Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 15, pp. 1–22.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.15

NULL-CONTROLLABILITY FOR 1-D DEGENERATE

QUASILINEAR PARABOLIC EQUATIONS

PITÁGORAS P. DE CARVALHO, REGINALDO DEMARQUE,

JUAN LÍMACO, LUIZ VIANA

Abstract. In this article, we prove local null-controllability for one-dimensional

degenerate quasilinear parabolic equations. We apply a well-known local in-
version argument used by Fursikov and Imanuvilov. The strategy is to use

Carleman estimates, previously obtained for weak and strong degenerate par-

abolic problems.

1. Introduction

In this article, we investigate the controllability of the quasilinear degenerate
parabolic system

ut − ℓ(au)
(
a(x)ux

)
x
+ f(t, x, u) = hχω, (t, x) ∈ Q,

u(t, 1) = 0, in (0, T ),
u(t, 0) = 0, (Weak), t ∈ (0, T ),

or

(aux)(t, 0) = 0 (Strong), t ∈ (0, T ),

u(0, x) = u0(x), x ∈ (0, 1),

(1.1)

where T > 0 is given, Q := (0, T ) × (0, 1), ω = (α, β) ⊂ (0, 1), u0 ∈ L2(0, 1)
and h ∈ L2(Qω) is a control that acts on the system through Qω := (0, T ) × ω.
During this section, we will specify some conditions on the functions a : [0, 1] → R,
ℓ : R → R and f : [0, T ] × [0, 1] × R → R, under which the discussion will be
developed.

Assumption 1.1. Let a ∈ C([0, 1])∩C1((0, 1]) be a nondecreasing function satis-
fying a(0) = 0 and a > 0 on (0, 1]. Additionally, suppose that there exists K ∈ R
such that

xa′(x) ≤ Ka(x), ∀x ∈ [0, 1], (1.2)

2020 Mathematics Subject Classification. 35K65, 35K59, 93B05, 35K55.
Key words and phrases. Degenerate parabolic equations; quasilinear parabolic equations;

controllability; nonlinear parabolic equations.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted June 2, 2024. Published February 19, 2025.

1



2 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

where K ∈ [0, 1), for the weakly degenerate case (WDC), and K ∈ [1, 2), for the
strongly degenerate case (SDC). Only for the (SDC), we also assume that

∃θ ∈ (1,K] such that θa ≤ xa′ near zero, if K > 1;

∃θ ∈ (0, 1) such that θa ≤ xa′ near zero, if K = 1.
(1.3)

Next, we provide some examples and comments about assumption 1.1.

(a) For γ ∈ (0, 1) and α ≥ 0, putting β = arctan(α), the function a1(x) =
xγ cos(βx) satisfies (1.2) for the (WDC). On the other hand, if γ ∈ (1, 2),
then a1 becomes an example for the (SDC);

(b) For each p ∈ (0, 1), the function a2(x) = xp + x satisfies (1.2) for the
(WDC). Analogously, if p ∈ (1, 2), then a3(x) = xp + x satisfies (1.2) for
the (SDC).

Since our main results are associated with (1.1), we should make some comments
about the controllability of one-dimensional degenerate or quasilinear problems.
Many applied phenomena are closely related to degenerate parabolic equations,
calling a notorious attention to their mathematical point of view. Motivated by
the properties already known for the uniformly parabolic case, a complete quali-
tative investigation for degenerate operators is also expected (see a well-posedness
result in [7], for instance). This brief comment certainly includes control theory,
where much more development is still desired. In one dimension, it seems to us
that [12] and [13] are the two first articles dealing with the controllability of de-
generate parabolic equations, which clearly inspired much relevant work since then
(see [3, 6, 9, 14, 15, 22, 24, 29, 33] and the references therein). On the other hand,
to our best knowledge, there are not many controllability results involving quasi-
linear equations, where the second-order differential operator is associated with
a nonlinearity which depends on the state (see [28], for instance). So that, in
this paper, the main intention is a investigation about the controllability of one-
dimensional degenerate quasilinear equations. To be more precise, we will prove
a local null-controllability result for (1.1), at any time, with controls acting on a
small subinterval ω ⊂ (0, 1). In other words, given any time T > 0 and a sufficiently
small initial data u0, there exists a state-control pair (uh, h) for (1.1), such that
uh(T, ·) = 0 in [0, 1]. The proof will be based on [30], where a meticulous local
inversion argument is developed, using Lyusternik’s Theorem. This goal passes by
a certain linearization of (1.1), for which a global null-controllability result and
some additional estimates will also be obtained.

In the current literature, it is undeniable the strength of the Carleman estimates
method, because it provides a refined technique that makes the one-dimensional
degenerate controllability field well-understood (see [1, 8, 10, 11, 32] and the ref-
erences aforementioned). In [28], the local null-controllability result, proved for
nondegenerate quasilinear equations, also follows Carleman’s approach. To sum-
marize, up to this moment, the controllability of quasilinear equations, where the
diffusion term depends nonlinearly on the state, has not been widely investigated,
even for the nondegenerate case. It is exactly the motivation for the current re-
search, where we would like to contribute providing a controllability study for the
degenerate quasilinear problem (1.1).

To complement the state of the art associated with degenerate problems, we
also mention [4], where the boundary null controllability of the degenerate heat
equation was obtained as the limit of internal controllability. To be more precise,



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 3

taking ω = ωε := (1 − ε, 1), for each ε ∈ (0, 1), is built a family of state-control
pairs {(uε, hε); ε ∈ (0, 1)} solving (2.2), with c ≡ 0 and g ≡ 0, with the following
property: (uε, hε) → (u, g) in a suitable functional space, as ε → 0, where (u, g)
solves the boundary null controllability problem for the degenerate heat equation.
However, this kind of question keeps not understood if ω = (0, ε), as explained in [4,
Section 5]. On the other hand, following this direction, some effort has been made
to prove an analogous fact for the degenerate wave equation (see [5]). Naturally,
it would also be very interesting to analyze the asymptotic behavior of families
of state-control pairs corresponding to nonlinear evolution PDEs (degenerate and
nondegenerate cases. The discussion above is completely related to the relevance of
degenerate operators in Partial Differential Equations, including the Control Theory
setting. Specifically talking about this theme, in the presence of nonlinearities,
we should mention [20, 21, 19], where the authors have obtained the local null-
controllability for a class of degenerate parabolic problems with nonlocal terms,
dealing with theoretical and numerical aspects. We emphasize that this work relies
on those Carleman estimates achieved in [20] and [19] for the (WDC) and the
(SDC), respectively.

Next, we present some important functional spaces, introduced in [1], which are
closely related to the initial data of (1.1) and its linearization. Other than that, it
also has to do with the statement of our main result.

Definition 1.2 (Weighted Sobolev spaces). Let us consider a real function a =
a(x), as in (1.1).

(a) For the (WDC), we set H1
a :=

{
u ∈ L2(0, 1) such that u is absolutely

continuous in [0, 1],
√
aux ∈ L2(0, 1), and u(1) = u(0) = 0

}
, equipped with

the natural norm

∥u∥H1
a
:=

(
∥u∥2L2(0,1) + ∥

√
aux∥2L2(0,1)

)1/2

.

(b) For the (SDC), we set H1
a :=

{
u ∈ L2(0, 1) such that u is absolutely

continuous in (0, 1],
√
aux ∈ L2(0, 1), and u(1) = 0

}
, with the same norm

taken for the (WDC);
(c) In both situations, the (WDC) and the (SDC),

H2
a := {u ∈ H1

a : aux ∈ H1(0, 1)}

with the norm

∥u∥H2
a
:= (∥u∥2H1

a
+ ∥(aux)x∥2L2(0,1))

1/2.

