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Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations

  • This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.

    Citation: Zhe Pu, Maohua Ran, Hong Luo. Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations[J]. Networks and Heterogeneous Media, 2023, 18(1): 291-309. doi: 10.3934/nhm.2023011

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  • This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.



    We consider systems of semilinear elliptic equations

    Δu(x)+Fu(x,u)=0

    where

    (F1) FC2(Rn×Rm;R) is 1-periodic sin all its variable, n,m1.

    When n=1 and m1, (PDE) are particular cases of the dynamical systems considered in the Aubry-Mather Theory ([9,23,24]). When n>1 and m=1 equations like (PDE) were studied by Moser in [25] (indeed in a much more general setting), and then by Bangert [13] and Rabinowitz and Stredulinsky [31], extending some of the results of the Aubry-Mather Theory for partial differential equations. These studies show the presence of a very rich structure of the set of minimal (or locally minimal) entire solutions of (PDE). In particular, when m=1 the set M0 of minimal periodic solutions of (PDE) is a non empty ordered set and if M0 is not a continuum then there exists another ordered family M1 of minimal entire solutions which are heteroclinic in one space variable to a couple of (extremal) periodic solutions u<v (a gap pair in M0). If M1 is not a continuum the argument can be iterated to find more complex ordered classes of minimal heteroclinic type solutions and the process continues if the corresponding set of minimal heteroclinics contains gaps. Variational gluing arguments were then employed by Rabinowitz and Stredulinsky to construct various kinds of homoclinic, heteroclinic or more generally multitransition solutions as local minima of renormalized functionals associated to (PDE), see [31]. Other extensions of Moser's results, including changing slope or higher Morse index solutions, have been developed by Bessi [10,11], Bolotin and Rabinowitz [12], de la Llave and Valdinoci [17,33]. Recently, in a symmetric setting and correspondingly to the presence of a gap pair in M0 symmetric with respect to the origin, entire solutions of saddle type were found by Autuori, Alessio and Montecchiari in [2].

    All the above results are based on the ordered structure of the set of minimal solutions of (PDE) in the case m=1 and a key tool in their proofs is the Maximum Principle, which is no longer available when m>1.

    The study of (PDE) when n,m>1 was initiated by Rabinowitz in [29,30]. Denoting L(u)=12|u|2+F(x,u) and Tn=Rn/Zn, periodic solutions to (PDE) were found as minima of the functional J0(u)=TnL(u)dx on E0=W1,2(Tn,Rm) showing that

    M0={uE0J0(u)=c0:=infE0J0(u)}.

    Paul H. Rabinowitz studied the case of spatially reversible potentials F assuming (¯F2) F is even in xi for 1in and proved in [29] that if M0 is constituted by isolated points then for each vM0 there is a v+M0{v} and a solution uC2(R×Tn1,Rm) of (PDE) that is heteroclinic in x1 from v to v+. These solutions were found by variational methods minimizing the renormalized functional

    J(u)=pZJp,0(u):=pZ(Tp,0L(u)dxc0), (1)

    (where Tp,0=[p,p+1]×[0,1]n1) on the space

    Γ(v,v+)={uW1,2(R×Tn1,Rm)uv±L2(Tp,0,Rm)0 as p±}.

    In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of M0 are not degenerate critical points of J0 and that the set M1(v,v+) of the minima of J on Γ(v,v+) is constituted by isolated points. This research line was continued by Montecchiari and Rabinowitz in [26] where, via variational methods, multitransition solutions of (PDE) were found by glueing different integer phase shifts of minimal heteroclinic connections.

    The proof of these results does not use the ordering property of the solutions and adapts to the study of (PDE) some of the ideas developed to obtain multi-transition solutions for Hamiltonian systems (see e.g. [3], [28] and the references therein). Aim of the present paper is to show how these methods, in particular a refined study of the concentrating properties of the minimal heteroclinic solutions to (PDE), can be used in a symmetric setting to obtain saddle type solutions to (PDE).

