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Advances in Nonlinear Analysis

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On the weakly degenerate Allen-Cahn equation

Maicon Sônego
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  • Instituto de Matemática e Computação – Universidade Federal de Itajubá, MG, Itajubá, Brazil
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Published Online: 2019-05-16 | DOI: https://doi.org/10.1515/anona-2020-0004


In this paper we consider a one-dimensional Allen-Cahn equation with degeneracy in the interior of the domain and Neumann boundary conditions. We allow the diffusivity coefficient vanish at some point of the space domain and we are addressed on the existence of stable non-constant solution.

Keywords: degenerate problem; Allen-Cahn equation; stable solution

MSC 2010: Primary: 35A15; Secondary: 35B35; 35A01

1 Introduction

Consider the following semi-linear problem


where f(u) = uu3 and a(⋅) ∈ C([0, 1]) satisfies

  • (H1)

    a(x0) = 0 at some x0 ∈ (0, 1);

  • (H2)

    a(x) > 0 for x ∈ [0, 1] ∖ {x0} and

  • (H3)

    1/aL1(0, 1).

With these conditions, the parabolic problem (1.1) is called weakly degenerate (see [1], for instance) with degeneracy in the interior of the space domain. A typical example for a(⋅) satisfying the conditions (H1)–(H3) is given by a(x) = ∣xx0α (0 < α < 1).

In this paper we are concerned to ensure the existence of solution to the evolution problem (1.1) and, mainly, to study the existence of stable solutions of the variational problem where the critical points of energy functional are stationary solutions of (1.1); that is


Roughly speaking, we study the role of the degeneracy of function a(⋅) at x0 to the existence of a local minimizer of the energy functional associated to (1.2). This class of solutions (which we call stable solutions, see Definition 3.1 and Remark 3.2), in general, enjoys better qualitative and quantitative properties (monotonicity and symmetry, for instance) than the other solutions. In particular, when a minimizer is isolated – in this case we say asymptotically stable – it can describe the whole dynamics of the corresponding parabolic problem. We refer to the excellent monograph [2] for a comprehensive and complete presentation of the main results available on stable solutions.

Degenerate problems have always attracted the attention of many authors, [1, 3, 4, 5, 6, 7] and references therein. In these works it is assumed that the function a(⋅) degenerates at the boundary or in the interior of the space domain and the results are mainly related to the theory of control. In particular, in [1] the authors study degenerate parabolic problems with interior degeneracy, under Dirichlet boundary conditions, and show that under suitable assumptions, they generate analytic semi-groups. In addition, some applications to linear and semi-linear parabolic problems are provided. Similar results were achieved in [7], under Neumann boundary conditions, whose focus was to obtain Carleman estimates. With regard to existence of solution to the problem (1.1), the main results of the present work are based on these last two articles.

Here we will focus on a particular reaction term related to Allen-Cahn problem, f(u) = uu3. The Allen-Cahn equation has its origin in the theory of phase transitions ([8]) and it is used as a model for some nonlinear reaction-diffusion processes. For instance, assume that there are two populations 𝓐 and 𝓑 and that u is a density measuring the percentage of the two populations at every point; that is, if u(x) = 1 (u(x) = −1) at a point x, we have only population 𝓐 (population 𝓑) at x and u(x) = 0 means that at x we have 50% of 𝓐 and 50% of 𝓑. The non-homogeneity of the medium is expressed by the space dependence of the diffusion coefficient a(⋅). If this coefficient vanishes at some point, then this will lead to the interruption of the migration and/or interaction of the species. Evidently, u ≡ 1 or u ≡ −1 are two stable states of the system, however, our goal is to obtain existence of stable non-constant solutions.

The study of existence or non-existence of stable solutions to semi-linear problems (in particular those of the Allen-Cahn type), with a(x) ≡ 1 or a(⋅) strictly positive, is the subject of numerous articles and books. In the face of an extensive literature, we cite [9, 10, 11, 12, 13, 14]and references therein. Of course, allowing a(⋅) vanishes at some point brings with it several technical difficulties. Since the operator Au := (aux)x is no longer elliptic (sometimes called degenerate elliptic), some basic analysis tools – such as the Maximum Principles, Hopf’s Lemma and Spectral Theory – can not be used. To the best of our knowledge, the present work is the first to study the role of a degenerate diffusion coefficient related to the existence of stable solutions.

