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A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart

  • Received: 01 August 2020 Revised: 01 December 2020 Published: 15 March 2021
  • Primary: 35B08, 35B53; Secondary: 35B40

  • We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p1u which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent p is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.

    Citation: Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart[J]. Electronic Research Archive, 2021, 29(5): 2829-2839. doi: 10.3934/era.2021016

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  • We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p1u which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent p is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.



    We consider classical solutions to the semilinear equation

    ut=Δu+|u|p1u,  xRN, t0, (1)

    with p>1. For obvious reasons, such solutions are frequently called ancient. Our interest will be in conditions which imply u0, a Liouville type theorem that is.

    In the past few decades there have been intensive studies of Liouville type theorems for the equation in (1), either when t0, tR (entire solutions) or t0 (global solutions). At the same time, these have emerged as a fundamental tool in deriving various qualitative properties of the solutions to the corresponding Cauchy problem in a general domain or for a nonlinearity that behaves like a power as u. The best general reference here is the monograph [19]. For a recent account of the theory and some further developments, we refer to [13,18].

    The following three exponents play an important role in the study of the equation in (1):

    Serrin's exponent  psg=NN2  if N3,  psg= if N=1,2;
    the critical Sobolev exponent  pS=N+2N2  if N3,  pS= if N=1,2;
    the Joseph-Lundgren exponent  pJL={(N2)24N+8N1(N2)(N10)if N>10,if N10.

    We note that psg<pS<pJL if N3. These exponents arise naturally in the study of the ordinary differential equation that is satisfied by the positive radial steady states [9,19]. In this regard, let us list some well known properties which we will need in the sequel. First, for p>psg there exists an explicit radial singular steady state

    φ(x)=L|x|2/(p1)  with  L=(2p1(N22p1))1/(p1). (2)

    For any p>1, the radial ODE for the steady states admits a unique solution Φ such that Φ(0)=1, Φr(0)=0. This solution is defined in a maximal interval of the form [0,Rmax) with 0<Rmax and is decreasing as long as it stays positive. The following qualitative properties of Φ will be useful for our purposes.

    ● If p(1,pS), then Φ has a first root ρ>0. Actually, there are no positive steady states in this regime (see [7]). In fact, under the further restriction that p(psg,pS), it intersects twice with φ in (0,ρ) (see in particular [9,Fig. 2]). We point out that these intersections are transverse thanks to the uniqueness of solutions to the corresponding IVP;

    ● If p=pS, then Rmax=, Φ>0, Φ()=0 and Φ has exactly two intersections with φ. We point out that this Φ has a simple explicit formula and decays to zero faster than φ as r (see also the discussion following (9) below);

    ● If p(pS,pJL), then Rmax=, Φ>0, Φ()=0 and Φ has infinitely many intersections with φ. Moreover, Φ/φ1 as r;

    ● If ppJL, then Rmax=, Φ>0, Φ()=0 and Φ<φ. Moreover, Φ/φ1 as r (detailed information on this asymptotic behaviour can be found in [8]).

    We emphasize that the rescaling

    φλ(x)=λΦ(λ(p1)/2|x|),  λ(0,), (3)

    furnishes a family of radial steady states such that φλ(0)=λ. Actually, this family includes all the positive radial steady states. We point out that φ is invariant under the above scaling. Therefore, in light of the above, if p>pS we see that φλφ pointwise in RN{0} as λ. For completeness, let us note that for 0<λ<μ< the following hold: φλ has a unique radial intersection with φμ if p=pS; φλ has infinitely many radial intersections with φμ if p(pS,pJL); φλ<φμ if ppJL.

    We are now in position to state our main result.

    Theorem 1.1. If u satisfies (1) with p(psg,pJL) and

    |u(x,t)|φ(x), xRN{0}, t0, (4)

    then u0.

