+ f (t, M, ρ).
α−γ+1

4

(2.1)

In order to derive our firsta priori estimate, we multiply this equation by

1
(α−γ+1)−1
− ρ and integrate (formally) over Ω to get
α−γ+1 M |M |



1
(α−γ+1)−1
M |M |
−ρ
α−γ+1
 
2 !


1
(α−γ+1)−1
γ 
M |M |
− ρ 
= − |M | , ∇
α−γ+1


1
+ f (t, M, ρ),
M |M |(α−γ+1)−1 − ρ
α−γ+1


1
≤ f (t, M, ρ),
M |M |(α−γ+1)−1 − ρ .
α−γ+1




α−γ+2 2
1
d

⇔
|M | 2 − (M, ρ)
dt (α − γ + 1)(α − γ + 2)


1
(α−γ+1)−1
≤ f (t, M, ρ),
M |M |
− ρ − (ρ,
˙ M)
α−γ+1
M˙ ,

(2.2)

and we multiply the equation (1.2) by (ρ+ρ−1)
˙
(which we find more appropriate,
in contrast to the previous study in [4]) in the same sense as above to get
1 d
1 d
||ρ − 1||2 = −
||∇ρ||2 − ||∇ρ||2 − (g(t, M, ρ), ρ˙ + ρ − 1) ⇔
2 dt
2 dt


1 d
2
||∇ρ||2 + ||ρ − 1||2 = −||∇ρ||2 − kρk
˙ − (g(t, M, ρ), ρ˙ + ρ − 1). (2.3)
2 dt
2

kρk
˙ +

Adding the inequalities (2.2) and (2.3) together, we obtain




α−γ+2 2
d
1
1
1

2
2
2
|
||∇ρ||
+
||ρ
−
1||
−
(M,
ρ)
+
|M

dt (α − γ + 1)(α − γ + 2)
2
2


1
2
≤ f (t, M, ρ),
M |M |(α−γ+1)−1 − ρ − ||∇ρ||2 − (ρ,
˙ M ) − kρk
˙
α−γ+1
− (g(t, M, ρ), ρ˙ + ρ − 1).

(2.4)

We consider first the term containing g(t, M, ρ) = g1 (t)ρ + g2 (t, ρ)M . It holds:

1


1 d
g1 ||ρ||2 + g˙1 ||ρ||2 − g1 ||ρ||2 − (1, ρ)
2 dt
2




1 d
2
≤ −
g1 ||ρ|| − g1 ||ρ||2 − (1, ρ)
2 dt
(1.7)


1 d
1
≤ −
g1 ||ρ||2 − (1 − )g1 ||ρ||2 + g1
2 dt
4


1 d
1
≤ −
g1 ||ρ||2 − (1 − )g1 ||ρ||2 + g1 (0)
(2.5)
2
dt
4
(1.7)

−(g1 ρ, ρ˙ + ρ − 1) = −

and
1
||g2 (t, ρ)M ||2
2
1
2
≤  kρk
˙ + ||ρ − 1||2 + |g3 |2 ||M ||2 .
2
(1.8)
2

−(g2 (t, ρ)M, ρ˙ + ρ − 1) ≤  kρk
˙ + ||ρ − 1||2 +

5

(2.6)

By combining (2.5) and (2.6) with the inequality
2

−(ρ,
˙ M ) − kρk
˙ ≤
and by choosing  ≤

1
2

1
1
2
||M ||2 − kρk
˙
2
2

(2.7)

we have

2

− (ρ,
˙ M ) − kρk
˙ − (g(t, M, ρ), ρ˙ + ρ − 1)


1 d
≤−
g1 ||ρ||2 − (1 − )g1 ||ρ||2 + ||ρ − 1||2 +
2 dt


1
1
+
+ |g3 |2 ||M ||2
2 2


1 d
g1 ||ρ||2 − (1 − )g1 ||ρ||2 + ||ρ − 1||2 +
≤ −
1
2 dt
≤ 2


1
1
2
+
+ |g3 | ||M ||2 .
2 2

1
g1 (0) −
4




1
2
−  kρk
˙
2

1
g1 (0)
4
(2.8)

Further, we can estimate the terms with f from (2.4) in the following way:




f (t, M, ρ), M |M |(α−γ+1)−1 ≤ −F2 M 2 + f3 |M |, |M |(α−γ+1)−1
(1.6)





α−γ+2 2
α−γ+1 2


= − F2 |M | 2 + f3 |M | 2 ,
(2.9)



ξ 2
1 2

2
− (f (t, M, ρ), ρ)) ≤ ||ρ|| + f1 1 + |M |
4
(1.5)