    Saddle solutions were first studied by Dang, Fife and Peletier in [16]. In that paper the authors considered Allen-Cahn equations Δu+W(u)=0 on R2 with W an even double well potential. They proved the existence of a (unique) saddle solution vC2(R2) of that equation, i.e., a bounded entire solution having the same sign and symmetry of the product function x1x2 and being asymptotic to the minima of the potential W along any directions not parallel to the coordinate axes. The saddle solution can be seen as a phase transition with cross interface.

    We refer to [14,15,6,7,27] for the study of saddle solutions in higher dimensions and to [1,20,8] for the case of systems of autonomous Allen-Cahn equations. Saddle solutions can be moreover viewed as particular k-end solutions (see [4,18,22,19]).

    In [5] the existence of saddle type solutions was studied for non autonomous Allen-Cahn type equations and this work motivated the paper [2] where solutions of saddle type for (PDE) were found in the case m=1.

    In the present paper we generalize the setting considered in [2] to the case m>1. Indeed we consider to have potentials F satisfying (F1) and the symmetry properties

    (F2) F is even in all its variables;

    (F3) F has flip symmetry with respect to the first two variables, i.e.,

    F(x1,x2,x3,...,xn,u)=F(x2,x1,x3,...,xn,u) on Rn×Rm.

    By [29] the set M0 of minimal periodic solution of (PDE) is not empty. The symmetry of F implies that any vM0 has components whose sign is constant on Rn and if vM0 then (ν1v1,,νmvm)M0 for every (ν1,,νm){±1}m (see Lemma 2.2 below). In this sense we can say that M0 is symmetric with respect to the constant function v00.

    As recalled above, in [2], where m=1, a saddle solution was found when M0 has a gap pair symmetric with respect to the origin. In the case m>1 we generalize this gap condition asking that 0M0 and, following [30], we look for saddle solutions of (PDE) when any vM0 is not degenerate for J0. We then assume

    (N) 0M0 and there exists α0>0 such that

    J0(v)hh=[0,1]n|h|2+Fu,u(x,v(x))hhdxα0h2L2([0,1]n,Rm)

    for every hW1,2([0,1]n,Rm) and every vM0.

    The assumption (N) and the symmetries of F allow us to find heteroclinic connections between elements of M0 which are odd in the variable x1. More precisely for vM0 these solutions are searched as minima of the functional J (see (1)) on the space

    Γ(v)={uW1,2(R×Tn1,Rm)u is odd in x1,limp+uvL2([p,p+1]×Tn1,Rm)=0}.

    In §4, setting

    c(v)=infuΓ(v)J(u) for vM0

    we show that

    Mmin0={v0M0c(v0)=minvM0c(v)}

    and that Mmin0 is such that if v0Mmin0 then

    M(v0)={uΓ(v0)J(u)=c(v0)}

    is not empty and compact with respect to the W1,2(R×Tn1,Rn) metric. The elements uM(v0) are classical solutions to (PDE), odd in x1, even and 1-periodic in x2,...,xn and satisfy the asymptotic condition

    uv0W1,2([p,p+1]×Tn1,Rm)0 as p+.

    Our main result can now be stated as follows

    Theorem 1.1. Assume (F1), (F2), (F3) and (N). Then, there exists a classical solution w of (PDE) such that every component wi (for i=1,m) satisfies

    (i) wi0 for x1x2>0;

    (ii) wi is odd in x1 and x2, 1-periodic in x3,...,xn;

    (iii) wi(x1,x2,x3,...,xn)=wi(x2,x1,x3,...,xn) in Rn.

    Moreover there exists v0Mmin0 such that the solution w satisfies the asymptotic condition

    distW1,2(Rk,Rm)(w,M(v0))0,ask+, (2)

    where Rk=[k,k]×[k,k+1]×[0,1]n2.

    Note that by (i) and (ii) any component of w has the same sign as the product function x1x2. Moreover by (2), since w is asymptotic as x2+ to the compact set M(v0) of odd heteroclinic type solutions, the symmetry of w implies that w is asymptotic to v0 or v0 along any direction not parallel to the planes x1=0, x2=0. In this sense w is a saddle solution, representing a multiple transition between the pure phases v0 and v0 with cross interface.

