2021 UNC Greensboro PDE Conference,

Electronic Journal of Differential Equations, Conference 26 (2022), pp. 139–149.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

WELLPOSEDNESS OF KELLER-SEGEL SYSTEMS IN MIXED

NORM SPACES

TIMOTHY ROBERTSON

Abstract. We study the well-posedness of the Cauchy problem for the Keller-

Segal system in the setting of mixed norm spaces. We prove existence of mild
solutions in scaling invariant spaces and uniqueness in a special case. These

results allow for existence and uniqueness when the initial data has anisotropic

properties. In particular, persistence of anisotropic properties under the evo-
lution is demonstrated which could be of biological interest.

1. Introduction

In this article, we study the following Cauchy problem for the Keller-Segel system
of parabolic equations

ut = ∆u−∇ · (u∇v), in Rn × (0, T ),

vt = ∆v −A(t)v +B(x, t)u in Rn × (0, T ),

u(0, ·) = u0(·) in Rn,
v(0, ·) = v0(·) in Rn,

(1.1)

where in (1.1) the functions u, v : Rn × (0, T ) → R are unknown solutions with
T ∈ (0,∞] and n ∈ N, the functions u0, v0 : Rn → (0,∞) are given measurable
initial data, and the functions A : (0, T ) → [0,∞) and B : Rn × (0, T ) → (0,∞)
are given and measurable. The Keller-Segel system was proposed in [7] to describe
chemotactic aggregation of cellular slime molds which move preferentially towards
relatively high concentrations of a chemical secreted by the amoebae themselves.
In this context, u(x, t) represents the cell density of the slime molds at position x
and time t, and similarly the concentration of the chemical substance at position x
and time t is represented by v(x, t).

Our goal is to develop the existence and uniqueness of solutions of (1.1) in the
setting of anisotropic spaces. The motivation of the study comes from applications
where initial data and solutions can concentrate and behave differently in differ-
ent spatial variables. Mathematically, our results demonstrate the persistence of
anisotropic properties under the evolution of the Keller-Segal system, which does
not seem to be trivial. To put our study into perspective, we recall the definition
of the anisotropic (mixed norm) space. For a given ~p = (p1, . . . , pn) ∈ [1,∞)n,

2020 Mathematics Subject Classification. 35A01, 35A02, 92C17.

Key words and phrases. Mixed norm spaces; Keller-Segel model; semigroup methods.
©2022 This work is licensed under a CC BY 4.0 license.
Published August 25, 2022.

139



140 T. ROBERTSON EJDE-2022/CONF/26

the mixed norm Lebesgue space L~p(Rn) is the set of all measurable functions
f : Rn → R such that the mixed norm

‖f‖~p =

(∫
R
· · ·
(∫

R

(∫
R
|f(x1, x2, . . . , xn)|p1dx1

) p2
p1
dx2

) p3
p2 · · · dxn

)1/pn

<∞.

A similar definition can be formed if pi =∞ for some i = 1, 2, . . . , n. Similarly, for
q ∈ [1,∞) and ~p = (p1, . . . , pn) ∈ [1,∞)n, we write Lq((0, T );L~p(Rn)) to represent
the space consisting of all u : (0, T )→ L~p(Rn) such that

‖u‖T,q,~p =
(∫ T

0

‖u(·, t)‖q~pdt
)1/q

<∞.

The definition of mixed norm Sobolev space follows naturally from the definition
of the mixed norm.

Definition 1.1. Let ~I = (I1, . . . , In) ∈ [1,∞)n. The Sobolev space H
~I(Rn) con-

sists of all measurable functions f : Rn → R such that the weak derivative Df
exists and

‖f‖H~I(Rn) := ‖f‖L~I(Rn) + ‖Df‖L~I(Rn) <∞.

Now, the mild solutions to (1.1) are defined as follows.

Definition 1.2. A pair of functions (u, v) : Rn × (0,∞)→ R2 is said to be a mild
solution to (1.1) if

u(x, t) = et∆u0 −
∫ t

0

∇ · e(t−s)∆(u · ∇v)ds,

v(x, t) = e−tet∆v0 +

∫ t

0

e−Ā(t−s)e(t−s)∆(Bu(s))ds

for all t ∈ (0, T ), provided the integrals are well-defined.