The technique we use here stems from the one presented in [11] (see also [13, 14, 15]). After finding a invariant set for the flux of (1.1) – hence the need to ensure the existence of solution – we use some variational techniques, in a convenient weighted Sobolev space, to show existence of a local minimizer of the energy functional in this invariant set.

Finally, let us remark that even our result of existence of solution to the problem (1.1), based on the works [1, 7], it is new and although it is essential to our main result, it may have interest of its own.

2 Existence of solution

In this section we deal with the existence of solution to the problem (1.1). It is important to note that the results of this section can easily be adapted to more general semi-linear equations.

Firstly, we should note that Au = (aux)x, in a suitable domain, generates a analytic semigroup. For this purpose, we introduce the following weighted spaces (sometimes, we use ′ to denote the derivative with respect to x):

Ha1(0,1):={uL2;u absolutely continuous in [0,1] and auL2(0,1)}(2.1)

with the norm






Definition 2.1

If u0L2(0, 1), a function u is said to be a weak solution of (1.1) if




for all ϕH1(0, T; L2(0, 1)) ∩ L2(0, T; Ha1(0, 1)).

Now we define the operator A by D(A) := {uHa2(0, 1); u′(0) = u′(1) = 0} and for any uD(A), Au = (au′)′.

The proof of the next two results can be found in [1, Lemma 2.1] (see also [7, Lemma 2.1]) and [7, Theorem 2.1], respectively.

Lemma 2.2

For all (u, v) ∈ D(A) × Ha1(0, 1) one has


Theorem 2.3

The operator A : D(A) → L2(0, 1) is self-adjoint, nonpositive on L2(0, 1) and it generates an analytic contraction semigroup.

Now we proceed as in [1]. Since A is a generator, and setting B(t)u := u, working in the spaces considered above, we can prove that the problem below (with cL(ℝ+ × (0, 1))) is well-posed in the sense of semigroup theory using some well-known perturbation technique (see [16], for instance).


Hence, for a fixed T > 0 we get the following result.

Theorem 2.4

If c(⋅, x) ∈ C1(ℝ+) for all x ∈ [0, 1] and u0D(A) then there is a unique weak solution


of (2.4) and


for a positive constant C.

Next result can be found in [1, Theorem 5.4]

Lemma 2.5

The set H1(0, T; L2(0, 1)) ∩ L2(0, T; Ha2(0, 1)) is compactly imbedded in C([0, T]; L2(0, 1)) ∩ L2(0, T; Ha1(0, 1)).

We are now in position to state the main result of this section. The proof follows the steps of [1, Theorem 4.12], however, some modifications are necessary because we consider Neumann boundary conditions and the specific nonlinear term f(u) = uu3.

Theorem 2.6

If u0(x) ∈ Ha1(0, 1) then (1.1) has a solution



We set X := C([0, T]; L2(0, 1)) ∩ L2(0, T; Ha1(0, 1)) and for any (x, v) ∈ (0, 1) × X, cv(t, x) := d(t, x, v(t, x)) where d(t, x, u) = 1-u2. Now, we consider the function


where uv is the unique solution of


We use Theorem 2.4 to ensure that (2.7) has a unique weak solution uX. Now, we will prove that 𝓣 has a fixed point uv (that is, 𝓣(uv) = uv) to conclude that uv is a solution of (1.1).

By Schauder’s Theorem, it is sufficient to prove that

  1. 𝓣:BXBX,

  2. 𝓣 is a compact function and

  3. 𝓣 is a continuous function,



CT it is the same constant of Theorem 2.4 and


The items (i) and (ii) are consequence of Theorem 2.4 and Lemma 2.5, respectively.