Now, let us state the properties of the functions ℓ : R → R and f : [0, T ]× [0, 1]×
R → R, both mentioned in (1.1).

Assumption 1.3. Let ℓ : R → R be a C1 function with bounded derivative and
suppose that ℓ(0) = 1. We should observe that our results remain the same if we
just suppose that ℓ(0) > 0.

Assumption 1.4. We assume that f : [0, T ] × [0, 1] × R → R is a C1 function,
with bounded derivatives, such that f(t, x, 0) ≡ 0 and c(t, x) = ∂3f(t, x, 0) belongs
to L∞(Q), where (t, x) ∈ [0, T ]× [0, 1].

To state our main result, we recall the important concept below:



4 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Definition 1.5. System (1.1) is said locally null-controllable at a given time T > 0
if there exists ε > 0 with the following property: whenever u0 ∈ H1

a and ∥u0∥H1
a
≤ ε,

we can find a control function h ∈ L2(Qω), associated with a state u, such that

u(T, x) = 0, for every x ∈ [0, 1].

Having in mind the considerations above, we state our main result.

Theorem 1.6 (Local Null-Controllability). Under assumptions 1.1, 1.3 and 1.4,
the nonlinear system (1.1) is locally null-controllable at any time T > 0, in the
sense of Definition 1.5, provided that one of the following conditions holds:

(a) K ̸= 1;
(b) K = 1 and θ ≥ 1/2.

Conditions (a) and (b) in Theorem 1.6 are both sufficient to assure that au ∈
L∞(0, 1), for any u ∈ H1

a . This will play a very important role in the proofs of
Lemma 3.2 and Proposition 3.4. A complete explanation about this will be given
in Appendix 5.

The remainder of this paper is organized as follows: in Section 2, we present
useful notation and preliminary results. The first part brings some explanation
about a local inversion argument, while the second one is concerned with Carleman
and observability estimates, valid for both the (WDC) and the (SDC).

In Section 3, we verify some properties of a mapping H : E → F , set in (2.1),
which will allow us to apply Lyusternik’s Theorem (stated as Theorem 2.1) to
achieve the local null-controllability of (1.1). At this point, the key information
comes from the global-null controllability of the linearization of (1.1), given in
(2.2), as well as from some additional regularity results.

In Section 4, we prove the main result of this paper (Theorem 1.6), where the
local null-controllability of (1.1) is obtained. Additionally, we include some further
comments related possible future directions. The last section is Appendix 5, which
complements the content studied in Section 3.

2. Preliminary results

In this section, we introduce some notation and useful auxiliary results, which
will help us to prove our main result.

2.1. Notation and results related to the local inversion argument. The
first part of this section is devoted to a brief explanation about the most impor-
tant strategies for proving our main results. Our approach relies on a version of
Lyusternik’s Inverse Mapping Theorem (see [2, 30], for instance), whose statement
is given below.

Theorem 2.1 (Lyusternik). Let E and F be two Banach spaces, consider H ∈
C1(E,F ) and put η0 = H(0). If H ′(0) ∈ L(E,F ) is onto, then there exist r > 0

and H̃ : Br(η0) ⊂ F → E such that

H(H̃(ξ)) = ξ, ∀ξ ∈ Br(η0),

which means that H̃ is a right inverse of H in Br(η0). In addition, there exists
K > 0 such that

∥H̃(ξ)∥E ≤ K∥ξ − η0∥F ,∀ξ ∈ Br(η0).



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 5

Next, let us indicate how the proof of Theorem 1.6 can be seen as an application
of Theorem 2.1. Even though we have not set the desired Hilbert spaces E and F
yet, let us put

H(u, h) = (H1(u, h),H2(u, h)), (2.1)

where

H1(u, h) := ut − ℓ(au)(aux)x + f(t, x, u)− hχω, H2(u, h) := u(0, ·).
Notice that for u0 ∈ H1

a , the first and the second relations in (1.1) are satisfied if,
and only if, there exists (u, h) ∈ E solving

H(u, h) = (0, u0).

From this point, we realize that, among other properties, E and F must be built:

• considering the boundary conditions mentioned in (1.1);
• having some imposition on its elements, assuring that u(T, ·) ≡ 0. It will
be done having in mind some weights which appear in (2.13);

• having in mind that we want H′(0, 0) ∈ L(E,F ) to be onto. In fact, it is
equivalent to say that, given any (g, u0) ∈ F , the linear system

ut − (a(x)ux)x + c(t, x)u = hχω + g, (t, x) ∈ Q;

u(t, 1) = 0, in (0, T ),
u(t, 0) = 0, (Weak), t ∈ (0, T )

or

(aux)(t, 0) = 0, (Strong), t ∈ (0, T )

u(0, x) = u0(x), x ∈ (0, 1),

(2.2)

is globally null-controllable at the time T > 0, where h ∈ L2(Qω) is the
control function and a satisfies assumption 1.1. Hence, it seems that E
should contain some information involving the well-posedness (and addi-
tional regularity) of the linear system (2.2).

The well-posedness of (2.2) was proved in [1], with the following statement.

Proposition 2.2. For each g ∈ L2(Q), h ∈ L2(Qω) and u0 ∈ L2(0, 1), there exists
a unique weak solution u ∈ C0([0, T ];L2(0, 1)) ∩ L2(0, T ;H1

a) of (2.2). Moreover,
if u0 ∈ H1

a , then

u ∈ U := H1(0, T ;L2(0, 1)) ∩ L2(0, T ;H2
a) ∩ C0([0, T ];H1

a),

and there exists a constant CT > 0 such that

sup
t∈[0,T ]

(∥u(t)∥2H1
a
) +

∫ T

0

(
∥ut|2L2(0,1) + ∥(aux)x∥2L2(0,1)

)
≤ CT

(
∥u0∥2H1

a
+ ∥g∥2L2(Q) + ∥h∥2L2(Qω)

)
.

(2.3)

Definition 2.3. Let δ = δ(t, x) and f = f(t, x) be two real-valued measurable
functions defined in Q, where δ is non-negative. We say that f belongs to L2(Q; δ)

if
√
δf ∈ L2(Q). Moreover, the natural norm in L2(Q; δ) will be denoted by ∥ · ∥δ,

that is,

∥f∥δ =
(∫ T

0

∫ 1

0

δf2 dx dt
)1/2

for each f ∈ L2(Q; δ).



6 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Let us take ω′ = (α′, β′) ⊂⊂ ω and consider ψ ∈ C2([0, 1];R) satisfying

ψ(x) :=

{∫ x
0

y
a(y)dy, x ∈ [0, α′);

−
∫ x
β′

y
a(y)dy, x ∈ [β′, 1].

(2.4)

Also, let us set the functions

η(x) := eλ(|ψ|∞+ψ), ηr(x) := eλ(|ψ|∞+ψ) − eλr|ψ|∞ . (2.5)

where (t, x) ∈ (0, T )× [0, 1] and λ, r ∈ (0,+∞). In addition, consider the constants:

η̂ = min
x∈[0,1]

η(x), η∗ := max
x∈[0,1]

η(x), η̂r = min
x∈[0,1]

ηr(x), η∗r := max
x∈[0,1]

ηr(x). (2.6)

It is important to notice that, if r ∈ (3,+∞) is sufficiently large, then ηr(x) < 0,
for any x ∈ [0, 1], and 3η∗r < 2η̂r. In this case, putting η̄r := 3η∗r − 2η̂r, we can
see that ηr(x) ≤ η̄r < 0, for any x ∈ [0, 1]. Next, we consider m ∈ C∞([0, T ];R)
satisfying

m(t) ≥ t4(T − t)4, t ∈ (0, T/2];

m(t) = t4(T − t)4, t ∈ [T/2, T ] ;

m(0) > 0,

to define

τ(t) :=
1

m(t)
, ζ(x, t) := τ(t)η(x), ζ∗(t) := τ(t)η∗,

A(t, x) := τ(t)ηr(x), Ā(t) := τ(t)η̄r,

(2.7)

where (t, x) ∈ [0, T ) × [0, 1] (see Remark 2.4). Finally, we can mention the weight
functions

ρ0 = e−sAζ−5/6, ρ̂ = e−s(A+Ā)/2(ζ∗)−11/6, ρ∗ = e−sĀ(ζ∗)−17/6, (2.8)

associated with the desired spaces E and F . We observe that ρ̂2 ≤ ρ0ρ∗ and that
there exists a constant CT > 0, only depending on T , such that 0 < CT ≤ ρ∗ ≤
Cρ̂ ≤ Cρ0. Thus,

L2(Q; ρ20) ↪→ L2(Q; ρ̂2) ↪→ L2(Q; ρ2∗) ↪→ L2(Q).

