We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p−1u 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
[1] | Christos Sourdis . A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, 2021, 29(5): 2829-2839. doi: 10.3934/era.2021016 |
[2] | Junsheng Gong, Jiancheng Liu . A Liouville-type theorem of a weighted semilinear parabolic equation on weighted manifolds with boundary. Electronic Research Archive, 2025, 33(4): 2312-2324. doi: 10.3934/era.2025102 |
[3] | Kelei Wang . Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29(6): 3805-3816. doi: 10.3934/era.2021062 |
[4] | Massimo Grossi . On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29(6): 4215-4228. doi: 10.3934/era.2021080 |
[5] | Fanqi Zeng, Wenli Geng, Ke An Liu, Boya Wang . Differential Harnack estimates for the semilinear parabolic equation with three exponents on $ \mathbb{R}^{n} $. Electronic Research Archive, 2025, 33(1): 142-157. doi: 10.3934/era.2025008 |
[6] | David Cheban, Zhenxin Liu . Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29(4): 2791-2817. doi: 10.3934/era.2021014 |
[7] | S. Bandyopadhyay, M. Chhetri, B. B. Delgado, N. Mavinga, R. Pardo . Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary. Electronic Research Archive, 2022, 30(6): 2121-2137. doi: 10.3934/era.2022107 |
[8] | Mustafa Aydin, Nazim I. Mahmudov, Hüseyin Aktuğlu, Erdem Baytunç, Mehmet S. Atamert . On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay. Electronic Research Archive, 2022, 30(3): 1016-1034. doi: 10.3934/era.2022053 |
[9] | Yong Zhou, Jia Wei He, Ahmed Alsaedi, Bashir Ahmad . The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $. Electronic Research Archive, 2022, 30(8): 2981-3003. doi: 10.3934/era.2022151 |
[10] | Weiwei Qi, Yongkun Li . Weyl almost anti-periodic solution to a neutral functional semilinear differential equation. Electronic Research Archive, 2023, 31(3): 1662-1672. doi: 10.3934/era.2023086 |
We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p−1u 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|p−1u, x∈RN, t≤0, | (1) |
with
In the past few decades there have been intensive studies of Liouville type theorems for the equation in (1), either when
The following three exponents play an important role in the study of the equation in (1):
Serrin's exponent psg=NN−2 if N≥3, psg=∞ if N=1,2; |
the critical Sobolev exponent pS=N+2N−2 if N≥3, pS=∞ if N=1,2; |
the Joseph-Lundgren exponent pJL={(N−2)2−4N+8√N−1(N−2)(N−10)if N>10,∞if N≤10. |
We note that
φ∞(x)=L|x|−2/(p−1) with L=(2p−1(N−2−2p−1))1/(p−1). | (2) |
For any
● If
● If
● If
● If
We emphasize that the rescaling
φλ(x)=λΦ(λ(p−1)/2|x|), λ∈(0,∞), | (3) |
furnishes a family of radial steady states such that
We are now in position to state our main result.
Theorem 1.1. If
|u(x,t)|≤φ∞(x), x∈RN∖{0}, t≤0, | (4) |
then
For
|u(x,t)|≤(1+ε)φ∞(x), x∈RN, t≤0, | (5) |
where
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
Remarkably, it was shown in [6] that the equation in (1) admits positive, entire solutions of homoclinic and heteroclinic type for
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
Theorem 1.2. If
Δu+|u|p−1u=0, z=(x,y)∈RN+M, with p∈(psg(N),pJL(N)), N≥3, M≥0, | (6) |
and
|u(x,y)|≤φ∞(x), x∈RN∖{0}, y∈RM, | (7) |
then
If
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
In the case of the critical Sobolev exponent
u(x)=|x|−N−22v(ln|x|), | (8) |
where
−v″+(N−2)24v−vN+2N−2=0 in R. | (9) |
Besides of the constant solution
Armed with the above information and by suitably adapting our approach, we can complement our main results with the following.