To prove (iii) we take vkX such that vkv in X, as k → ∞. We will prove that 𝓣(vk) = uvk := uk → 𝓣(v) = uv in X, as k → ∞. Recall that uk and uv are the solutions of (2.7) associated to vk and v, respectively. As D(A) is dense in Ha1(0, 1) ([7]), (2.5) and (2.6) occurs for u0Ha1(0, 1). Hence, ukBY where Y = H1(0, T; L2(0, 1)) ∩ L2(0, T; Ha2(0, 1)) and, up to a sub-sequence, uk converges weakly to some ū in Y. By Lemma 2.5, uk converges strongly to ū in X.

Multiplying the equation


by a test function ϕH1(0, T; L2(0, 1)) ∩ L2(0, T; Ha1(0, 1)) and integrating over (0, T) × (0, 1) (recall the Lemma 2.2) we get


We recall that uxk(t,1)=uxk(t,0)=0 and our next step is to prove that

  1. limk01uk(T,x)ϕ(T,x)dx=01u¯(T,x)ϕ(T,x)dx;

  2. limk0T01ϕt(t,x)uk(t,x)dxdt=0T01ϕt(t,x)u¯(t,x)dxdt;

  3. limk0T01a(x)uxk(t,x)ϕx(t,x)dxdt=0T01a(x)ux¯(t,x)ϕx(t,x)dxdt;

  4. limk0T01cvk(t,x)uk(t,x)ϕ(t,x)dxdt=0T01cv(t,x)u¯(t,x)ϕ(t,x)dxdt.

Since uk converges strongly to ū in X, it is immediate to prove (a)-(c). In order to prove (d) we recall that cvk(t, x) = 1 − (vk)2(t, x) and vk converges strongly to v in X. Therefore, we can conclude that vk converges to v a.e. as well as uk converges to ū a.e.. Thus,


and (d) holds by an application of Lebesgue Theorem.

We proved that ū is the unique weak solution of (2.4) in YX associated to v; that is ū = uv and (iii) is proved. It follows that 𝓣 has a fixed point uvY which is a solution of (1.1). The theorem is proved.□

3 Existence of stable solutions

We start by defining an energy functional E : Ha1(0, 1) → ℝ by




It is not difficult to verify that E is twice continuously differentiable and a simple computation give us that its critical points are weak stationary solutions of (1.1) (i.e. weak solutions of (1.2)).

Definition 3.1

Let u be a weak solution to (1.2). We say that u is stable if


for any ϕHa1(0, 1).

Remark 3.2

Note that (3.2) is equivalent to the second variation of the energy functional E(⋅) at u to be non-negative. Therefore, local minimizers of E are stable solutions of (1.2).

In order to state our next results, we set Il, Ir two sub-intervals of (0, 1) such that Il ⊂ (0,x0) and Ir ⊂ (x0, 1).

Lemma 3.3

(Poincaré-type inequality). There exists a constant 𝓒j (j = l, r), depending only on Ij and aIj, such that


for each function uHa1(Ij), where uj¯=1|Ij|Ijudx.


For brevity, we omit the sub-indices j = l, r. We argue by contradiction; that is, we suppose that for each k ∈ ℕ, there exists ukHa1(0, 1) such that


where uk¯:=1|I|Iukdx.

We renormalize by defining


It follows that vk¯:=1|I|Ivkdx=0,vkL2(I) = 1 and by (3.4)


In particular the functions {vk}k∈ℕ are bounded in Ha1(I). As Ha1(I) is compactly imbedded in L2(I) (the proof is analogous to that present in [1, Theorem 5.1]), there exist a sub-sequence {vkj}j∈ℕ and a function vL2(I) such that

vkjv in L2(I).(3.7)

From (3.5)

v¯:=1|I|Ivdx=0 and ||v||L2(I)=1.(3.8)

If ϕC0(I) we use (3.6) to conclude that


Hence vH1(I) and v′ = 0 a.e. in I. It follows that v is constant in I which is a contradiction with (3.8).□

Now, we set the positive number


where 𝓒j (j = l, r) is the optimal constant in (3.3).

Our main result is stated as follow.