ξ 2
1
1

≤2||ρ − 1||2 + 2 + f12 + f12 |M | 2 .
4
4
2

(2.10)

Using the inequalities (2.8)-(2.10) we conclude from (2.4):



α−γ+2 2
d
1

|M | 2 − (M, ρ)
dt (α − γ + 1)(α − γ + 2)

1
1
1
2
2
2
+ ||∇ρ|| + ||ρ − 1|| + g1 ||ρ||
2
2
2



2

2

α−γ+2
α−γ+1
ξ 2
1 2
1
1





2
2
2
2
+ |g3 | ||M ||2
≤ − F2 |M |
+ f3 |M |
+ f1 |M | +
4
2 2
1
1
− ||∇ρ||2 − (1 − )g1 ||ρ||2 + 3||ρ − 1||2 + 2 + g1 (0) + f12 .
(2.11)
4
4
In order to shorten the formulas, we introduce a new variable:


α−γ+2 2
1

ϕ :=
|M | 2 − (M, ρ)
(α − γ + 1)(α − γ + 2)
1
1
1
+ ||∇ρ||2 + ||ρ − 1||2 + g1 ||ρ||2 + C
2
2
2

6

(2.12)

where the constant C can be chosen in such a way that ϕ ≥ 1 holds. Indeed,
α−γ+2
|M | 2
is the leading M -power present in the expression (2.12) due to the
assumptions made on α, γ and ξ, and we also have the estimate
(M, ρ) ≤ ||ρ||2 +

1
||M ||2
4

(2.13)

valued for all  > 0. Moreover, applying the Poincar´e and the H¨older inequalities
and adjusting the constants C and  we can deduce from (2.11) the inequality
ϕ˙ ≤ −A1 ϕ + a2 ϕθ

(2.14)

for some A1 ∈ R+ and a2 ∈ L1b (R+
0 ), a2 ≥ 0 and
n
o
ξ
max α−γ+1
,
2
2
θ :=
∈ (0, 1).
α−γ+2
2

A simple calculation shows that any solution ϕ of the inequality (2.14) satisfies
the inequality

ϕ(t) ≤

ϕ1−θ
e−A1 (1−θ)t + (1 − θ)
0

Z

t

e−A1 (1−θ)(t−τ ) a2 dτ

1
 1−θ

.

(2.15)

0

In particular, it is bounded because a2 ∈ L1b (R+
0 ) holds and as a consequence of
that and the inequality (2.13), we finally achieve our first uniform estimate:
sup kM (t)kα−γ+2 , sup||ρ(t)||H01 (Ω) < ∞.
t>0

(2.16)

t>0

Remark 2.1. We emphasize on the fact that, unlike in [4], the nonlinearity g is
not necessarily non-negative. Thus we cannot relay on the comparison principle
and have to work with both equations (1.1) and (1.2) simultaneously to get the
desired estimates.
In what follows we use (2.16) to obtain some intermediate estimates for M and
ρ, which in turn lead to L∞ -bounds in both time and space. The following
observation, which is an implication from the theory of abstract parabolic evolution equations (see [15]) and will be helpful in further.
Having a δ ∈ (2, ∞) fixed, consider the unbounded operator
∆ : Lδ (Ω) → Lδ (Ω)
equipped with the domain
n
o
D(∆) := u ∈ W01,δ : ∆u ∈ Lδ (Ω) .
It is known (see [15]) that this operator generates an analytic semigroup et∆
and it’s spectrum lies entirely in the left half plane {λ ∈ C : λ ≤ −β} for some
β > 0. As such it has the following properties:
(−∆)µ et∆ = et∆ (−∆)µ ,
t∆

||e

(2.17)

µ,δ −βt −µ

µ

(−∆) ||δ ≤ A
7

e

t

(2.18)

for all t > 0 and µ > 0 for some constants Aµ,δ . Now, the equation (1.2) can be
rewritten in the following way:
d
(ρ − 1) = ∆(ρ − 1) − g(t, M, ρ)
dt
and can thus be regarded as an abstract parabolic evolution equation with
respect to ρ − 1. Therefore for all t > 0 holds:
Z t
ρ(t) − 1 = et∆ (ρ(0) − 1) −
e(t−τ )∆ g(τ, M (τ ), ρ(τ )) dτ.
(2.19)
0

and applying the operator ∇ to both sides of (2.19) and making use of the
property (2.17) we obtain
Z t 

t∆
∇ρ(t) = e ∇ρ0 −
∇ e(t−τ )∆ g(τ, M (τ ), ρ(τ )) dτ.
(2.20)
0

The initial value ρ(0) is assumed to be sufficiently smooth, so that holds
||∇ρ0 ||δ < ∞.