    For p[pS,pJL)(psg,pJL) the above theorem was proven previously by Fila and Yanagida in [6] by a different approach. Roughly, they 'squeezed' u between two forward self-similar solutions. We note that forward self-similar solutions exist also for subcritical p, and the required properties of theirs that are needed to show the Liouville property are well known (see [12,Thm. 1.1 and Cor. 1.2]). So, the proof in [6] applies in the above subcritical range as well. In fact, the approach in the aforementioned reference even yields the nonexistence in the class of functions satisfying the weaker condition

    |u(x,t)|(1+ε)φ(x), xRN, t0, (5)

    where ε>0 is given by the properties of the forward self-similar solutions. Loosely speaking, time can be considered as the 'squeezing parameter' in their proof.

    In contrast, our proof does not make use of (time dependent) similarity variables. Instead of using time dependent solutions as barriers, we will plainly use φ after appropriately 'covering' its singularity with a piece of φλ from (3) (a 'surgery' type of argument in some sense). The result is a weak supersolution of (1). Our 'squeezing parameter' will plainly be λ>0 through the use of Serrin's sweeping principle (see [11,Thm. 9] for the elliptic case) in the spirit of the sliding method [1]. However, to be able to start such a continuity argument, we need that u is bounded. Thankfully, as it turns out, this can be assumed without loss of generality in light of the scaling and doubling arguments of [15]. On the other hand, most likely, our approach cannot be used to prove the nonexistence in the case of the weaker condition (5).

    Remarkably, it was shown in [6] that the equation in (1) admits positive, entire solutions of homoclinic and heteroclinic type for p(pS,pL) and p[pS,pJL), respectively, where pL>pJL stands for Lepin's exponent.

    Our approach, being elliptic in nature, carries over with only minor modifications to establish the following elliptic counterpart of Theorem 1.1 (as we will point out, the solutions in the latter can be extended for tR). In contrast, the approach of [6] is intrinsically parabolic and seems to be inapplicable for this purpose.

    Theorem 1.2. If u satisfies

    Δu+|u|p1u=0, z=(x,y)RN+M, with p(psg(N),pJL(N)), N3, M0, (6)

    and

    |u(x,y)|φ(x), xRN{0}, yRM, (7)

    then u0.

    If M=0, the assumption (7) implies that u is a stable solution of (6) in RN{0}1 (see [4,Ch. 1] for the definition). Indeed, as in [4,Prop. 1.3.2], it is easy to check that the difference φ|u| is a positive weak supersolution of the linearized operator Δp|u|p1 in RN{0}. Consequently, by the obvious weak version of [4,Prop. 1.2.1], we infer that u is stable in RN{0}. Therefore, in the special case M=0, our result follows from [5,Thm. 2] which asserts that in that case (6) cannot have a nontrivial solution that is stable outside a compact set. On the other hand, we note that this viewpoint cannot be applied for general M>0 because the exponent pJL(K) is decreasing with respect to K.

    1 We were informed of this property by L. Dupaigne after the first version of the paper, we borrow his argument.

    An analogous Liouville type result to Theorem 1.1 for ppJL, which takes into account that φλ<φ, λ(0,), can be found in our recent paper [21]. In the aforementioned work we have extended, again with a sweeping argument, the Liouville type result of Polacik and Yanagida from [16] who relied on (time dependent) similarity variables and invariant manifold ideas. A version of Theorem 1.2 for ppJL(N) is contained in an extended remark in the same paper of ours.

    In the case of the critical Sobolev exponent p=pS, a famous result of Caffarelli, Gidas and Spruck [2] asserts that all positive solutions of the steady state problem in RN{0} are radial (whether they have a removable singularity at the origin or not). Using this information, Schoen [20] observed that all such solutions with a singularity at the origin can be completely classified by standard ODE phase-plane analysis. They are of the form

    u(x)=|x|N22v(ln|x|), (8)

    where v is a positive periodic solution of

    v+(N2)24vvN+2N2=0 in R. (9)