We note that the following assumption on the coefficients A,B are used through-
out this article: there is Λ > 0 such that The coefficients A,B are assumed to satisfy
the condition that there is Λ > 0 such that

0 ≤ Ā(t) :=

∫ t

0

A(s) ds and 0 ≤ B(x, t) ≤ Λ, ∀(x, t) ∈ Rn × (0, T ). (1.2)

We now state our main results. Our first result is on the local time existence of
solutions in the mixed-norm space.

Theorem 1.3. Let ~I = (I1, . . . , In), ~J = (J1, . . . , Jn) ∈ (1,∞)n satisfy

1

2

n∑
k=1

1

Ik
=

n∑
k=1

1

Jk
= 1.

Also let ~r = (r1, . . . , rn), ~R = (R1, . . . , Rn) be in (1,∞)n and 1 < q < Q <∞ such
that 1 < rk < Rk <∞, Ik < rk, Jk < Rk for k = 1, 2, . . . , n and

1

q
+

1

2

n∑
k=1

1

rk
= 1,

1

Q
+

1

2

n∑
k=1

1

Rk
=

1

2
. (1.3)

Then under assumption (1.2) and for a pair functions u0, v0 : Rn → R satisfying

u0 ∈ L~I(R
n), v0 ∈ H

~J(Rn)



EJDE-2018/CONF/26 MIXED NORM WELLPOSEDNESS 141

there exist T > 0 and a pair of functions

u ∈ Lq((0, T );L~r(Rn)) and v ∈ LQ((0, T );H
~R(Rn))

that solves (1.1) in the mild sense.

Observe that the conditions on q,Q,~r and ~R in Theorem 1.3 gives invariance
under the heat scaling. Precisely, for a given pair of solutions (u, v) and for λ > 0,
let

uλ(x, t) = λ2u(λx, λ2t), vλ(t, x) = v(λx, λ2t), x ∈ Rn, t ∈ (0, T/λ).

Then

‖uλ‖T/λ,q,~r = ‖u‖T,q,~r, ‖∇vλ‖T/λ,Q,~R = ‖∇v‖T,Q,~R, ∀λ > 0 (1.4)

if and only if

1 =
1

q
+

1

2

n∑
k=1

1

rk
,

1

2
=

1

Q
+

1

2

n∑
k=1

1

Rk
.

Observe that the condition (1.4) must be the correct scaling condition since (uλ, vλ)
is a solution of (1.1) when A(t) is replaced by A(λ2t)/λ2 and B(x, t) is replaced by
B(λx, λ2t). For a more extensive discussion of scaling see [5].

Our second result Theorem 1.4 shows uniqueness of mild solutions under a special
case of Theorem 1.3.

Theorem 1.4. Suppose ~I = (I1, . . . , In), ~J = (J1, . . . , Jn) in (1,∞)n satisfy 1 =∑n
k=1 1/Jk with Ik = 1

2Jk for k = 1, . . . , n and

u0 ∈ L~I(R
n), ∇v0 ∈ L ~J(Rn).

If (u, v) is a mild solution of (1.1) and satisfies

u ∈ C((0, T );L~I(R
n)), ∇v ∈ C((0, T );L ~J(Rn)) (1.5)

then (u, v) is the only mild solution that satisfies (1.5) on Rn × (0, T ).

The wellposedness of the Keller-Segel model in its various forms has been ex-
tensively studied in the literature. Perhaps the most notable result thus far is the
classical result that for n = 2 the Keller-Segel system has a solution if and only
if the critical mass is less than 8π, as discussed in [3] and [4]. Unfortunately, the
problem of existence is not so clear when n ≥ 3. We do not present an exhaustive
review here and refer interested readers to the recent review paper [1]. However,
we do highlight several pertinent results. Global existence for small initial data

u0 ∈ Ln/2w and v0 ∈ BMO was shown in [8]. Small time existence was shown for
initial data u0 ∈ Ln/2 and v0 ∈ H1,n/2 in [9]. Our results extend the latter result
to mixed norm spaces using techniques established in [12, 13].