Theorem 3.4

If there is δ > 0 such that δ < 2ϵ0, Qδ := [x0δ, x0+δ] ∩ (IlIr) = ∅ and


then (1.2) admits a non-constant stable solution.

For each t > 0, we consider


defined by


where u(t, x) is the solution of (1.1) with u(0, x) = u0(x) given by Theorem 2.6. Then tT(t)[u0(x)] is continuous for any u0Ha1(0, 1). Moreover, T(t) is a compact operator because T(t)[u0(x)] ∈ Ha2(0, 1) and Ha2(0, 1) is compactly imbedded in Ha1(0, 1) (see [1, Theorem 5.2]). For simplicity, we denote T(t)[u0(x)] by T(t)u0.

Proposition 3.5

Consider the set


If u0Λ then T(t)u0Λ for all t > 0.


  • Claim 1

    −1 ≤ T(t)u0 = u(t, x) ≤ 1 in [0, 1] for all t > 0.

    If there is (, ) such that u(, ) > 1 then if T >


    and uM = u(, ) at some (, ) ∈ (0, T] × [0, 1]. We have three possibilities:

    1. ∈ (0, 1) ∖ {x0},

    2. = x0 or

    3. ∈ {0, 1}.

    If (i) holds we can apply the Maximum Principle in a sub-interval that does not contain x0. Hence, we have a contradiction since u(0, x) = u0(x) ≤ 1. For (iii), we use a one-dimensional version of Hopf’s Lemma and that ux(,0) = ux(, 1) = 0 to get a contradiction. Finally, we note that


    and then ut(t,x0) < 0 for t near of . It follows that (ii) does not occur. Therefore u(t, x) ≤ 1 (t > 0) and analogously we prove −1 ≤ u(t, x) for all t > 0 which proves the Claim 1.

  • Claim 2

    E(u(t, x)) < ϵ014.

    Indeed, it is true because

    ddtE(u(t,x))=01(ut(t,x))2dx<0 and E(u(0,x))<ϵ014.

  • Claim 3

    Ilu(t, x)dx < 0 and Iru(t, x)dx > 0 for all t > 0.

    By contradiction, let t1 > 0 be such that u1(x) := u(t1, x) satisfies


    Then, by Lemma 3.3


    Now, note that

    0f(s)s, for 1s00f(s)s for 0s1.





    We also have


    Therefore, as F(u1) ≤ F(1) = 1/4



    • if 1Cl1 then ϵ0 > |Il|4 or

    • if 1Cl<1 then ϵ0 > |Il|4Cl.

      In both cases we have a contradiction. Similarly we prove that Iru(t, x)dx > 0 for all t > 0. It is proved that Λ is invariant under T(t) for t ≥ 0.□

Proposition 3.6

If Λ ≠ ∅ then (1.1) has at least one non-constant stationary solution uΛ which is stable in Ha1(0, 1).


If vΛ, by Proposition 3.5, γ(v) := {T(t)v; t ≥ 0} ⊂ Λ and because the system is gradient (E is a functional of Lyapunov) γ(v) is compact. It follows that

ω(v):=limtnT(tn)v=u, for some real sequence (tn).

If 𝓔 is the set of all equilibrium solutions to (1.1) then ω(v) ⊂ 𝓔. Hence, if uω(v) then −1 ≤ u ≤ 1, E(u) ≤ E(v) < ϵ0 − (1/4) and, as before, it is possible to prove that Il u < 0, Ir u > 0; that is, ω(v) ⊂ Λ.

Therefore, if vΛ, Λ ∩ 𝓔 ≠ ∅. Moreover Λ ∩ 𝓔 is compact in Ha1(0, 1). Indeed, we note that Λ ∩ 𝓔 is bounded in Ha1(0, 1) because for any uΛ ∩ 𝓔, −1 ≤ u ≤ 1 and

a(x)u(x)=0xu3(s)u(s)ds,x(0,1) (see (2.2)).

It is not difficult to see that 𝓔 is closed in Ha1(0, 1). Now, as T(t)[Λ ∩ 𝓔] = Λ ∩ 𝓔 and T(t) is a compact operator, we conclude that Λ ∩ 𝓔 is compact in Ha1(0, 1).