Remark 2.4. In (2.7), we define τ = τ(t) satisfying limt→0+ τ(t) = τ(0) > 0. It
plays a crucial role in order to guarantee that (1.1) is locally null-controllable at
the time T > 0, as stated in Theorem 1.6. Precisely, each function given in (2.7) is
based on the weights which will appear in (2.13). As a result, since ρ0(t) → +∞, as
t→ T−, and ρ0(0) > 0 (since m(0) > 0), it is possible to conclude that u(T, x) = 0
for any for u ∈ L2(Q; ρ20). Hence, it seems reasonable to require that, if (u, h) ∈ E,
then u must belong to L2(Q; ρ20).

Now, we are ready to define E and F . Let us consider

U := H1(0, T ;L2(0, 1)) ∩ L2(0, T ;H2
a) ∩ C0([0, T ];H1

a),

as in Proposition 2.2, and put Lu := ut − (aux)x for each u ∈ U . Under all these
notations, we set, for the (WDC), the Hilbert spaces

E :=
{
(u, h) ∈ U × L2(Qω; ρ

2
∗) : ρ0u, ρ0(Lu− hχω) ∈ L2((0, T )× (0, 1))

}
, (2.9)

and

F := L2(Q; ρ20)×H1
a , (2.10)



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 7

equipped with the norms

∥(u, h)∥E :=
(
∥u∥2ρ20 + ∥hχω∥2ρ2∗ + ∥Lu− hχω∥2ρ20 + ∥u(0, ·)∥2H1

a

)1/2

,

and

∥(g, v)∥F :=
(
∥g∥2ρ20 + ∥v∥2H1

a

)1/2

,

respectively. We observe that, for the (SDP), the definition of E must also contain
the condition aux(t, 0) ≡ 0, a.e. in [0, T ], while the definition of F remains the
same.

Since we have already defined the weight functions, and identified the Hilbert
spaces E and F , as well as the mapping H : E → F , given in (2.1), we are supposed
to verify that H satisfies the hypotheses of Theorem 2.1 (Section 3). To do that,
we need to establish a Carleman estimate that will guarantee those hypotheses.

2.2. Carleman inequality. In this second part of Section 2, we present a key
Carleman estimate, closely related to those properties of H : E → F that we
expect to prove. We start taking into consideration the adjoint system associated
with (2.2), given by

−vt − (a(x)vx)x + c(t, x)v = F, (t, x) ∈ Q,
v(t, 0) = 0, t ∈ (0, T )

or

(avx)(t, 0) = 0, t ∈ (0, T )

v(T, x) = vT (x), x ∈ (0, 1),

(2.11)

where F ∈ L2(Q) and vT ∈ L2(0, 1). Now, we consider the functions and the
constants given in (2.5) and (2.6), and define

σ(x, t) :=
η(x)

[t(T − t)]4
, and φ(x, t) :=

ηr(x)

[t(T − t)]4
,

where (t, x) ∈ (0, T ) × [0, 1]. Our desired Carleman and observability inequalities,
mentioned above, will be achieved as consequences of the next lemma, whose proof
can be found in [20], for the (WDC), and in [19], for the (SDC).

Lemma 2.5. There exist C > 0 and λ0, s0 > 0 such that every solution v of (2.11)
satisfies, for all s ≥ s0 and λ ≥ λ0, the estimate∫ T

0

∫ 1

0

e2sφ
(
(sλ)σav2x + (sλ)5/3σ5/3v2

)
≤ C

(∫ T

0

∫ 1

0

e2sφ|F |2 + (λs)17/3
∫ T

0

∫
ω

e2sφσ17/3v2
)
,

(2.12)

where the constants C, λ0 and s0 only depend on ω, a, ∥c∥L∞(Q) and T .

The functions in (2.8) were not directly based on the weights which appear in
(2.12), because

lim
t→0+

1

[t(T − t)]4
= +∞.

Instead of that, we have taken τ = τ(t) in order to build ρ0, ρ̂ and ρ∗ (recall Remark
2.4).



8 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Next, we will state a new version of (2.12), involving the weights given in (2.8),
whose proof can be done following the same steps of [19, Prop. 3.6].

Proposition 2.6 (Carleman Inequality). There exist C > 0 and λ0, s0 > 0 such
that every solution v of (2.11) satisfies, for all s ≥ s0 and λ ≥ λ0, the estimate∫ T

0

∫ 1

0

e2sA
[
sλζa|vx|2 + (sλ)5/3ζ5/3|v|2

]
≤ C

(∫ T

0

∫ 1

0

e2sA|F |2 + (sλ)17/3
∫ T

0

∫
ω

e2sA(ζ∗)17/3|v|2
)
,

(2.13)

where the constants C, λ0 and s0 only depend on ω, a, ∥c∥L∞(Q) and T .

We would like to complete this section making some brief comments about
Lemma 2.5 and Proposition 2.6.

(a) For the (WDC), Lemma 2.5 and Proposition 2.6 are both detailed in [20].
In that paper, the discussion is organized under the presentation of several
technical lemmas, whose proofs rely on energy estimates, as well as on the
crucial Hardy-Poincaré inequality proved in [1];

(b) For the (SDC), we follow the same strategy used for the (WDC). However,
considering K ∈ [1, 2), as in assumption 1.1, the case K = 1 deserves a
special attention (precisely, see [19, Lemma 3.2]);

(c) The definition of ρ0 in (2.8) is inspired by the integral

(sλ)5/3
∫ T

0

∫ 1

0

e2sAζ5/3|v|2,

which appears in (2.13), according to a standard argument. Likewise, ρ̂ and
ρ∗ are set having in mind the definition of ρ0, however, there are technical
reasons to consider Ā = Ā(t) and ζ∗ = ζ∗(t) in their expressions;

(d) It is well-known that (2.13) implies the observability inequality

∥v(0)∥2L2(0,1) ≤ C

∫ T

0

∫
ω

e2sA(sλ)17/3(ζ∗)
17/3|v|2, (2.14)

valid for any solution v of (2.11), with F ≡ 0. In fact, this inequality holds
if λ > 0 and s > 0 are sufficiently large.

3. Properties of the mapping H

This section we prove the properties defined in (2.1), which required to ap-
ply Lyusternik’s Theorem, namely: H′(0, 0) must be onto and H must belong to
C1(E,F ). These tasks will be done in the next two subsections. However, before
presenting them, let us state a global null-controllability result for the linearized
system (2.2), as well as some additional regularity of this system that, as we have
already pointed out in Section 2, will be necessary to check the required hypotheses
over H. Its proof will be given at the end of this section.