The proof of Theorem 1.3 follows from the standard fixed point argument using
Picard’s iteration technique. To implement this method, we need to derive several
estimates of the heat semi-group in mixed norm spaces which could be of indepen-
dent interest. In Section 2 we introduce and review several analysis results and
estimates on Sobolev imbedding theorems and semi-groups in mixed-norm spaces.
In particular, in Lemma 2.2 below, the smoothing estimate of the heat semi-group
in mixed-norm spaces is introduced, and this result seems to be new. The proof
of Lemma 2.2 follows from the Marcinkiewicz interpolation theorem. The proof
of Theorem 1.3 is then given in Section 3. To prove Theorem 1.4, we take the



142 T. ROBERTSON EJDE-2022/CONF/26

difference between the two solutions and then control it. The challenging part is to
control the nonlinear terms in the mixed-norm spaces. Delicate details with algebra
in mixed-norms are then required and all are presented in Section 4.

2. Preliminary inequalities and estimates in mixed-norm spaces

The proofs of Theorems 1.3, 3.1 rely on several technical lemmas stated below.
The proof of Lemma 2.1 is given in [12] so it is omitted here.

Lemma 2.1. Let ~p = (p1, . . . , pn) and ~q = (q1, . . . , qn) be in [1,∞]n such that
pk ≤ qk for k = 1, 2, . . . , n. Then there exists C = C(~p, ~q, n) > 0 such that

‖et∆f‖~q ≤ Ct−
1
2

∑n
k=1( 1

pk
− 1
qk

)‖f‖~p,

‖∇et∆f‖~q ≤ Ct−
1
2−

1
2

∑n
k=1( 1

pk
− 1
qk

)‖f‖~p,
(2.1)

for all f ∈ L~p(Rn) and for all t > 0.

Lemma 2.2 is an interpolation result yielding control of norms of semi-group
operators. Here we follow the approach of the recent work [13, Lemma 3.2]. The
proof is included for completeness.

Lemma 2.2. Let q ∈ (1,∞), ~p = (p1, . . . , pn) and ~r = (r1, . . . , rn) be in (1,∞)n

satisfying pk < rk for k = 1, . . . , n and

1

q
=
l

2
+

1

2

n∑
k=1

( 1

pk
− 1

rk

)
.

Then for l = 0, 1 there is Cl = C(~p, q, n, l) > 0 such that(∫ t

0

‖Dles∆f‖q~rds
)1/q

≤ Cl‖f‖~p, for all f ∈ L~p(Rn).

Proof. Let ~p′ = (p1, . . . , pn−1). To employ the Marcinkiewicz interpolation theorem
consider the space

X = {f : measurable f : R→ L~p′(R
n−1)},

with ‖ · ‖X := ‖ · ‖L~p′ , and the operator Tl : L~r(X)→ Lαw defined by

Tlf : f → ‖Dlet∆f‖~r,

where Lαw is the weak Lebesgue space with parameter α. For fixed l Lemma 2.1
yields

‖Tl(f)‖Lαw <∞
if

1

α
=
l

2
− 1

2

n∑
k=1

( 1

pk
− 1

rk

)
.

We define

α(pn) =
( l

2
− 1

2

n−1∑
k=1

( 1

pk
− 1

rk

)
+

1

2rn
− 1

2pn

)−1

.

Note that p̂ and p̃ may be chosen such that

1 < p̂ < pn < p̃ < rn,



EJDE-2018/CONF/26 MIXED NORM WELLPOSEDNESS 143

and by an appropriate choice of θ ∈ (0, 1) the equality

1

pn
=

1− θ
p̂

+
θ

p̃

holds. Clearly,

1

q
=

1

α(pn)
=

1− θ
α(p̂)

+
θ

α(p̃)
.

Since Tl is of weak type (p̂, α(p̂)) and (p̃, ˆ̃p), by Marcinkiewicz interpolation Tl must
be of strong type (pn, q). Observing that

‖Tl‖q ≤ C‖f‖Lpn (X) = C‖f‖~p

completes the proof. �

Finally, this section concludes by recalling several function space inequalities that
will be used in the uniqueness proof. Lemma 2.3 generalizes Hölder’s inequality,
while Lemma 2.4 generalizes Young’s inequality to the Lorentz space setting. Recall
that Lp,q is the Lorentz space with parameters p and q, with norm

‖f‖Lp,q =
(∫ ∞

0

p1/qsq−1m({x : |f(x)| ≥ s})
q
p ds
)1/q

.