    Besides of the constant solution (N22)N22, which gives rise to the self-similar singular solution φ, there is a family of periodic solutions that can be uniquely parametrized, up to translations, by their minimal value which spans the interval (0,(N22)N22). These periodic solutions have a unique local maximum and minimum per period. In fact, they are symmetric with respect to their local extrema. The singular solutions of (1) that they produce via (8) are frequently called of Delaunay-type in comparison with Delaunay surfaces which are singly periodic, rotationally symmetric surfaces with constant mean curvature. We point out that each Delaunay-type singular solution has infinitely many radial intersections with φ. Actually, the radial regular steady state Φ of (1) is given by (8) with v an appropriate translation of the positive, even homoclinic solution of (9). Remarkably, the latter solution can be computed explicitly and is equal to (N(N2))N24(2cosh())N22. Let us note in passing that the translation invariance of (9) echoes the scaling invariance (3) of (1). It is worth mentioning that an analogous transformation to (8) also applies for ppS. However, the corresponding second order autonomous ODE for v is dissipative and thus has no nonconstant periodic or homoclinic solutions (it has, however, heteroclinic solutions for p(psg,pS) that give rise to fast decaying singular solutions, see Remark 2 and the references therein).

    Armed with the above information and by suitably adapting our approach, we can complement our main results with the following.

    Theorem 1.3. If p=pS(N), the assertions of Theorems 1.1 and 1.2 hold with the righthand side of (4) and (7), respectively, being an arbitrary Delaunay-type singular solution.

    To illustrate the delicacy of our result, at least in the parabolic case, we remark that the previously mentioned heteroclinic solutions of [6] connect φλ, λ(0,), as t to the trivial solution as t+ and are decreasing in time.

    The rest of the paper is essentially devoted to the proofs of our main results in the next section. In Subsection 2.1, we will prove Theorem 1.1. After its proof, in Remark 1, we will hint at a perhaps unexpected connection between our supersolution and a well known argument from the theory of minimal surfaces. As we have already mentioned, the proof of Theorem 1.2 requires only minor modifications and will therefore be omitted. In Subsection 2.2, we will prove Theorem 1.3. Subsequently, in Remark 2, we will give a partial analog of this theorem for subcritical exponents. Lastly, for the reader's convenience, in Appendix A we will state a reduced version of the doubling lemma from [14] that is needed for our results.

    In this section we will prove Theorems 1.1 and 1.3. In order to avoid confusion, we mention again that the proof of Theorem 1.2 will be omitted as it requires only minor adaptations.

    Proof. The main idea of the proof is to construct a family of weak supersolutions of (1) by appropriately modifying the singular solution φ around the origin. Our construction will hinge on the fact that, as we have already mentioned, the radial regular steady state Φ intersects at least once with φ since p(psg,pJL). We denote by r1>0 the smallest radius at which such an intersection takes place, and define a function Z:RNR with radial profile given by

    Z(r)={Φ(r),0rr1,φ(r),r>r1. (10)

    Clearly, Z is continuous by our choice of r1. The point is that it is a weak supersolution of (1) (see for instance [10,Ch. 5] for the definition) because

    Φ(r1)>φ(r1) (11)

    holds. Next, according to (3), we let

    zλ(x)=λZ(λ(p1)/2|x|)={φλ(r),0rsλ,φ(r),r>sλ, r=|x|, λ>0,

    where we have denoted

    sλ=r1λ(p1)/2.

    We emphasize that we have used that φ is invariant under the above rescaling. We point out that zλ uniformly on |x|sλ as λ. On the other hand, zλ0 as λ0, uniformly in RN. Clearly, zλ is still a weak supersolution to (1). Actually, we will not use any weak form of the maximum principle in the sequel. Nevertheless, the fact that zλ is a weak supersolution of (1) will serve as an important guideline.