Proposition 3.1. If T > 0 and (u0, g) ∈ H1
a × L2(Q; ρ20), then there exists a

state-control pair (u, h) ∈ L2(Q; ρ20) × L2(Qω; ρ
2
∗) such that the null-controllability

of (2.2), at time T > 0, holds.
Furthermore, we have
√
aux ∈ L2(Q; ρ̂2) : ut, (aux)x ∈ L2(Q; ρ̂2), ρ̂u, ρ∗

√
aux ∈ L∞(0, T ;L2(0, 1)),



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 9

and there exists C > 0 such that

sup
[0,T ]

∥ρ̂u(t, ·)∥2L2(0,1) + sup
[0,T ]

∥ρ∗
√
aux(t, ·)∥2L2(0,1)

+ ∥
√
aux∥2ρ̂2 + ∥ut∥2ρ2∗ + ∥(aux)x∥2ρ2∗

≤ C(∥u∥2ρ20 + ∥hχω∥2ρ2∗ + ∥g∥2ρ20 + ∥u0∥2H1
a
).

(3.1)

3.1. Surjectiveness of H′(0,0). Before we establish the properties of H, let us
verify that H is well defined. To check that, it will be essential to know that
au ∈ L∞(0, 1), for each u ∈ H1

a . It is always true for K ̸= 1, where K ∈ [0, 2) is
mentioned in Hypothesis 1.1. For the (WDC), it comes from the continuous embed-
ding H1

a ↪→ L∞(0, 1). For the (SDC) it comes from the fact that a ∈ W 1,∞(0, 1).
For the caseK = 1, we could just prove it for θ ≥ 1/2, where θ is given in (1.3). The
case 0 < θ < 1/2 remains open. A detailed proof of these facts will be presented in
Appendix 5 (see Propositions 5.2 and 5.1).

Lemma 3.2. The mapping H : E → F , given in (2.1), is well defined, recalling
that the spaces E and F are defined in (2.9) and (2.10), respectively.

Proof. For each (u, h) ∈ E, let us check that H(u, h) ∈ F . Clearly, H2(u, h) =
u(·, 0) ∈ H1

a . Also, recalling assumptions 1.1, 1.3 and 1.4, we have∫ T

0

∫ 1

0

ρ20|H1(u, v, h)|2

=

∫ T

0

∫ 1

0

ρ20 |ut − ℓ(au)(aux)x + f(t, x, u)− hχω|2

≤ 3

∫ T

0

∫ 1

0

ρ20|Lu− hχω|2 + 3

∫ T

0

∫ 1

0

ρ20 |ℓ(au)− ℓ(0)|2 |(aux)x|2

+ 3

∫ T

0

∫ 1

0

ρ20|f(t, x, u)− f(t, x, 0)|2

≤ 3∥(u, h)∥2E + C

∫ T

0

∫ 1

0

ρ20|au|2|(aux)x|2 + C

∫ T

0

∫ 1

0

ρ20|u|2

≤ C∥(u, h)∥2E + C

∫ T

0

∫ 1

0

ρ20|au|2|(aux)x|2.

At this point, we must estimate I :=
∫ T
0

∫ 1

0
ρ20|au|2|(aux)x|2. We start recalling

that A = τ(t)ηr(x) ≥ τ(t)η̂r and au ∈ L∞(0, 1) to obtain

I =

∫ T

0

∫ 1

0

e−2sAζ−5/3|au|2|(aux)x|2

≤
∫ T

0

∫ 1

0

e−2sτη̂rη−5/3τ−5/3|au|2|(aux)x|2

≤ C

∫ T

0

e−2sτη̂rτ−5/3

∫ 1

0

|au|2|(aux)x|2

≤ C

∫ T

0

e−2sτη̂rτ−5/3(∥u∥2L2(0,1) + ∥
√
aux∥2L2(0,1))

∫ 1

0

|(aux)x|2

= C
(∫ T

0

e−2sτη̂rτ−5/3∥u∥2L2(0,1)∥(aux)x∥
2
L2(0,1)



10 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

+

∫ T

0

e−2sτη̂rτ−5/3∥
√
aux∥2L2(0,1)∥(aux)x∥

2
L2(0,1)

)
=: I1 + I2 . (3.2)

Recall that η̄r = 3η∗r−2η̂r < 0 which implies κ := η̂r−2η̄r > 0 and, consequently,

e−2sτη̂rτ−5/3ρ−4
∗

= e−2sτη̂rτ−5/3e4sĀ(ζ∗)34/3

= e−2sτ(η̂r−2η̄r)τ29/3 = e−2sκττ29/3 ≤ C.

(3.3)

Since ρ∗ ≤ Cρ̂, from inequality (3.1), we have

I1 =

∫ T

0

(e−2sτη̂rττ−5/3ρ−4
∗ )(ρ2∗∥u∥2L2(0,1))(ρ

2
∗∥(aux)x∥2L2(0,1))

≤ C sup
t∈[0,T ]

∥ρ̂u(t, ·)∥2L2(0,1)∥(aux)x∥
2
ρ2∗

≤ C∥(u, h)∥4E
(3.4)

and

I2 =

∫ T

0

(e−2sτη̂rττ−5/3ρ−4
∗ )(ρ2∗∥

√
aux∥2L2(0,1))(ρ

2
∗∥(aux)x∥2L2(0,1))

≤ C sup
t∈[0,T ]

∥ρ∗
√
aux(t, ·)∥2L2(0,1)∥(aux)x∥

2
ρ2∗

≤ C∥(u, h)∥4E .

(3.5)

As a conclusion, H1(u, v) ∈ L2(Q, ρ20) and the proof is complete. □

Proposition 3.3. H′(0, 0) ∈ L(E;F ) is onto.

Proof. Take (g, u0) ∈ F . By Propositions 3.1 and 2.2, there exists (u, h) ∈ E that
solves (2.2). In other words,

H′(0, 0)(u, h) = (H′
1(0, 0)(u, h),H′

2(0, 0)(u, h))

= (ut − (a(x)ux)x + c(t, x)u− hχw, u(·, 0))
= (g, u0).

This completes the proof. □

3.2. H is continuously differentiable. In this subsection, we will prove that
H ∈ C1(E,F ). The proof will rely on the additional regularity described in (3.1).

Proposition 3.4. The mapping H is continuously differentiable.

Proof. It is clear that H2 ∈ C1(E,F ). So that, the proof is focused on checking
that H1 has a continuous Gateaux derivative on E.

For (u, h), (ū, h̄) ∈ E and λ > 0, set

b := ℓ(au)a(x), bλ := ℓ(a(u+ λū))a(x), f := f(t, x, u),

fλ := f(t, x, u+ λū), f3 := D3f(t, x, u).

Claim 1: Given (u, h) ∈ E, the linear mapping L : E → L2(Q; ρ20), defined by

L(ū, h̄) := ūt − ℓ′(au)aū(aux)x + ℓ(au)(aūx)x + f3ū− h̄χω,

is the Gateaux derivative of H1 at (u, h) ∈ E.
Indeed, for each (ū, h̄) ∈ E, we have∥∥ 1

λ
(H1(u+ λū, h+ λh̄)−H1(u, h))− L(ū, h̄)

∥∥



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 11

=
∥∥∥ūt − [ℓ(a(u+ λū))(a(u+ λū)x)x − ℓ(au)(aux)x

λ

]
+

1

λ
(fλ − f)− h̄χω − L(ū, h̄)

∥∥∥
≤

∥∥∥[ℓ(a(u+ λū))− ℓ(au)

λ
− ℓ′(au)aū

]
(aux)x

∥∥∥
+ ∥[ℓ(a(u+ λū)) + ℓ(au)](aūx)x∥ρ20

+ ∥ 1
λ
(fλ − f)− f3ū∥ρ20

=: B1 +B2 +B3.