The proof for both lemmas is given in [10].

Lemma 2.3. Let q,Q ∈ [1,∞] and p, P ∈ (1,∞) such that 1
p + 1

P < 1. Then there

exists C = C(p, P, q,Q) > 0 such that

‖fg‖
L

1
p

+ 1
P
,min{q,Q} ≤ C‖f‖Lp,q‖g‖LP,Q

for f ∈ Lp,q and g ∈ LP,Q.

Lemma 2.4. Let z, Z ∈ [1,∞] and p, q ∈ (1,∞) such that 1 < 1
p + 1

q < 2. Then

for r defined by

1

r
=

1

p
+

1

q
− 1

there exists C = C(p, q, z, Z) > 0 such that

‖f ∗ g‖Lr,min{z,Z} ≤ C‖f‖Lp,z‖g‖Lp,Z .

for f ∈ Lp,z and g ∈ Lq,Z .

The last lemma is a special case of the Sobolev Embedding Theorem in the mixed
norm setting found in [2].

Lemma 2.5. Let ~I, ~J ∈ (1,∞)n and ~p = (p1, . . . , pn) be defined by pk = ( 1
Ik

+
1
Jk

)−1. Then there exists a constant C = C(n, ~I, ~J) > 0 such that

‖f‖~I ≤ C
[
‖∇f‖~p + ‖f‖~p

]
for all f ∈ H~p(Rn).



144 T. ROBERTSON EJDE-2022/CONF/26

3. Proof of Theorem 1.3

The proof of Theorem 1.3 combines Picard iteration with estimates of the heat
semigroup. Concluding the iteration argument requires the following elementary
lemma, the proof of which is provided in the appendix.

Lemma 3.1. Let a1, b1, C,K > 0 be given real numbers. Suppose 0 < a1 <
(b1C−1)2

4BC2

and 0 < b1 <
1
C . Then the sequences {an}n and {bn}n defined recursively by

an+1 = a1 + Canbn

bn+1 = b1 + CKan
(3.1)

converge.

We now turn to the proof of Theorem 1.3.

Proof. We define ~p = (p1, . . . , pn) by pk = ( 1
rk

+ 1
Rk

)−1 for k = 1, . . . , n and z by

z = ( 1
q + 1

Q )−1. Let a1 = ‖et∆u0‖q,~r. Then it follows from Lemma 2.2 that

a1 ≤ N‖u0‖~p
for N = N(n,~r, q). Let

um+1 = et∆u0 −
∫ t

0

∇ · e(t−s)∆(um · ∇vm)ds,

vm+1(x, t) = e−tet∆v0 +

∫ t

0

e−Ā(t−s)e(t−s)∆(Bum(s))ds.

Note that the initial data u0 and v0 are the first terms of the sequences {um}∞m=0

and {vm}∞m=0, respectively. Applying Minkowski’s inequality and Lemma 2.1 yields∥∥∫ t

0

∇ · e(t−s)∆(um∇vm)ds‖~r ≤
∫ t

0

‖∇ · e(t−s)∆(um∇vm)‖~rds

≤
∫ t

0

(t− s)−
1
2−

1
2

∑n
k=1( 1

pk
− 1
rk

)‖um∇vm‖~pds

≤
∫ t

0

(t− s)−
1
2−

1
2

∑n
k=1

1
Rk ‖um‖~r‖∇vm‖~Rds

Recalling that

1

q
=

1

2
− 1

2

n∑
k=1

( 1

pk
− 1

rk

)
,

the Hardy-Littlewood-Sobolev inequality and Hölder’s inequality give∥∥∫ t

0

∇ · e(t−s)∆(um∇vm)ds
∥∥
q,~r
≤
∥∥ ∫ t

0

(t− s)−
1
2−

1
2

∑n
k=1

1
Rk ‖um‖~r‖∇vm‖~Rds

∥∥
q

≤ C‖‖um‖~r‖∇vm‖~R‖z
≤ C‖um‖q,~r‖∇vm‖Q,~R

for some C = C(n, q,Q, ~R). Letting am = ‖um‖q,~r and bm = ‖∇vm‖Q,~R it follows

that
‖um+1‖q,~r ≤ a1 + C‖um‖q,~r‖∇vm‖Q,~R ≤ a1 + Cambm.