    The proof of Theorem 1.1 uses a variational approach similar to the one already used in previous papers like [5,2]. To adapt this approach to the case m>1 and so to avoid the use of the Maximum Principle we need a refined analysis of the concentrating properties of the minimizing sequences. For that a series of preliminaries results is given in §2, §3, §4 while the proof of Theorem 1.1 is developed in §5.

    In this section we recall some results obtained by Rabinowitz in [29], on minimal periodic solutions to (PDE). Moreover, following the argument in [2], we study some symmetry properties related to the assumptions (F2) and (F3). Here and in the following we will work under the not restrictive assumption

    (F4) F0 on Rn×Rm.

    Let us introduce the set

    E0=W1,2(Tn,Rm)={uW1,2(Rn,Rm)u is 1-periodic in all its variables}

    with the norm

    uW1,2([0,1]n,Rm)=(mi=1[0,1]n(|ui|2+|ui|2)dx)12.

    We define the functional J0:E0R as

    J0(u)=[0,1]n12|u|2+F(x,u)dx=[0,1]nL(u)dx. (3)

    and consider the minimizing set

    M0={uE0|J0(u)=c0} where c0=infuE0J0(u)

    Then in [29], [30] it is shown

    Lemma 2.1. Assume (F1), then M0. Moreover, setting [u]=[0,1]nudx, we have that

    1. ˆM0={uM0[u][0,1]m} is a compact set in E0;

    2. if (uk)kE0, with [uk][0,1]m, is a minimizing sequence for J0, then there exists uˆM0 such that uku in E0 up to subsequences;

    3. For every ρ>0 there exists β(ρ)>0 such that if uE0 is such that

    distW1,2([0,1]n,Rm)(u,M0):=infvM0uvW1,2([0,1]n,Rm)>ρ,

    then J0(u)c0β(ρ);

    4. If (F2) holds, then any uM0 minimizes also I(u)=[0,12]nL(u)dx on W1,2([0,12]n,Rm). As a consequence, every uM0 is symmetric in xi about xi=0 and xi=12 for every index i and u is even in xi for every index i;

    5. If (F2) holds, there results c0=infuW1,2([0,1]n,Rm)J0(u). Furthermore, if uW1,2([0,1]n,Rm) verifies J0(u)=c0, then for every i=1,2,...,n, u is symmetric in xi about xi=12 and hence uM0.

    Assumption (F2), in particular the even parity of F with respect to the components of u, provides that the elements in M0 have components with definite sign, thanks to the unique extension property (see [29], Proposition 3).

    Lemma 2.2. Assume (F1), (F2) and 0M0. If u=(u1,,um)M0 then, for every i=1,...,m, one has either ui0, or ui0 on [0,1]n and u does not vanish on open sets. Moreover, (ν1u1,,νmum)M0 for every (ν1,,νm){±1}m.

    Proof. It is sufficient to observe that if u=(u1,,um)M0 then, since F is even with respect to the components of u, we have

    ⅰ) ˉu=(|u1|,,|um|)M0 and

    ⅱ) (ν1u1,,νmum)M0 for every (ν1,,νm){±1}m.

    Property (ⅱ) gives the second part of the statement while by (i) and the unique extension property proved in [29], we obtain that the components of u do not change sign. If u vanishes on an open set, the unique continuation property gives u0, giving a contradiction and concluding the proof.

    On the other hand, assumption (F3) gives more structure on the set M0: its elements have a flip symmetry property. Indeed, setting T+={x[0,1]n|x1x2}, for every uW1,2(T+,Rm), let us define ˜uW1,2([0,1]n,Rm) as

    ˜u(x)={u(x),xT+,u(x2,x1,x3,,xn),x[0,1]nT+. (4)

    Then, we have

    Lemma 2.3. If uM0 then, u˜u in [0,1]n.

    Proof. Given uM0, without loss of generality, we assume

    T+L(u)dx[0,1]nT+L(u)dx.

    Since ˜uW1,2([0,1]n,Rm) by Lemma 2.1-(5) we have J0(˜u)c0. By the previous inequality we get

    c0=J0(u)=T+L(u)dx+[0,1]nT+L(u)dx2T+L(u)dx=J0(˜u)c0.