We will see that Bi → 0, as λ→ 0, for any i = 1, 2, 3.
Firstly, from assumption 1.4, for each (t, x) ∈ (0, 1)×(0, T ), we apply mean value

theorem to obtain u∗λ = u∗λ(t, x) ∈ R such that

B2
3 ≤

∫ T

0

∫ 1

0

ρ20 |(D3f(t, x, u
∗
λ)−D3f(t, x, u))ū|2 → 0,

as λ→ 0, where this convergence comes from Lebesgue’s theorem.
Secondly, applying assumption 1.3 and the mean value theorem again, there

exists sλ = sλ(t, x) ∈ R such that

B2
1 =

∫ T

0

∫ 1

0

ρ20

∣∣∣∣ℓ(a(u+ λū))− ℓ(au)

λ
− ℓ′(au)aū

∣∣∣∣2 |(aux)x|2
=

∫ T

0

∫ 1

0

ρ20|ℓ′(sλ)− ℓ′(au)|2|aū|2|(aux)x|2 → 0,

as λ→ 0. Since we can argue as in (3.2), (3.4) and (3.5) we obtain∫ T

0

∫ 1

0

ρ20|ℓ′(sλ)− ℓ′(au)|2|aū|2|(aux)x|2 ≤
∫ T

0

∫ 1

0

ρ20|aū|2|(aux)x|2

≤ C
(

sup
t∈[0,T ]

∥ρ̂ū(t, ·)∥2L2(0,1)∥(aux)x∥
2
ρ2∗

+ sup
t∈[0,T ]

∥ρ∗
√
aūx(t, ·)∥2L2(0,1)∥(aux)x∥

2
ρ2∗

)
≤ C∥(u, h)∥2E∥(ū, h̄)∥2E .

Analogously, there exists uλ = uλ(t, x) ∈ R such that

B2
2 =

∫ T

0

∫ 1

0

ρ20|ℓ(a(u+ λū)) + ℓ(au)|2|(aux)x|2

=

∫ T

0

∫ 1

0

ρ20|ℓ′(uλ)|2|aλū|2|(aūx)x|2

≤ Cλ2
∫ T

0

∫ 1

0

ρ20|aū|2|(aūx)x|2

≤ Cλ2∥(ū, h̄)∥2E → 0,

as λ→ 0. Thus, the Claim 1 is concluded.
Claim 2: The Gateaux derivative H′

1 : E → L(E;L2(Q; ρ20)) is continuous. Take
(u, h) ∈ E and let ((un, hn))∞n=1 be a sequence such that ∥(un, hn)− (u, h)∥E → 0.



12 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

We will prove that ∥H′
1(u

n, hn) −H′
1(u, h)∥L(E;L2(Q;ρ20))

→ 0. In fact, we consider

(ū, h̄) on the unit sphere of E. Since

H′
1(u, h)(ū, h̄) = ūt − ℓ′(au)aū(aux)x + ℓ(au)(aūx)x + f3ū− h̄χω

and

H′
1(u

n, hn)(ū, h̄) = ūt − ℓ′(aun)aū(aunx)x + ℓ(aun)(aūx)x +D3f(t, x, u
n)ū− h̄χω,

we obtain

∥(H ′
1(u

n, hn)−H ′
1(u, h))(ū, h̄)∥2ρ20

≤ C

∫ T

0

∫ 1

0

ρ20|ℓ′(aun)|2|aū|2|[a(u− un)x]x|2

+ C

∫ T

0

∫ 1

0

ρ20|ℓ′(aun)− ℓ′(au)|2|aū|2|(aux)x|2

+ C

∫ T

0

∫ 1

0

ρ20|ℓ(aun)− ℓ(au)|2|(aūx)x|2

+ C

∫ T

0

∫ 1

0

ρ20|ū|2|D3f(t, x, u
n)−D3f(t, x, u)|2

=: C(J1 + J2 + J3 + J4).

Once again, arguing as in (3.2), (3.4) and (3.5), we obtain

J1 ≤
∫ T

0

∫ 1

0

ρ20|aū|2|(a(u− un)x)x|2

≤ C
(

sup
t∈[0,T ]

∥ρ̂ū(t, ·)∥2L2(0,1)∥[a(u− un)x]x∥2ρ2∗

+ sup
t∈[0,T ]

∥ρ∗
√
aūx(t, ·)∥2L2(0,1)∥[a(u− un)x]x∥2ρ2∗

)
≤ C∥(ū, h̄)∥2E ∥(un, hn)− (u, h)∥2E .

Next, applying relation (3.3), we have

J2 ≤
∫ T

0

(e−2sτη̂rτ−5/3ρ−4
∗ )(ρ2∗∥aū∥2∞)

∫ 1

0

η−5/3ρ2∗|ℓ′(aun)− ℓ′(au)|2|(aux)x|2

≤ C
[ ∫ 1

0

ρ2∗(∥ū∥2L2(0,1) + ∥
√
aūx∥2L2(0,1))

∫ 1

0

ρ2∗|ℓ′(aun)− ℓ′(au)|2|(aux)x|2
]

≤ C
[

sup
t∈[0,T ]

∥ρ̂ū(t, ·)∥2L2(0,1) + sup
t∈[0,T ]

∥ρ∗
√
aūx(t, ·)∥2L2(0,1)

]
×
∫ T

0

∫ 1

0

ρ2∗|ℓ′(aun)− ℓ′(au)|2|(aux)x|2

≤ C∥(ū, h̄)∥2E
∫ T

0

∫ 1

0

ρ2∗|ℓ′(aun)− ℓ′(au)|2|(aux)x|2 → 0,

as n → +∞, where the convergence is a consequence of Lebesgue’s theorem. In a
very similar way,

J3 ≤
∫ T

0

(e−2sτη̂rτ−5/3ρ−4
∗ )(ρ2∗∥a(un − u)∥2∞)

∫ 1

0

η−5/3ρ2∗|(aūx)x|2



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 13

≤ C
[

sup
t∈[0,T ]

∥ρ̂(un − u)(t, ·)∥2L2(0,1) + sup
t∈[0,T ]

∥ρ∗
√
a(un − u)x(t, ·)∥2L2(0,1)

]
×
∫ T

0

∫ 1

0

ρ2∗|(aūx)x|2

≤ C∥(un, hn)− (u, h)∥2E∥(ū, h̄)∥2E .

At last, applying Hypothesis 1.3 and (3.1), we obtain

J4 =
(∫ T

0

∫ 1

0

ρ20|ū|2|D3f(t, x, u
n)−D3f(t, x, u)|2

)
≤ sup

(t,x)∈Q
|D3f(t, x, u

n)−D3f(t, x, u)|2
∫ T

0

∫ 1

0

ρ20|ū|2

≤ ∥(un, hn)− (u, h)∥2E∥(ū, h̄)∥2E ,

where we have also used the continuous embedding C([0, T ];H1
a) ↪→ C(Q). There-

fore,

H′
1(u

n, hn) → H′
1(u, h)

in L(E;L2(Q; ρ20)), which means that H′
1 : E → L(E;L2(Q; ρ20)) is a continuous

mapping, as stated in Claim 2. This completes the proof. □

3.3. Proof of Proposition 3.1.

Proof. Given T > 0 and (u0, g) ∈ H1
a × L2(Q; ρ20), let us consider the problem

ut − (a(x)ux)x + c(t, x)u = h+ g, (t, x) in Q,

u(t, 1) = 0, t ∈ (0, T ),
u(t, 0) = 0, (Weak), t ∈ (0, T )

or

(aux)(t, 0) = 0, (Strong), t ∈ (0, T )

u(0, x) = u0(x), xin (0, 1),

(3.6)

where h ∈ L2(Q). Observe that (3.6) is similar to (2.2), where we are replacing
hχω, with support in Qω, just by h. Our aim is to define, for each n ∈ N∗, a

functional Jn :
[
L2(Q)

]2 → R, minimizing each one of them subject to the natural
constraint determined by (2.2). It will allow us to obtain a sequence ((un, hn))

∞
n=1

of solutions to (2.2) converging, in some sense, to

(u, h) ∈ L2(Q; ρ20)× L2(Qω; ρ
2
∗),

which is also a solution to (2.2).
To do so, for each n ∈ N∗, let us define

An(t, x) =
A(T − t)4

(T − t)4 + 1
n

, barAn(t) =
Ā(T − t)4

(T − t)4 + 1
n

,

where (t, x) ∈ [0, T ]× [0, 1]. We also consider

ρn = e−sAn , ρ̄n = e−sĀn , ρ0,n = ρnζ
−5/6, ρ∗,n = ρ̄nζ

∗−17/6mn,

where

mn(x) =

{
1, x ∈ ω,

n, x /∈ ω.