Similarly, we observe that

b1 = ‖∇e−Ātet∆v0‖Q,~R =
∥∥e−Āt∇et∆v0

∥∥
Q,~R
≤ ‖∇et∆v0‖Q,~R ≤ N‖v0‖~P .



EJDE-2018/CONF/26 MIXED NORM WELLPOSEDNESS 145

Using the same estimation scheme as above we obtain that

‖vm+1‖Q,~R ≤ b1 + CΛ‖um‖q,~r.

In detail

‖vm+1‖Q,~R ≤ b1 +
∥∥∫ t

0

‖∇e−Ā(t−s)e(t−s)∆(Bum)‖~Rds
∥∥
Q

≤ b1 + ΛC
∥∥∫ t

0

(t− s)−
1
2−

1
2

∑n
k=1( 1

rk
− 1
Rk

)‖um‖~rds
∥∥
Q

≤ b1 + ΛC‖um‖q,~r
≤ b1 + ΛCam.

where the Hardy-Littlewood inequality justifies the second inequality. The constant
C may not be the same constant as in the estimate for um, but we take C to be
the maximum of the two constants.

Applying Lemma 3.1 gives the convergence of {‖um‖q,~r}m and {‖∇vm‖Q,~R}m.

Let

X = sup
m
‖um‖T∗,q,~r, Y = sup

m
‖∇vm‖T∗,Q,~R.

Observe that for T = T ∗ > 0 small enough X < 1
4C2B and Y < 1

4C . Thus, for each
m ≥ 2,

‖um+1 − um‖T∗,q,~r = ‖
∫ t

0

∇ · e(t−s)∆(um∇vm − um−1∇vm−1)ds‖T∗,~r

≤ CY ‖um − um−1‖T∗,q,~r + CX‖∇vm −∇vm−1‖T∗,Q,~R)

≤ CY ‖um − um−1‖T∗,q,~r + C2BX‖um − um−1‖T∗,q,~r)

≤ 1

2
‖um − um−1‖T∗,q,~r.

Clearly, {um}m is Cauchy in Lq((0, T ∗), L~r(Rn)), which in turn implies that {vm}m
is Cauchy in LQ((0, T ∗), L~R(Rn)). Replacing um with its limit in the above estimate
and using a similar strategy for vm shows that the limits are indeed mild solutions.

Recall that this scheme holds provided that 0 < b1 <
1
C and 0 < a1 <

(b1C−1)2

4ΛC2 ,
which holds if T ∗ is chosen sufficiently small. Although there are many choices, it
suffices for to T ∗ small enough that b1 <

1
2C and a1 <

1
16Λ . �

To demonstrate the utility of this result we now give a short example.

Example 3.2. Choose Ik > 1 such that
∑n
k=1

1
Ik
< 4 and Ik 6= Ij for k 6= j. Then

we have

q =
(

1− 1

4

1

Ik

)−1

> 1

and by Theorem 1.3 the existence of solutions

u ∈ Lq((0, T );L2~I(R
n)) v ∈ L2q((0, T );H4~I(Rn))

for initial data u0 ∈ L~I(R
n) and v0 ∈ H2~I(Rn). This example case compares

naturally to [9, Theorem 1], but covers a distinctly different class of initial data.



146 T. ROBERTSON EJDE-2022/CONF/26

4. Proof of uniqueness

In this section the second main result Theorem 1.4 is proved. The idea of the
proof is to obtain an inequality of the form

‖u2 − u1‖T,p,~I ≤ F (T )‖u2 − u1‖T,p,~I
where F (T ) is a continuous function of T and u1, u2 are any two solutions to
(1.1). Provided F (T ) < 1 for small enough T , we obtain uniqueness for small time.
Extending the local uniqueness to the entire interval of existence is then possible
using a contradiction argument.