    Hence, again by Lemma 2.1-(5), ˜uM0. By the unique extension property of the solutions of (PDE) (cf. [29], Proposition 3), we have ˜uu in [0,1]n.

    As an immediate consequence, using Lemma 2.1-(5), we have the following.

    Lemma 2.4. There results

    minuW1,2(T+,Rm)T+L(u)dx=c02. (5)

    Moreover, if uW1,2(T+,Rm) verifies T+L(u)dx=c02, then ˜uM0.

    Remark 1. Lemma 2.3 tells us that the elements of M0 are symmetric with respect to the diagonal iperplane {xRnx1=x2} and by Lemma 2.4 they can be found by minimizing T+L(v)dx on W1,2(T+,Rm). Analogously, setting T=[0,1]nT+, we can find the elements of M0 by minimizing TL(v)dx on W1,2(T,Rm) or, by periodicity, by minimizing TL(v)dx on W1,2(T,Rm) whenever T=p+T± with pZn. For future references it is important to note that this property implies in particular that uM0 if and only if u is a minimizer of the functional σ0L(v)dx on W1,2(σ0,Rm) where

    σ0={xR×[0,1]n1|x21x1x2}.

    More precisely we have c0=infvW1,2(σ0,Rm)σ0L(v)dx and uM0 if and only if σ0L(u)dx=c0. From Lemma 2.1-(3) we recover an analogous property in W1,2(σ0,Rm): for any r>0 there exists β(r)>0 such that if uW1,2(σ0,Rm) verifies σ0L(u)dxc0+β(r), then distW1,2(σ0,Rm)(u,M0)r.

    Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that (N0) ˆM0 is a finite set and 0ˆM0,

    where we recall that ˆM0={uM0[u][0,1]m} and note that M0=ˆM0+Zm.

    Note finally that by (N0), setting

    r0:=min{uvL2(Tn,Rm)u,vM0,uv}, (6)

    we have r0>0.

    This section is devoted to introduce the variational framework to study solutions of (PDE) which are heteroclinic between minimal periodic solutions. We follow some arguments in [29], [26], introducing the renormalized functional J and studying some of its basic properties.

    Let us define the set

    E={uW1,2loc(Rn,Rm)u is 1-periodic in x2,xn}.

    For any uE we consider the functional

    J(u)=pZJp,0(u),

    where, denoting Tp,0=[p,p+1]×[0,1]n1,

    Jp,0(u)=Tp,0L(u)dxc0,pZ.

    Denoting briefly u(+p) the shifting of the function u with respect to the first coordinate (that is, where ), note that by periodicity we have

    Then, by Lemma 2.1, we have for any and , from which is non-negative on .

    Lemma 3.1. The functional is weakly lower semicontinuous.

    Proof. Consider a sequence such that weakly in . Then, for every , by the weak lower semicontinuity of , and hence of , we have . If , then we obtain easily . So, let us assume , then for any we have that there exists such that . We get

    thus finishing the proof.

    Using the notation introduced above, note that if is such that , then as , that is, the sequence is such that as . Hence, by Lemma 2.1-(3), there exist such that, up to a subsequence, as in . Using this remark and the local compactness of given by , we are going to prove some concentration properties of the minimizing sequence of the functional .

    First of all, let us consider the functional for a certain fixed integer . Notice that, by Lemma 2.1-(5),

    and the set of minima coincide with . We introduce the following distance

    Remark 2. Let us fix some constants that will be used in rest of the paper. By Lemma 2.1-(3), we have that for any there exists such that

    (7)

    It is not restrictive to assume that the function with is non-decreasing.

    On the other hand for every if we set

    we get and that is non-decreasing. Moreover, for every , since if for a certain , then , we obtain for every , so that holds. Hence, recalling the definition of in (6), we can fix satisfying . Finally, we can define

    (8)

    which is non-decreasing and . Then we fix such that for every .

    We say that a set is a set of consecutive integers if it is of the form or for a and . If is such that is small enough for some consecutive integers , then, using , we can prove that, in the corresponding sets , is ``near'' to an element of , the same for all . Indeed we have

    Lemma 3.2. Given , and a set of consecutive integers , if for any , then there exists such that , for every .