14 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

These weight functions are built in such a way that

• ρ0,n and ρ∗,n are bounded from below by a positive constant only depending
on T ;

• ρ0,n and ρ∗,n are bounded from above by another positive constant depend-
ing on n and T .

For each n ∈ N∗, we set the functional Jn :
[
L2(Q)

]2 → R, given by

Jn(u, h) =
1

2

∫ T

0

∫ 1

0

ρ20,n|u|2 +
1

2

∫ T

0

∫ 1

0

ρ2∗,n|h|2,

for each (u, h) ∈ [L2(Q)]2. Since each Jn is lower semi-continuous, strictly convex
and coercive (see [21]), we can apply [23, Proposition 1.2] to obtain a unique (un, hn)
satisfying

J(un, hn) = min{J(u, h); (u, h) ∈ C}

where C = {(u, h) ∈
[
L2(Q)

]2
; (u, h) solves (2.2)}. Consequently, by Lagrange’s

Principle, for each n ∈ N∗, there exists a function pn solving the system

−pnt − (apnx)x + c(t, x)pn = −ρ20,nun, (t, x) ∈ Q,

pn(t, 1) = 0, t ∈ (0, T ),
pn(t, 0) = 0, (Weak), t ∈ (0, T )

or

(apnx)(t, 0) = 0, (Strong), t ∈ (0, T )

pn(T, x) = 0, x ∈ (0, 1),

pn = ρ2∗,nhn, (t, x) ∈ Q.

(3.7)

By standard arguments, (3.7) can help us to prove that Jn(un, hn) ≤ C
√
Jn(un, hn)

for all n ∈ N∗, i.e., (Jn(un, hn))
∞
n=1 is a numerical bounded sequence.

Since ρ20,n ≥ CT and ρ2∗,n ≥ CTmn, we deduce that

∥un∥2L2 +

∫ T

0

∫
ω

|hn|2 + n

∫ T

0

∫
[0,1]\ω

|hn|2 ≤ CJn(un, hn) ≤ C,

whence there exists (u, h) ∈ L2(Q)× L2(Qω), such that

un ⇀ u, in L2(Q) and hn ⇀ hχω in L2(Q),

up to subsequences. From this, we have

ρ0,nun ⇀ ρ0u and ρ∗,nhn ⇀ ρ∗hχω in L2(Q). (3.8)

Consequently, u ∈ L2(Q; ρ20) and h ∈ L2(Qω; ρ
2
∗). Recalling that (un, hn) is a

solution of (2.2), for each n ∈ N∗, a passing to the limit argument implies that
(u, h) also solves (2.2).

Thinking about a better presentation, the estimates mentioned in (3.1) will be
established in two subsequent lemmas. □

Lemma 3.5. Under the assumptions of Proposition 3.1, we have that
ρ̂u ∈ L∞(0, T ;L2(0, 1)),

√
aux ∈ L2(Q; ρ̂2), and there exists C > 0 such that

sup
t∈[0,T ]

∥ρ̂u(t, ·)∥2L2(0,1) + ∥
√
aux∥2ρ̂2 ≤ C(∥u∥2ρ20 + ∥hχω∥2ρ2∗ + ∥g∥2ρ20 + ∥u0∥2H1

a
).



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 15

Proof. Multiplying the PDE in (2.2) by ρ̂2u, integrating in [0, 1] and using the two
relations

1

2

d

dt

∫ 1

0

ρ̂2u2 =

∫ 1

0

ρ̂2utu+

∫ 1

0

ρ̂ρ̂tu
2

and ∫ 1

0

ρ̂2(aux)xu = −2

∫ 1

0

ρ̂ρ̂xauux −
∫ 1

0

ρ̂2au2x,

we obtain

1

2

d

dt

∫ 1

0

ρ̂2u2 +

∫ 1

0

ρ̂2au2x

= −
∫ 1

0

ρ̂2cu2 +

∫ 1

0

ρ̂2uhχω +

∫ 1

0

ρ̂2gu+

∫ 1

0

ρ̂ρ̂tu
2 − 2

∫ 1

0

ρ̂ρ̂xauux

=: I1 + I2 + I3 + I4 + I5.

(3.9)

Above, we have also used u(t, 0) = u(t, 1) ≡ 0 for (WDP), and u(t, 1) = aux(t, 0) ≡
0 for (SDP). Now, since ρ∗ ≤ Cρ̂ ≤ Cρ0 and ρ0ρ∗ ≥ ρ̂2, we obtain

I1 ≤ C

∫ 1

0

ρ20|u|2,

I2 ≤ C
(1
2

∫ 1

0

ρ2∗|hχω|2 +
1

2

∫ 1

0

ρ20|u|2
)
,

I3 ≤ C
(1
2

∫ 1

0

ρ20|g|2 +
1

2

∫ 1

0

ρ20|u|2
)
.

Let us estimate I4. Firstly, rewriting A and Ā as A(t, x) = ζ(t, x)η̃(x), where
η̃ = ηr/η, and Ā(t, x) = ζ(t, x) η̄rη , we have

|ρ̂t| =
∣∣∣− s(

ηr + η̄r
2η

)ζte
−s(A+Ā

2 )(ζ∗)−11/6 − 11

6
e−s(

A+Ā
2 )(ζ∗)−11/6ζ∗t

∣∣∣
≤ e−sA

[
sη̄r(ζ

∗)−11/6|ζt|+
11

6
(ζ∗)−17/6|ζ∗t |

]
.

Secondly, we obtain

|ρ̂ρ̂t| ≤ e−2sA
[
sη̄r(ζ

∗)−11/6|ζt|+
11

6
(ζ∗)−17/6|ζ∗t |

]
≤ Ce−2sA(ζ−2|ζt|+ ζ−3|ζt|)ζ−5/3

≤ Cρ20,

for all t ∈ [0, T ], following that I4 ≤ C
∫ 1

0
ρ20|u|2. Next, using

|ρ̂x| =
∣∣− s(

Ax + Āx
2

)e
−s

(
A+Ā

2

)
(ζ∗)−11/6

∣∣
≤ Ce−sAζe−s(

A+Ā
2 )(ζ∗)−11/6

≤ e−sA(ζ)−5/6

= ρ0,

we obtain

I5 ≤ 1

2

∫ 1

0

ρ̂2au2x + 2

∫ 1

0

ρ̂2xau
2 ≤ 1

2

∫ 1

0

ρ̂2au2x + 2

∫ 1

0

ρ20u
2.



16 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Hence, (3.9) gives us

d

dt

∫ 1

0

ρ̂2|u|2 +
∫ 1

0

ρ̂2a|ux|2 ≤ C
(∫ 1

0

ρ20|u|2 +
∫ 1

0

ρ2∗|hχω|2 +
∫ 1

0

ρ20|g|2
)
.