Proof. Our strategy is to estimate the difference between two solutions satisfying
(1.5) and show that it must be zero. Namely, let (u1, v1) and (u2, v2) be two mild
solutions that satisfy (1.5). It is convenient to define the following:

(1) G(t; f, g) =
∫ t

0
∇e(t−s)∆(fg)ds for f : Rn×(0, T )→ R and g : Rn×(0, T )→

Rn
(2) w1(t) =

∫ t
0
e−Ā(t−s)e(t−s)∆(∇(Bu1))ds

(3) w2(t) = −G(t;u2, v2)
(4) u = u1 − u2

(5) v = v1 − v2

Since u1 and u2 are both mild solutions subtraction and substitution yields:

u(x, t) = −G(t;u1,∇v1) +G(t;u2,∇v2)

= −G(t;u1,∇v1) +G(t;u2,∇v1)−G(t;u2,∇v1) +G(t;u2,∇v2)

= −G(t;u1 − u2,∇v1)−G(t;u2,∇v1 −∇v2)

= −G(t;u,∇v1)−G(t;u2,∇v)

= −G(t;u, e−Ātet∆∇v0 + w1(t))−G(t; et∆u0 + w2(t))

= −G(t;u, e−Ātet∆∇v0)−G(t;u,w1(t))

−G(t; et∆u0,∇v)−G(t;w2(t),∇v)

:= T1 + T2 + T3 + T4.

(4.1)

It remains to estimate each Tk. Formally, for p ∈ (2,∞) with ~z = (z1, . . . , zn) given
by zk = ( 1

Ik
+ 1

Jk
)−1 for k = 1, . . . , n:

‖(−∆)−1/2(uw1)‖T,p,~I ≤ C‖uw1‖T,p,~z
≤ C‖‖u‖~I‖w1‖ ~J‖Lp

≤ C‖u‖T,p,~I
(

sup
s∈(0,T )

‖w1‖ ~J
)
.

(4.2)

The first inequality is justified by Lemma 2.5 and the fact that uw1 is integrable,
and the second by Hölder’s inequality. To justify the third inequality note that

‖u‖T,p,~I ≤ T‖u‖C((0,T );L~I(Rn)) <∞.

Since

‖e−Ātet∆v0‖ ~J ≤ C‖Gt‖L∞(Rn)‖∇v0‖ ~J ≤ C‖∇v0‖ ~J ,

it follows that ‖w1‖C((0,T ),L~I(Rn)) <∞.



EJDE-2018/CONF/26 MIXED NORM WELLPOSEDNESS 147

Applying the results found in [6], [11] and using (4.2) to estimate T2 yields:

‖G(t;u,w1)‖T,p,~I ≤
∥∥ ∫ t

0

(−∆)1/2e(t−s)∆(uw1)ds
∥∥
T,p,~I

≤
∥∥ ∫ t

0

(−∆)e(t−s)∆(−∆)−1/2(uw1)ds
∥∥
T,p,~I

≤ C‖(−∆)−1/2(uw1)‖T,p,~I
≤ C‖u‖Lp((0,T );L~I)

(
sup

s∈(0,T )

‖w1‖ ~J
)

(4.3)

Now applying the same process to T4 it follows that w2(t) ∈ C((0, T );L~I(R
n)), and

‖∇v‖T,p, ~J <∞, and so

‖(−∆)−1/2w2∇v‖T,p,~I ≤ C‖∇v‖T,p, ~J
(

sup
s∈(0,T )

‖w2‖~I
)
.

Hence,

‖G(t;w2,∇v)‖T,p,~I ≤ C‖∇v‖T,p, ~J
(

sup
s∈(0,T )

‖w2‖~I
)
. (4.4)

Turning our attention to T1, Lemma 2.1 yields

sup
s∈(0,∞)

s1/2
∥∥es∆∇v0

∥∥
L∞(Rn)

≤ C sup
s∈(0,∞)

s1/2s−1/2‖∇v0‖ ~J = C‖∇v0‖ ~J .