    Proof. Let be such that . Then and, by Remark 2 and the definition of , . Then, by and the choice of in (6), we can find such that

    from which for . If , repeating the argument with the couple of indices and we find such that for . By the choice of in (6), we conclude that and the lemma follows.

    Moreover, using the notations introduced above, we have

    Lemma 3.3. If then

    Proof. Setting , we have

    and so there exists such that

    On the other hand, by Hölder inequality,

    completing the proof.

    By the previous lemmas we obtain that the elements in the sublevels of satisfy the following boundeness property.

    Lemma 3.4. For every there exists such that for every satisfying one has for any .

    Proof. Let be such that . We define and note that the number of elements of is at most , where denotes the integer part. Then, the set is constituted by sets of consecutive elements of , , with . By the triangular inequality, for any , we obtain

    (9)

    where the first term in (9) follows by the application of Lemma 3.3, since

    while the second one follows by the definition of and Lemma 3.2.

    Since and , the lemma follows by choosing .

    The following lemma states the weak compactness of the sublevels of the functional .

    Lemma 3.5. Given any , let be a sequence such that for every and let be a sequence of integers. Assume that there exist and such that for all . Then, there exists with such that, up to a subsequence, weakly in .

    Proof. First note that, by Lemma 3.4, there exists such that if and then for any . If for some and , by triangular inequality for any we obtain

    Consider now a sequence as in the statement, setting for , we get

    Hence, is bounded in for any and, by a diagonal argument and the weak lower semicontinuity of , the statement follows.

    By Lemma 3.2 we also deduce the following result concerning the asymptotic behaviour of the functions in the sublevels of .

    Lemma 3.6. If , there exist such that

    Proof. Since , we have as and there exists such that for any . Thus, by Lemma 3.2, there exists such that for and for .

    Hence the sequence is such that for every and as . Then, by Lemma 2.1, as . Analogously we obtain that as

    By Lemma 3.6, if satisfies we can view it as an heteroclinic or homoclinic connection between two periodic solutions and belonging to . Hence, we can consider elements of belonging to the classes

    where .

    We note that by Lemma 3.5, every sequence with for all , admits a subsequence which converges weakly to some . Indeed, since as for every , fixed there exists such that and since , by Lemma 3.5, there exists such that, up to a subsequence, weakly as .

    In particular, given and setting

    as in [29], we obtain that for any there exist and such that . Moreover, it can be proved that any such that is a classical solution of (PDE) (see Theorem 3.3 in [29]).

    Finally, we have that as a consequence of the following lemma.

    Lemma 3.7. For every with , we have . Moreover, as .

    Proof. Assume that there exists satisfying . Then for every , so that by Lemma 3.2 there exists such that for every . Since we know that as and as , so that by (6) we would have giving a contradiction.

    In order to prove the second part of the statement, assume the existence of two sequences and in such that is bounded while as . Since is bounded, we can find and a sequence , with , such that , for every index . Hence, by Lemma 3.4, there exists such that for every and . Moreover, for every and , since , there exist such that and for every . In particular we get

    since, by periodicity, for any , . Finally, since for every , we recover in contradiction with .

    We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of , , so let us introduce the set

    In what follows, when we will consider functions we often present their properties for , avoiding to write the corresponding ones for . In particular, for every we have , where

    For any let

    In this setting we can rewrite Lemma 3.6 as follows.

    Lemma 4.1. For every for which there exists such that as , that is .

    We are going to look for minimizer of in the set . So, for every we set

    (10)

    Notice that for any we have holds and, by Lemma 3.7 since by , , we have the following.

    Lemma 4.2. For any , , and as .

    Moreover, note that, by assumption (), the intersection between and a bounded set consists of a finite number of elements. Hence, from the previous lemma, the minimum

    (11)

    is well defined and the set

    (12)

    is nonempty and consists of a finite number of elements. In particular, we have

    (13)

    The following lemma provides a concentration property for such that is close to the value : the elements of the sequence remain far from only for a finite number of indexes . Moreover, approaches an element only once. Indeed, recalling the notation introduced in Remark 2, we have

    Lemma 4.3. For any there exists , with as with the following property: if is such that then

    if for every in a set of consecutive integers, then ,

    if for some index and , then for every , and .