    By making partial use of our supersolution, we will first show that u can be extended as a solution of the equation in (1) for tR. To this end, the standard existence and uniqueness theory for the corresponding Cauchy problem (it is well-posed in L(RN), see [19,Prop. 51.40]) guarantees that u can be extended in a maximal time interval of the form (,T) for some T(0,]. Moreover, by the strong maximum principle for linear parabolic equations [10,Ch. II], we assert from (4) that

    |u|<φ, xRN{0}, t(,T). (12)

    Since u(,0) is a bounded function, there exists a λ1 such that

    u(x,0)<φλ(x),  |x|sλ.

    Let ε(0,T) be arbitrary. By virtue of the above two relations, since u and φλ are bounded on {|x|sλ, t[0,Tε]}, the parabolic maximum principle [10,Lem. 2.3] (applied to the linear equation for the difference of these two solutions of (1)) yields

    uφλ  for |x|sλ, t[0,Tε].

    Since ε>0 is arbitrary, we obtain uφλ for |x|sλ, t[0,T). Applying the same argument with u in place of u, and keeping in mind (12), we conclude that u remains bounded as tT. This means that T= as desired (if not, then u could be continued further as a solution in contradiction to the maximality of T).

    Having disposed of this preliminary step, we can now turn our attention to the Liouville property. By nowadays standard doubling and scaling arguments [15], we can assume that u is bounded. In fact, we can do better and assume that

    |u|1 in RN×R. (13)

    Indeed, let us suppose that |u(x0,t0)|>1 for some (x0,t0)RN×R. Motivated from [17], we will apply Lemma A.1 from Appendix A with X=RN×R, equipped with the parabolic distance

    d((x,t),(˜x,˜t))=|x˜x|+|t˜t|,

    and

    M(x,t)=|u|(p1)/2(x,t).

    For y=(x0,t0) and any kN, the aforementioned lemma provides (xk,tk) such that

    Mk:=|u|(p1)/2(xk,tk)|u|(p1)/2(x0,t0)

    and

    |u|(p1)/2(x,t)2Mk whenever |xxk|+|ttk|kMk.

    We note that (4) and the definition of Mk force

    Mk|xk|L(p1)/2.

    Hence, passing to a subsequence if necessary, we may assume that

    Mkxky for some yRN. (14)

    The rescaled functions

    vk(y,s)=ρ2/(p1)ku(xk+ρky,tk+ρ2ks), where ρk=12Mk,

    are entire solutions of (1) and satisfy |vk(0,0)|=22/(p1), |vk(y,s)|1 for |y|+|s|2k. The parabolic regularity theory [10,Chs. IV,VII] guarantees that the sequence {vk} is relatively compact in C2+θ,1+θ/2loc for some θ(0,1). Hence, using the usual diagonal argument, passing to a further subsequence if needed, we may assume that

    vkV in C2,1loc(RN×R),

    where V is an entire solution to (1) such that |V|1 and V(0,0)0. Furthermore, on account of (4), we have

    |vk(y,s)|Lρ2/(p1)k|xk+ρky|2/(p1)=L|xk/ρk+y|2/(p1)=L|2Mkxk+y|2/(p1), yxkρk.

    Thus, by letting k and using (14), we obtain

    |V(y,s)|L|2y+y|2/(p1), y2y.

    Now, the spatially shifted solution

    W(y,s)=V(y2y,s)

    satisfies |W|1, W(2y,0)0 and (4). Consequently, it is sufficient to prove the theorem for entire solutions that satisfy (4) with tR and (13). This task will take up the rest of the proof.

    The main tool in the proof is Serrin's sweeping principle (see [11,Thm. 9] for the elliptic case) using the family of supersolutions {zλ}. Since u is bounded and satisfies (4), there exists a ˉλ1 such that

    uzμ, xRN, tR, for any μˉλ.

    Starting from ˉλ, we proceed to decrease λ while keeping the above ordering. There are only two possibilities. Either we can continue all the way until we reach λ=0 or we will get 'stuck' at some first λ0>0 and cannot continue further. Our goal is to show that the latter scenario (to be described in more detail below) cannot happen. This will imply that u0. Then, the assertion of the theorem follows readily by carrying out the same procedure with u in place of u.