(2.21)

What remains is to estimate the δ-norm of the integral from (2.20) with help
of (2.18) and the assumptions on g. Choosing µ ∈ (0, 1) and δˆ ≥ 1 such that
ˆ
W µ,δ (Ω) ,→ W 1,δ (Ω) we obtain
Z t 



(t−τ )∆

∇
e
g(τ,
M
(τ
),
ρ(τ
))
dτ


0

δ

Z t




≤
(−∆)µ e(t−τ )∆ g(τ, M (τ ), ρ(τ )) ˆ dτ
δ
0
Z t


ˆ
≤Aµ,δ
e−βt (t − τ )−µ g1 (0)||ρ(τ )||δˆ + g3 (τ )||M (τ )||δˆ dτ.

(2.22)

0

Suppose for a moment that
sup||ρ(t)||δˆ, sup||M (t)||δˆ < ∞.

(2.23)

t>0

t>0

Due to (1.8) we have g3 ∈ Lηb (R+
0 ) and can therefore conclude from (2.21) and
(2.22) with the H¨
older inequality




1
µ,δˆ
e
sup k∇ρ(t)kδ ≤A
sup||ρ(t)||δˆ + sup||M (t)||δˆ , µ ∈ 0, 1 −
. (2.24)
η
t>0
t>0
t>0
For instance, δ˜0 := min {α − γ + 2, 6} and 43 are an appropriate choice of δˆ
and µ respectively. This is a consequence of the estimate (2.16), the fact that
H 1 (Ω) ,→ L6 (Ω), η > 4 and that due to α ≥ γ we have δe0 ≥ 2, so that
2µ − e3 ≥ 0 = 1 − 33 . Therefore it follows with (2.24)
δ0

B1 := sup kρ(t)kW 1,3 (Ω) < ∞.

(2.25)

t>0

It is worth mentioning that in contrast to the previous study ([4]) we have to be
more precise in our further derivations due to a more general assumption (1.8)
8

on the nonlinearity g2 which, at this point, has led to an W 1,3 -estimate for ρ
instead of an W 1,6 -estimate.
Getting back to the equation (1.1) we multiply this equation by M |M |δ−1 for an
arbitrary δ ≥ α − γ + 1, so that all occurring powers remain non-negative (note:
the assumption γ ≥ α2 + 1 won’t be necessary here), and (formally) integrate
over Ω:


M˙ , M |M |δ−1 = (∇ · (|M |α ∇M ) − ∇ · (|M |γ ∇ρ)


+f (t, M, ρ), M |M |δ−1 .
It follows:



δ+1 2
α+δ+1 2
4δ
1 d


2
|
|M | 2 = −
∇|M

δ + 1 dt
(α + δ + 1)2


α+δ+1
δ−1
α
2δ
+
∇|M | 2 , |M |γ− 2 + 2 ∇ρ
α+δ+1
+ (f (t, M, ρ), M |M |δ−1 ).
(2.26)
Denote ϑ(δ) :=

δ−1
γ− α
2+ 2
α+δ+1
2

. Then ϑ(δ) < 1 holds due to the assumption α ≥ γ

we made. Applying H¨
older’s inequality we obtain:

 

α+δ+1
δ−1
α+δ+1
α+δ+1
α
∇|M | 2 , |M |γ− 2 + 2 ∇ρ = ∇|M | 2 , |M |ϑ(δ) 2 ∇ρ



α+δ+1
α+δ+1 ϑ(δ)

≤ k1k 6 ∇|M | 2 |M | 2
k∇ρk3
1−θ(δ)
6


α+δ+1 1+ϑ(δ)

≤B2 ∇|M | 2
k∇ρk3 .
(2.27)
For the last inequality the embedding H 1 (Ω) ,→ L6 (Ω) has been used.
Further, we use once more the H¨older inequality and the assumptions on the
function f and write:




δ+1 2
δ 2


(f (t, M, ρ), M |M |δ−1 ) ≤ − F2 |M | 2 + f3 |M | 2
(2.28)
δ



 δ+1
δ+1 2
δ+1 2


≤ − F2 |M | 2 + f3 k1kδ+1 |M | 2
.
(2.29)
We can conclude from (2.26) using (2.27) and (2.29) that:



δ+1 2
α+δ+1 2
1 d
4δ


2
|
|M | 2 ≤ −
∇|M

δ + 1 dt
(α + δ + 1)2


α+δ+1 1+ϑ(δ)
2δ

B2 ∇|M | 2
+
k∇ρk3
α+δ+1
δ



 δ+1
δ+1 2
δ+1 2


2
2
− F2 |M | + f3 k1kδ+1 |M |
.

9