    Proof. Note that plainly follows from Lemma 2.1-(3), setting , where denotes the integer part.

    To prove , we consider defined for as

    Hence, and since in , while in , we obtain

    By definition, on we have and so . Since in , we have , so that, by Remark 2, we obtain and therefore

    Then and in particular for any . Hence, by Lemma 3.2, for any . Hence follows setting . Indeed, by Remark 2, we have and, since for all , we get for every .

    By the previous lemma we get

    Lemma 4.4. For any , if satisfies , then there exists such that and

    if for a certain index , then we have for every , and .

    if , then for every .

    Proof. Note that the existence of such that is ensured by Lemma 4.1 and plainly follows from Lemma 4.3-. To prove we argue by contradiction assuming that there exist , and such that . Again, by Lemma 4.3- we get for every which is in contradiction with , recalling the definition of in (6).

    As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.

    Lemma 4.5. For any there exists , with as , and such that if satisfies , then there exists such that and

    ;

    ;

    for every and .

    Proof. The existence of such that is again ensured by Lemma 4.1. By Lemma 4.3, given any , there exists such that . Then, if is such that , by Lemma 4.3-, there exists such that and hence a such that Therefore, by Lemma 4.3-, we obtain for all and since , we can conclude that and hence that for every . Moreover, again by Lemma 4.3-, we have . Hence and follows setting and .

    Finally, for every , and follows directly by Lemma 4.4 -.

    We are now able to prove the existence of a minimum of in the set for every , i.e., that for all .

    Theorem 4.6. Let , then there exists such that .

    Proof. Let be such that . Without loss of generality we can assume that for any . By Lemma 4.5, we obtain that for any ,

    (14)

    By Lemma 3.5, since is weakly closed, there exists such that, along a subsequence, weakly in . Finally, by (14) and the weakly lower semicontinuity of the distance we obtain

    (15)

    Therefore, by Lemma 3.6, we conclude that as , so that . Finally, by semicontinuity, .

    By Theorem 4.6 we know that for every , is nonempty. One can prove that consists of weak solutions of (PDE).

    Lemma 4.7. Given , with , then for any we have

    The proof can be adapted by the one of Lemma 3.3 of [4] or Lemma 5.2 of [6]. Therefore we get that any is a classical solution of (PDE) which is -periodic in the variables , .

    Finally, we now study further compactness properties for the functional that will be useful in the next section. They will be obtained as consequences of the nondegeneracy property of the elements of asked in (). In particular assumption () asks that, for every , the linearized operator about

    has spectrum which does not contain . This is the assumption made in [30] and it is indeed equivalent to require as in that

    () there exists such that

    for every and every .

    As a consequence of we obtain the following (see also Lemma 3.6 in [2]).

    Lemma 4.8. There exist and such that if , , verifies for some then

    (16)

    Proof. Notice that, by (), if and then

    where , and so

    We conclude that

    and so, using and setting , we obtain

    Since by Taylor's formula we have for all and , we obtain that there exists such that if verifies for some , then

    (17)

    On the other hand, again Taylor's expansion gives us

    and we deduce that there exists such that, taking smaller if necessary, if verifies , , then

    (18)

    The lemma follows by periodicity from (17) and (18) recalling that and that for all .

    Remark 3. In connection with Remark 1, arguing as in Remark 3.8 of [2], we can prove that (16) holds true also for the functional on , that is, if for some then

    (19)

    Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain

    Lemma 4.9. Let and be such that . Then there exists such that, up to a subsequence, as .

    In this section we prove our main theorem. To this aim, following and adapting the argument in [2], we will first prove the existence of a solution of (PDE) on the unbounded triangle

    satisfying Neumann boundary conditions on , which is odd in the first variable , asymptotic as to a certain heterocline where

    Then, by recursive reflections with respect to the hyperplanes , we will recover a solution of (PDE) on the whole .

    Let us introduce now some notations. We define the squares

    and the horizontal strips

    The intersection between the strip and the triangle consists of a bounded strip

    where .

    Figure 1. 

    The decomposition of the triangular set

    .