    Let us suppose, to the contrary, that there exists some λ0(0,ˉλ] where we get stuck in the sense that the set

    Λ={λ0 : zμu in RN×R for every μλ}

    coincides with [λ0,) (by its definition Λ is a semi infinite interval, while it is closed thanks to the continuity of zμ with respect to μ). Clearly, we have

    uzλ0, xRN, tR. (15)

    Keeping in mind that zλ depends nontrivially on λ only in the space-time cylinder {|x|<sλ, tR} (where it is equal to φλ), and (12) with T=, we get λk(0,λ0) such that λkλ0 as k, xkRN with |xk|<sλk, and tkR such that

    u(xk,tk)>φλk(xk), (16)

    (the reader should not be confused with the repeated use of notation in different contexts within the proof). The whole argument is actually reminiscent to the famous sliding method [1] for elliptic problems, when translating a compactly supported subsolution. We also note that zλ and zμ with λ<μ may intersect each other in |x|<sμ as is the case in the aforementioned procedure. Passing to a subsequence if necessary, we may assume that

    xkx for some xRN such that |x|sλ0. (17)

    If the sequence {tk} is bounded, passing to a further subsequence if needed, we may assume that tkt for some tR. From (15) and (16), it follows that u(x,t)=φλ0(x). In fact, thanks to (12) with T=, we see that |x|sλ0. Thus, by virtue of (15) and the parabolic strong maximum principle [10] (applied in the linear equation for the difference uφλ0 sufficiently close to (x,t)), we deduce that u coincides with φλ0 in some neighborhood of (x,t). In turn, by repeated applications of the strong maximum principle, we obtain uφλ0 which is clearly absurd on account of (4).

    It remains to deal with the case where, up to a subsequence, tk (the case where tk+ can be handled similarly). To this end, we consider the time translated solutions

    uk(x,t)=u(x,t+tk), xRN, tR.

    From (15) and (16) it follows that

    zλ0uk in RN×R and uk(xk,0)>φλk(xk),

    respectively. Since u is bounded, as before, by the usual diagonal-compactness argument, possibly up to a further subsequence, we have

    ukU in C2,1loc(RN×R),

    where U is an entire solution to (1) such that

    Uzλ0 in RN×R and U(x,0)φλ0(x).

    In particular, we get U(x,0)=φλ0(x). Intuitively, keeping in mind (11), it is clear that we have been led to a contradiction. The rigorous justification is easy. Indeed, by virtue of (11), the above imply that

    Uφλ0, |x|<sλ0+δ, |t|<1,

    for some sufficiently small δ>0. Then, as before, we deduce by the strong maximum principle that Uφλ0 which is impossible.

    Remark 1. In [21] we highlighted a heuristic connection of (1) to ancient solutions of the mean curvature flow. In that context, our time independent supersolution in (10) relates to the competitor that is used in order to show that the symmetric minimal cones are not area minimizers in low dimensions.

    Proof. The proof is similar to that of Theorems 1.1 and 1.2 apart from some technical modifications. We will give a sketch of the proof only for the parabolic problem (the elliptic case is analogous) and point out the main differences.

    Let us denote by

    ψ(r)=h(lnr)rN22, r=|x|>0,

    with h>0 a T-periodic solution of (9), a Delaunay-type singular solution that bounds the absolute value of u. For each λ(0,), the homoclinic solution of (9) that gives φλ via (8) intersects at least twice with h (this can be seen easily from the phase plane portrait). Hence, there exists a first radius τλ>0 at which φλ and ψ intersect. Clearly, we have τλ0 as λ and τλ as λ0. Moreover, since such an intersection is transverse (by the uniqueness of the IVP for the radial ODE), the implicit function theorem guarantees that τλ varies smoothly with respect to λ>0. Keeping in mind that ψ is not invariant under the scaling in (3) (unless hL of course), we now define our supersolution zλ directly as

    zλ(x)={φλ(r),0rτλ,ψ(r),r>τλ, r=|x|.