Integrating in time, we reach the desired estimate. □

Lemma 3.6. Under the assumptions of Proposition 3.1 we have

ρ∗
√
aux ∈ L∞(0, T ;L2(0, 1));ut, (aux)x ∈ L2(Q; ρ2∗),

and there exists C > 0 such that

sup
t∈[0,T ]

∥ρ∗
√
aux(t, ·)∥2L2(0,1) + ∥ut∥2ρ2∗ + ∥(aux)x∥2ρ2∗

≤ C(∥u∥2ρ20 + ∥hχω∥2ρ2∗ + ∥g∥2ρ20 + ∥u0∥2H1
a
).

Proof. Firstly, let us estimate the first and the second terms on the left side of the
desired inequality. Multiplying the PDE in (2.2) by ρ2∗ut and integrating in [0, 1],
we have∫ 1

0

ρ2∗u
2
t =

∫ 1

0

ρ2∗uthχω +

∫ 1

0

ρ2∗gut −
∫ 1

0

ρ2∗c(t, x)uut +

∫ 1

0

ρ2∗(aux)xut

= I1 + I2 − I3 + I4.

(3.10)

Using Young’s inequality with ε and ρ∗ ≤ Cρ̂ ≤ Cρ0 ≤ Cρ, we obtain

I1 ≤
∫ 1

0

ρ2∗|hχω||ut| ≤ ε

∫ 1

0

ρ2∗|ut|2 +
1

4ε

∫ 1

0

ρ2∗|hχω|2,

I2 ≤
∫ 1

0

ρ2∗|gut| ≤ ε

∫ 1

0

ρ2∗|ut|2 +
1

4ε

∫ 1

0

ρ2∗|g|2 ≤ ε

∫ 1

0

ρ2∗|ut|2 +
C

4ε

∫ 1

0

ρ20|g|2,

−I3 ≤
∫ 1

0

|c(t, x)|ρ2∗|uut|

≤ ε

∫ 1

0

ρ2∗|ut|2 +
∥c∥∞
4ε

∫ 1

0

ρ2∗|u|2

≤ ε

∫ 1

0

ρ2∗|ut|2 +
C

4ε

∫ 1

0

ρ20|u|2 .

Since ut(t, 0) = ut(t, 1) ≡ 0 for the (WDP) and aux(t, 0) = ut(t, 1) ≡ 0 for the
(SDP), we integrate by parts to obtain

I4 = ρ2∗auxut
∣∣x=1

x=0
−

∫ 1

0

(ρ2∗utxaux

= −1

2

d

dt

∫ 1

0

ρ2∗au
2
x +

1

2

∫ 1

0

(ρ2∗)tau
2
x

= −1

2

d

dt

∫ 1

0

ρ2∗au
2
x +

1

2
I41.

(3.11)

Hence ∫ 1

0

ρ2∗|ut|2 +
1

2

d

dt

∫ 1

0

ρ2∗a|ux|2 = I1 + I2 − I3 +
1

2
I41. (3.12)

At this point, we observe that

(ρ∗)t = −sτtη̄re−sĀ(ζ∗)−17/6 − 17

6
e−sĀ(ζ∗)−23/6τtη

∗



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 17

and, consequently,

|ρ∗(ρ∗)t| ≤ Ce−2sĀ[|τtη∗|(ζ∗)−17/3 + |τtη∗|(ζ∗)−20/3]

= Ce−2sĀ(ζ∗)−11/3[(ζ∗)−2 + (ζ∗)−3]|ζ∗t )|
≤ Cρ̂2.

So that

I41 ≤ C

∫ 1

0

ρ̂2au2x .

As a result, taking a sufficiently small ε > 0, we obtain∫ 1

0

ρ2∗u
2
t +

1

2

d

dt

∫ 1

0

ρ2∗au
2
x

≤ C
(∫ 1

0

ρ2∗|hχω|2 +
∫ 1

0

ρ20g
2 +

∫ 1

0

ρ20u
2 +

∫ 1

0

ρ̂2au2x

)
,

which implies

sup
t∈[0,T ]

∥ρ∗
√
aux(t, ·)∥2L2(0,1) + ∥ut∥2ρ2∗ ≤ C(∥u∥2ρ20 + ∥hχω∥2ρ2∗ + ∥g∥2ρ20 + ∥u0∥2H1

a
).

To estimate ∥(aux)x∥2ρ2∗ , we proceed analogously, multiplying the PDE in (2.2)

by −ρ217(aux)x and integrating in [0, 1]. The details can be seen in [20, Lemma
4.3]. □

4. Main result and further comments

Proof of Theorem 1.6. In Section 3, we have proved that H : E → F is a continu-
ously differentiable mapping, whose derivative H′(0, 0) ∈ L(E;F ) is onto (Lemma
3.2, and Propositions 3.3 and 3.4). As a result, Theorem 2.1 can be applied in order

to obtain a sufficiently small ε > 0 and a right inverse mapping H̃ : Bε(0) ⊂ F → E
of H. Hence, taking u0 ∈ H1

a satisfying ∥u0∥H1
a
< ε, we can see that

(u, h) := H̃(0, u0)

solves
ut − ℓ(au)(a(x)ux)x + f(t, x, u) = hχω, (t, x) ∈ Q,

u(t, 1) = 0, in (0, T ),
u(t, 0) = 0, (Weak), t ∈ (0, T )

or

(aux)(t, 0) = 0, (Strong), t ∈ (0, T )

u(0, x) = u0, x ∈ (0, 1),

u(T, x) = 0, x ∈ (0, 1),

(4.1)

where the last condition comes from Remark 2.4. It completes the proof. □

In the context of degenerate equations, there are many important questions
which have not been dealt yet, or for which much more investigation should be
performed. Among them, we would like to emphasize the following ones: the
controllability of linear problems in higher-dimensional spatial domains, using the
method of moments; the boundary controllability obtained as the limit of internal
controllability, in nonlinear cases



18 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Specifically talking about some numerical perspective regarding this current pa-
per, we include some comments pointing out possible future works. In order to
solve the proposed controllability problem numerically, some different approaches
can be combined to perform an approximate and reliable analysis of the system. A
practical and well-established strategy in the literature involves the use of the finite
element method (FEM). This methodology makes possible a complete discretization
of the problem into a finite-dimensional space, allowing numerical approximations
through well-defined iterative processes. So that, a natural future study could be
the comparison between two approaches: the primal method and the dual one.
The primal method is more straightforward, focusing directly on the finite element
formulation by discretizing the spatial and temporal domains, updating the solu-
tion iteratively in an approximate solution space Vh, with dim(Vh) < ∞. On the
other hand, the dual method introduces some pre-programming complexity, since
it is incorporated dual variables into the weak formulation of the problem. This
process reformulates the original problem as a constrained optimization problem
within a variational framework, where iterative algorithms are employed to solve
both primal and dual variables, simultaneously. The advantage and disadvantage
of each on of this methods are properly discussed in [27]. At this point, we should
say that some initial numerical insights into the class of problems proposed here
can be found in [19], where iterative algorithms, adapted for nonlinear parabolic
problems, are presented. Besides, in [19], an effective approach for numerical it-
erations is considered, by adjusting quasi-Newton method for null controllability
problems. The whole numerical analysis of null-controllability problems is com-
pletely associated with well-chosen weight functions, such those defined in Section
2. We think that numerical simulations for the null-controllability of degenerate
quasilinear equations could be based on [17, 18, 19, 28, 30] and [31].

5. Appendix: Essential boundedness of au

This appendix shows that, for each u ∈ H1
a , we have au ∈ L∞(0, 1). This fact

is essential in to prove that the mapping H : E → F , set in (2.1), is well defined
and continuously differentiable. For the whole discussion, let us consider K ∈ [0, 2)
mentioned in assumption 1.1 and θ ∈ R given in (1.3).