    For every we define the sets of functions

    and the normalized functionals on the bounded strips as

    for every , where .

    Remark 4. Notice that for every , . Indeed, we can view the restriction as a traslation of a function in and the restriction on can be treated similarly using Lemma 2.4, the symmetry of and Remark 1. Moreover, we note that the functional is lower semicontinuous with respect to the weak topology for every .

    Then, we can set

    We plainly obtain that and that the sequence is increasing. Moreover, , evaluating on a function with . Moreover, the non degeneracy assumption () permits us to obtain as in [2] (see Lemma 4.2) the following stronger result.

    Lemma 5.1. We have , in particular as .

    We can now introduce on the set

    the functional

    Notice that for every . Indeed, the restriction and so for any . Moreover, is lower semicontinuous in the weak topology of . By Lemma 5.1 we readily obtain that is finite for at least one .

    Lemma 5.2. If for some , then .

    We now look for a minimum of the functional on , thus we set

    Lemma 5.2, gives that and we can prove the existence of the minimum applying the direct method of the Calculus of Variations (see e.g. the proof of Proposition 4.4 in [2]).

    Proposition 1. We have .

    Arguing as in [2,4,6] (see e.g. the argument in Lemma 3.3 of [4] or Lemma 5.2 of [6]), we can prove that if then it is a weak solution of (PDE) on with Neumann boundary condition on . Then we can conclude that every is indeed a classical solution of (PDE). Finally, using (), we can recursively reflect with respect to the hyperplanes , obtaining an entire solution of (PDE) (see e.g. [2]). By construction, it is odd both in and , symmetric with respect to the hyperplanes and it is 1–periodic in . Hence, it satisfies hypotheses - of Theorem 1.1.

    In the next lemma we finally characterise the asymptotic behavior of the solution .

    Lemma 5.3. Let be the function obtained by recursive reflection of a given . Then there exists such that

    Proof. Let be as in the statement, we start proving that there exists such that

    (20)

    We have . Hence, as so that, by Lemma 5.1, as . Therefore, we can find a sequence , with such that as , and in particular . By Lemma 2.1-(3), we get as thus giving the existence of such that

    Now, for every , we define in the horizontal strip the following interpolation between and :

    A computation gives so that

    Now, consider defined on . We have , therefore

    and hence, since and , we obtain as . As a consequence, since , by (13), we can conclude that . Moreover we have

    and since , and , we obtain

    (21)

    In particular , so that , by the symmetry of with respect to . Summing up, using (21), we get , so we can apply Lemma 3.2 and conclude that

    (22)

    Let us now consider, for every , a different interpolation in the horizontal strip between and the periodic solution previously introduced:

    Arguing as above , so that, defining in we find {and hence, } . Since and we obtain, reasoning as above,

    thus giving .

    We now prove that the sequence is indeed a (definitively) constant sequence, i.e. for every sufficiently large. Being , we can assume and since and , we can apply Lemma 4.5 obtaining that

    ;

    ;

    As a consequence, by definition of and recalling that when we obtain and

    (23)

    provided that where . Consider now the vertical rectangle . We have for any in the set of consecutive integers , so that we can argue as in Lemma 3.2 and conclude that there exists such that

    (24)

    Finally, recalling (6), since both (23) and (24) holds, we must have for every . In particular, (22) gives the claim in (20).

    Moreover, we have proved that with and since , we can apply Lemma 4.9 to get that there exists for which, up to a subsequence,

    Hence we obtain that

    (25)

    Finally, for every we have

    Notice that since and using (20), we have

    Hence, by (25), we conclude

    The previous lemma gives the asymptotic estimate in Theorem 1.1 since .

    We can conclude now the proof of Theorem 1.1 proving the sign property . By Lemma 2.2, for any periodic solution we can define belonging to too, being easily verified. Now, by Theorem 4.6, there exists a heteroclinic solution . We can define the function , such that when , and verify that being .

    Finally, for any we can find as in Lemma 5.3. Similarly as above, we can define such that when . Then, we can verify that verifies Lemma 5.3 with the choice . By reflecting with rispect to the hyperplanes , we obtain the saddle-type solution satisfying in Theorem 1.1, thus completing the proof.



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