Because of the continuity of E, there is e0Λ ∩ 𝓔 such that E(e0) ≤ E(v) for all vΛ ∩ 𝓔. More than that, E(e0) ≤ E(v) for all vΛ since otherwise there would be v1Λ such that E(v1) < E(e0). As before ω(v1) ⊂ Λ and then, for all vω(v1),


which is a contradiction.

The next step is to prove that e0 is a local minimum of E in Ha1(0, 1). Let Λj (j = 1, …, 4) be the sets

  • Λ1 := {uHa1(0, 1); −1 < u < 1 a.e. in (0, 1)};

  • Λ2 := {uHa1(0, 1); Il u < 0};

  • Λ3 := {uHa1(0, 1); Ir u > 0};

  • Λ4 := {uHa1(0, 1); E(u) < ϵ014}.

  • Claim 1

    j=14Λj is an open set in Ha1(0, 1).

    Indeed, Λj (j = 2, …, 4) are open in Ha1(0, 1) by the continuity of the functionals E, El(u) := Il u and Er(u) := Ir u in Ha1(0, 1). It is not difficult to prove that Λ1 is open in Ha1(0, 1) using that Ha1(0, 1) ↪ C(0, 1). Claim 1 is proved.

  • Claim 2

    e0j=14Λj. By an application of Maximum Principle (recall that e0 ∈ 𝓔), it is possible to conclude that −1 < e0(x) < 1 for all x ∈ (0, 1) ∖ {x0}. This implies that e0Λ1. Clearly, Ile0 ≤ 0 and Ire0 ≥ 0 and if the equality occurs we get a contradiction as before. Thus e0Λ1Λ2. We have that E(e0) ≤ ϵ0 − (1/4) and if E(e0) = ϵ0 − (1/4) then for any vΛ,


    which contradicts E(e0) ≤ E(v) for all vΛ. Therefore e0Λ4 and this proves Claim 2. Thus, e0 is a local minimum of E in Ha1(0, 1) and then




    which proves that e0 is a stable non-constant (e0Λ) solution of (1.2).□

    Finally, we are in position to prove our main result.

Proof of the Theorem 3.4

We shall prove that Λ ≠ ∅ and the theorem follows by Proposition 3.6.

Consider the signed distance function defined in ℝ by


and ξ : ℝ → ℝ defined by


We will show that w0(x) := ξ(d(x, x0))∣(0,1)Λ. It is not difficult to see that w0Ha1(0, 1); −1 ≤ w0 ≤ 1; Ilw0 < 0 and Irw0 > 0.

Now, if we set aMδ:=maxxQδ{a(x)} and recalling that (d′(x, x0))2 = 1 and F(1) = F(−1) = 1/4,


By (3.10), E(w0) < ϵ0 − (1/4) and then w0Λ which proves the Theorem 3.4.□

Remark 3.7

It is common to say that the solution e0 obtained in this work “is trapped at the bottom of an energy well”. That is why we say that it is stable. A natural question is whether such a solution is asymptotically stable; that is, to know if the solutions of the corresponding parabolic problem (problem (1.1)) with the initial data near e0 tend to e0 as t → ∞. This type of stability – also called linearized stability or Lyapunov stability – in general, it is accomplished by studying the spectrum of the corresponding linearized problem. However, the degeneracy considered here makes it impossible to carry out such a study. This question has been studied when the degenerate operator has an uniformly elliptic direction; which, obviously, the operator considered here does not have. For all the details on this issue, we cite [17].

Remark 3.8

In the case where A = (aux)x is strongly degenerate; that is, aW1,∞(0, 1) satisfies (H1) and (H2) but 1/aL1(0, 1), can not be carried out in an analogous way. The hypotheses (H1)-(H3) are essential in all results of this work which makes the strongly degenerate case an open problem.


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About the article

Received: 2018-09-15

Accepted: 2018-11-29

Published Online: 2019-05-16

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 361–371, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0004.

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© 2020 M. Sônego, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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