    As before, we can use zλ as a barrier in order to show that u cannot blow up in finite time. Therefore, we may again assume that u is an entire solution to (1) that satisfies

    |u(x,t)|h(ln|x|)|x|N22,  xRN{0}, tR. (18)

    As in the proof of Theorem 1.1, by applying Serrin's sweeping principle, we can conclude that u0 under the additional assumption that it is bounded.

    It remains to verify that, by the doubling lemma as in the proof of Theorem 1.1, we can assume without loss of generality that (13) holds. To this end, assuming that this was not the case, we define M, (xk,tk), Mk, ρk and vk(y,s) analogously to the aforementioned proof. We quickly come across a minor difference which is that now we have

    Mk|xk|h2N2L(R).

    Nevertheless, up to a subsequence, we still have Mkxky for some yRN. Moreover, we still have the local convergence of vk to some bounded, nontrivial limiting solution V. However, the main differences arise when passing to the limit in the rescaled form of (18). More precisely, the latter gives

    |vk(y,s)|ρN22kh(ln|xk+ρky|)|xk+ρky|N22=h(ln|xk+ρky|)|xk/ρk+y|N22

    for yxk/ρk, sR. Based on the identity

    ln|xk+ρky|=lnρk+ln|xkρk+y|,

    we decompose lnρk as

    lnρk=mkT+dk,

    with mkZ and |dk|T. By passing to a further subsequence if necessary, we may assume that dkd for some dR. Since h is T-periodic, we obtain

    h(ln|xk+ρky|)=h(mkT+dk+ln|xkρk+y|)=h(dk+ln|xkρk+y|).

    Consequently, recalling the definition of Mk, we get

    |vk(y,s)|h(dk+ln|2Mkxk+y|)|2Mkxk+y|N22,  y2Mkxk.

    Letting k, we deduce that

    |V(y,s)|h(d+ln|2y+y|)|2y+y|N22=h(ln(ed|2y+y|))|2y+y|N22,  y2y.

    We remark that the righthand side of the above relation is plainly a rescaling (according to (3)) and a translation of the Delaunay-type singular solution ψ. In other words, after a translation, V satisfies (18) with h replaced by a positive (edT)-periodic solution of (9). Hence, V is a bounded solution that satisfies the assumptions of the theorem, which is what we wanted.

    Remark 2. If p(psg,pS), for any a>0, there exists a positive, radial singular solution ϕ to the steady state problem such that

    limr0r2p1ϕ(r)=L and limrrN2ϕ(r)=a

    (see [3,Prop. 2.2]). We note that these singular solutions decay faster than the self-similar one as |x|. We observe that φλ with λ(0,) must intersect at least twice with each such fast decaying singular solution. Indeed, if not then by the discussion following Theorem 1.2 we would have that φλ is a stable solution of the steady state problem in its support which is absurd (see for instance [4,Ex. 1.2.3]). In light of this property, by arguing as in the proof of Theorem 1.3 we can show that the Liouville property holds for bounded, ancient solutions to (1) that are smaller in absolute value than such a fast decaying singular steady state (one can also prove a corresponding elliptic result in the spirit of Theorem 1.2).

    In this small appendix, we will state for the reader's convenience the following reduced version of [14,Lem. 5.1] that we referred to in the proof of Theorem 1.1.

    Lemma A.1. Let (X,d) be a complete metric space and M:X[0,) be bounded on compact subsets of X. Fix a yX such that M(y)>0 and a real k>0. Then, there exists xX such that

    M(x)M(y)

    and

    M(z)2M(x)  whenever  d(z,x)kM(x).

    Remark 3. Our formulation of the doubling lemma is restricted to the whole metric space X. We also note that we assume M to be nonnegative instead of strictly positive, as was the case in the aforementioned reference. However, if M(y)>0 then throughout the proof of [14,Lem. 5.1] we observed that M is evaluated only at points where MM(y). Thus, there is no loss of generality.

    The author would like to thank IACM of FORTH, where this paper was written, for the hospitality.



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