Using Minkowski’s inequality, and the Hölder and Young inequalities for Lorentz
spaces we find that

‖G(t;u, e−Ātet∆∇v0)‖T,p,~I

≤
∥∥∫ t

0

‖∇e(t−s)∆(ue−Āses∆∇v0)‖~Ids
∥∥
Lp

≤ C
∥∥∫ t

0

s−1/2(t− s)−1/2‖s1/2es∆∇v0‖L∞‖u‖~Ids
∥∥
Lp

≤ C‖∇v0‖ ~J
∥∥ ∫ t

0

s−1/2(t− s)−1/2‖u‖~Ids
∥∥
Lp

≤ C‖s−1/2‖u‖~I‖L
2p
p+2

,p

≤ C‖‖u‖~I‖Lp,p
= C‖u‖Lp((0,T );L~I(Rn)).

(4.5)

To estimate T3 observe that because 1 =
∑n
k=1

(
1
Ik
− 1

Jk

)
, Lemma 2.1 gives

sup
s∈(0,∞)

s1/2‖es∆u0‖ ~J ≤ C‖u0‖~I .

Estimating as before yields

‖G(t; et∆u0,∇v)‖T,p,~I ≤ C
∥∥∫ ‖es∆u0‖ ~J‖∇v‖ ~J(t− s)−1/2s1/2s−1/2ds

∥∥
Lp

≤ C‖u0‖~I‖∇v‖ ~J
≤ C‖u0‖~I‖u‖L~I

(4.6)



148 T. ROBERTSON EJDE-2022/CONF/26

where the integral representation of v in terms of u and the Sobolev embedding
theorem to obtain the last inequality. Altogether, (4.5), (4.3), (4.6) and (4.4) give

‖u‖T,p,~I ≤ C(n, p)f(T )‖u‖T,p,~I .

Observing that f(T ) → 0 as T → 0 it follows that for T1 ≤ Tmax small enough
u = 0 and so v = 0, where Tmax is the maximal existence time. Let T1 be the
maximal value for which u = 0. By a linear shift the calculation above applies
equally well when the time interval (T1, T1 + δ) with T1 + δ ≤ Tmax, which is a
contradiction to the definition of T1. Hence T1 = Tmax, proving the theorem. �

5. Appendix

We provide the proof of Lemma 3.1 via induction and the Monotone Convergence
theorem.

Proof. Limiting values (a, b) = limn→∞(an, bn) must satisfy

a = a1 + Cab

b = b1 + CKa.

Thus, a solves the quadratic equation

0 = a1 + (b1C − 1)a+ CK2a2,

and so

a =
−(b1C − 1)±

√
(b1C − 1)2 − 4a1KC2

2KC2
.

Provided that −(b1C − 1) > 0 and (b1C − 1)2 − 4a1KC
2 > 0, it is clear that both

values of a are positive. Choose the smaller value of a, setting

a =
−(b1C − 1)−

√
(b1C − 1)2 − 4a1KC2

2KC2
.

Then

b = b1 +KC
(−(b1C − 1)−

√
(b1C − 1)2 − 4a1KC2

2KC2

)
Note that both {an}n and {bn}n are monotonically increasing if a1, b1 > 0, giving
(a, b) = limn→∞(an, bn) if an ≤ a and bn ≤ b for all n ∈ N. Observe that if ak ≤ a
and bk ≤ b for all k ≤ n then

an+1 = a1 + Canbn ≤ a1 + Cab = a,

bn+1 = b1 +KCan ≤ b1 +KCa = b.

By mathematical induction it suffices to show that a1 ≤ a and b1 ≤ b. The
inequality for b1 is trivially true, so all that remains is to show that a1 ≤ a. A
straightforward calculation shows that this inequality holds if 0 ≤ a2

1(2KC2)2,
which is clearly true. Observe that −(b1C − 1) > 0 iff b1 <

1
C and (b1C − 1)2 −

4a1KC
2 > 0 if and only if a1 <

(b1C−1)2

4KC2 . �



EJDE-2018/CONF/26 MIXED NORM WELLPOSEDNESS 149

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Timothy Robertson
Department of Mathematics, University of Tennessee, Knoxville, USA

Email address: trober41@vols.utk.edu


	1. Introduction
	2. Preliminary inequalities and estimates in mixed-norm spaces
	3. Proof of Theorem ??
	4. Proof of uniqueness
	5. Appendix
	References