Proposition 5.1. Given u ∈ H1
a , we have au ∈ L∞(0, 1) and Ca > 0, only

depending on the function a, such that

∥au∥L∞(0,1) ≤ Ca∥u∥H1
a
,

provided that one of the following conditions holds:

(a) K ̸= 1;
(b) K = 1 and θ ≥ 1/2.

The proof of this proposition will be a consequence of the four next lemmas.

Lemma 5.2. The continuous embedding H1
a ↪→ L∞(0, 1) holds for the (WDC). In

particular, au ∈ L∞(0, 1), for any u ∈ H1
a .

Proof. In fact, given u = u(x) ∈ H1
a , we can take

|u(x)| ≤
∣∣ ∫ 1

x

ux
∣∣ ≤ (∫ 1

x

1

a

)1/2(∫ 1

x

au2x

)1/2

≤ ∥1
a
∥2L1∥u∥H1

a
,



EJDE-2025/15 NULL-CONTROLLABILITY DEGENERATE QUASILINEAR EQUATIONS 19

for each x ∈ (0, 1]. Hence, there exists Ca > 0, only depending on the function a,
such that ∥u∥L∞ ≤ Ca∥u∥H1

a
. □

Lemma 5.3. If a ∈W 1,∞(0, 1), then au ∈ L∞(0, 1), for each u ∈ H1
a .

Proof. Indeed, given y ∈ (0, 1], since a ∈ C1([y, 1]) and u is absolutely continuous
in [y, 1], we have

|au(y)| ≤
∫ 1

y

|a′u| dx+

∫ 1

y

a|ux| dx

≤
∫ 1

0

|a′u| dx+

∫ 1

0

a|ux| dx

≤ ∥a′∥L∞(0, 1)∥u∥L2(0,1) +

∫ 1

0

√
a
√
a|ux| dx

≤ ∥a′∥L∞(0,1)∥u∥L2(0,1) + ∥a∥L∞(0,1)∥
√
aux∥L2(0,1)

≤ ∥a∥W 1∞(0,1)∥u∥H1
a
.

Since, y ∈ (0, 1] is arbitrary, the desired result follows. □

Lemma 5.4. If K > 1, then a ∈W 1,∞(0, 1). In particular, au ∈ L∞(0, 1).

Proof. We only need to prove that a′ ∈ L∞(0, 1). From (1.3), there exists ε > 0
such that θa ≤ xa′, ∀x ∈ (0, ε]. In particular a′ > 0 and, since θ > 1, the mapping
x 7→ a

x is increasing in (0, ε]. So that, using (1.2), we have

0 ≤ a′(x) ≤ Ka(x)

x
≤ Ka(ε)

ε
, ∀x ∈ (0, ε].

On the other hand, a′ ∈ C0([ε, 1]), following that a′ ∈ L∞(0, 1) and

∥a′∥L∞(0,1) ≤ max
{Ka(ε)

ε
, max
x∈[ε,1]

|a′(x)|
}
.

□

Lemma 5.5. If K = 1 and θ ≥ 1/2, then au ∈ L∞(0, 1), for any u ∈ H1
a .

Proof. From (1.3), there exists ε > 0 such that θa ≤ xa′ for all x ∈ (0, ε]. This
implies that a′ > 0 and

x 7→ a

xθ
is nondecreasing in (0, ε]. (5.1)

Since a ∈ C0([0, 1]) and u ∈ H1
a ↪→ H1(ε, 1) ↪→ C0([ε, 1]), we have that au ∈

L∞(ε, 1) and

|au(y)| ≤ ∥a∥L∞(0,1)∥u∥H1
a
, for all y ∈ [ε, 1].

We just need to prove au ∈ L∞(0, ε). Indeed, given y ∈ (0, ε], we have

a2u2(y) = a2u2(ε)−
∫ ε

y

(a2u2(x))x dx,

whence,

|au(y)|2 ≤ ∥a∥2L∞(0,1)∥u∥
2
H1

a
+ 2

∫ ε

y

a2|u||ux| dx+ 2

∫ ε

y

aa′|u|2 dx.



20 P. P. DE CARVALHO, R. DEMARQUE, J. LÍMACO, L. VIANA EJDE-2025/15

Let us estimate each one of the two last integrals. The first of them is easier, since
we just use Hölder inequality to obtain∫ ε

y

a2|u||ux| dx ≤ ∥a∥3/2L∞(0,1)∥u∥L2(0,1)∥
√
aux∥L2(0,1) ≤ ∥a∥3/2L∞(0,1)∥u∥

2
H1

a
.

To estimate the second integral, let us prove that aa′ is bounded in (0, ε]. In fact,
using (1.2) and (5.1), we have

aa′ ≤ a2

x
= (

a

xθ
)2x2θ−1 ≤ a(ε)

ε
, ∀x ∈ (0, ε],

since θ ≥ 1/2. Hence,∫ ε

y

aa′|u|2 dx ≤ a(ε)

ε

∫ 1

0

|u|2 dx =
a(ε)

ε
∥u∥2L2(0,1).

Therefore, au ∈ L∞(0, 1) and

∥au∥L∞(0,1) ≤ C∥u∥H1
a
,

where

C =
(
max

{
∥a∥2L∞(0,1), 2∥a∥

3/2
L∞(0,1),

2a(ε)

ε

})1/2

> 0.

□

Remark 5.6. The prototype function ã(x) = xα, with α ∈ [1, 2), belongs to
W 1,∞(0, 1), therefore ãu ∈ L∞(0, 1), for any u ∈ H1

a . Nevertheless, for any p ∈
(0, 1), consider the function a(x) = xp + x and note that

• a′ = pxp−1 + 1 ⇒ a ̸∈W 1,∞(0, 1);
• xa′ = pxp+x ≤ xp+x = a, that is, a satisfies assumption 1.1, with K = 1;
• aa′ = px2p−1 + (p+ 1)xp + x is bounded if, and only if, p ≥ 1/2.

Therefore, the proof given in Lemma 5.5 does not work for p < 1/2.

Acknowledgments. We want to thank the anonymous referees for their thor-
ough review and insightful comments, which significantly improved the quality of
this manuscript. Their expertise and constructive feedback greatly enhanced the
theoretical and numerical aspects, as well as the clarity and presentation of our
work.

P. P. De Carvalho was partially supported by CNPq grant No. 306541/2022-
0 (Brazil). R. Demarque was supported by FAPERJ (Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro) under grant No. E-
26/210.456/2024. J. Ĺımaco was partially supported by CNPq, grant No. 310860/
2023-7 (Brazil).

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Pitágoras P. de Carvalho
Coordenação de Matemática, Universidade Estadual do Piaúı, Teresina, PI, 64002-150,

Brazil
Email address: pitagorascarvalho@gmail.com

Reginaldo Demarque (corresponding author)

Departamento de Ciências da Natureza, Universidade Federal Fluminense, Rio das Os-
tras, RJ, 28895-532, Brazil

Email address: reginaldodr@id.uff.br

Juan Ĺımaco

Departamento de Matemática Aplicada, Universidade Federal Fluminense, Niterói, RJ,

24210-201, Brazil
Email address: jlimaco@id.uff.br

Luiz Viana
Departamento de Análise, Universidade Federal Fluminense, Niterói, RJ, 24210-2010,

Brazil

Email address: luizviana@id.uff.br


	1. Introduction
	2. Preliminary results
	2.1. Notation and results related to the local inversion argument
	2.2. Carleman inequality

	3. Properties of the mapping H
	3.1. Surjectiveness of H'(0,0)
	3.2. H is continuously differentiable
	3.3. Proof of Proposition ??

	4. Main result and further comments
	5. Appendix: Essential boundedness of au
	Acknowledgments

	References