Theorem 1.3. If
To illustrate the delicacy of our result, at least in the parabolic case, we remark that the previously mentioned heteroclinic solutions of [6] connect
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
Z(r)={Φ(r),0≤r≤r1,φ∞(r),r>r1. | (10) |
Clearly,
Φ′(r1)>φ′∞(r1) | (11) |
holds. Next, according to (3), we let
zλ(x)=λZ(λ(p−1)/2|x|)={φλ(r),0≤r≤sλ,φ∞(r),r>sλ, r=|x|, λ>0, |
where we have denoted
sλ=r1λ−(p−1)/2. |
We emphasize that we have used that
By making partial use of our supersolution, we will first show that
|u|<φ∞, x∈RN∖{0}, t∈(−∞,T). | (12) |
Since
u(x,0)<φλ∗(x), |x|≤sλ∗. |
Let
u≤φλ∗ for |x|≤sλ∗, t∈[0,T−ε]. |
Since
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|≤1 in RN×R. | (13) |
Indeed, let us suppose that
d((x,t),(˜x,˜t))=|x−˜x|+√|t−˜t|, |
and
M(x,t)=|u|(p−1)/2(x,t). |
For
Mk:=|u|(p−1)/2(xk,tk)≥|u|(p−1)/2(x0,t0) |
and
|u|(p−1)/2(x,t)≤2Mk whenever |x−xk|+√|t−tk|≤kMk. |
We note that (4) and the definition of
Mk|xk|≤L(p−1)/2. |
Hence, passing to a subsequence if necessary, we may assume that
Mkxk→y∞ for some y∞∈RN. | (14) |
The rescaled functions
vk(y,s)=ρ2/(p−1)ku(xk+ρky,tk+ρ2ks), where ρk=12Mk, |
are entire solutions of (1) and satisfy
vk→V in C2,1loc(RN×R), |
where
|vk(y,s)|≤Lρ2/(p−1)k|xk+ρky|2/(p−1)=L|xk/ρk+y|2/(p−1)=L|2Mkxk+y|2/(p−1), y≠−xkρk. |
Thus, by letting
|V(y,s)|≤L|2y∞+y|2/(p−1), y≠−2y∞. |
Now, the spatially shifted solution
W(y,s)=V(y−2y∞,s) |
satisfies
The main tool in the proof is Serrin's sweeping principle (see [11,Thm. 9] for the elliptic case) using the family of supersolutions
u≤zμ, x∈RN, t∈R, for any μ≥ˉλ. |
Starting from
Let us suppose, to the contrary, that there exists some
Λ={λ≥0 : zμ≥u in RN×R for every μ≥λ} |
coincides with
u≤zλ0, x∈RN, t∈R. | (15) |
Keeping in mind 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
xk→x∞ for some x∞∈RN such that |x∞|≤sλ0. | (17) |
If the sequence
It remains to deal with the case where, up to a subsequence,
uk(x,t)=u(x,t+tk), x∈RN, t∈R. |
From (15) and (16) it follows that
zλ0≥uk in RN×R and uk(xk,0)>φλk(xk), |
respectively. Since
uk→U in C2,1loc(RN×R), |
where
U≤zλ0 in RN×R and U(x∞,0)≥φλ0(x∞). |
In particular, we get
U≤φλ0, |x|<sλ0+δ, |t|<1, |
for some sufficiently small
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)r−N−22, r=|x|>0, |
with
zλ(x)={φλ(r),0≤r≤τλ,ψ(r),r>τλ, r=|x|. |
As before, we can use
|u(x,t)|≤h(ln|x|)|x|−N−22, x∈RN∖{0}, t∈R. | (18) |
As in the proof of Theorem 1.1, by applying Serrin's sweeping principle, we can conclude that
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
Mk|xk|≤‖h‖2N−2L∞(R). |
Nevertheless, up to a subsequence, we still have
|vk(y,s)|≤ρN−22kh(ln|xk+ρky|)|xk+ρky|N−22=h(ln|xk+ρky|)|xk/ρk+y|N−22 |
for
ln|xk+ρky|=lnρk+ln|xkρk+y|, |
we decompose
lnρk=mkT+dk, |
with
h(ln|xk+ρky|)=h(mkT+dk+ln|xkρk+y|)=h(dk+ln|xkρk+y|). |
Consequently, recalling the definition of
|vk(y,s)|≤h(dk+ln|2Mkxk+y|)|2Mkxk+y|N−22, y≠−2Mkxk. |
Letting
|V(y,s)|≤h(d∞+ln|2y∞+y|)|2y∞+y|N−22=h(ln(ed∞|2y∞+y|))|2y∞+y|N−22, y≠−2y∞. |
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
Remark 2. If
limr→0r2p−1ϕ(r)=L and limr→∞rN−2ϕ(r)=a |
(see [3,Prop. 2.2]). We note that these singular solutions decay faster than the self-similar one as
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
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
The author would like to thank IACM of FORTH, where this paper was written, for the hospitality.
[1] |
Monotonicity for elliptic equations in an unbounded Lipschitz domain. Comm. Pure Appl. Math. (1997) 50: 1089-1111. ![]() |
[2] |
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. (1989) 42: 271-297. ![]() |
[3] |
Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations. J. Geom. Anal. (1999) 9: 221-246. ![]() |
[4] |
(2011) Stable Solutions of Elliptic Partial Differential Equations. Boca Raton: Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, CRC Press. ![]() |
[5] |
On the classification of solutions of the Lane-Emden equation on unbouned domains of RN. J. Math. Pures Appl. (2007) 87: 537-561. ![]() |
[6] |
Homoclinic and heteroclinic orbits for a semilinear parabolic equation. Tohoku Math. J. (2011) 63: 561-579. ![]() |
[7] |
Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. (1981) 34: 525-598. ![]() |
[8] |
On the stability and instability of positive steady states of a semilinear heat equation in Rn. Comm. Pure Appl. Math. (1992) 45: 1153-1181. ![]() |
[9] |
Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. (1973) 49: 241-269. ![]() |
[10] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302
![]() |
[11] | Strong comparison theorems for elliptic equations of second order. J. Math. Mech. (1961) 10: 431-440. |
[12] |
An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations. Proc. Roy. Soc. Edinburgh Sect. A (2006) 136: 807-835. ![]() |
[13] |
Entire and ancient solutions of a supercritical semilinear heat equation. Discrete Cont. Dynamical Syst. (2021) 41: 413-438. ![]() |
[14] |
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems. Duke Math. J. (2007) 139: 555-579. ![]() |
[15] |
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations. Indiana Univ. Math. J. (2007) 56: 879-908. ![]() |
[16] |
A Liouville property and quasiconvergence for a semilinear heat equation. J. Differential Equations (2005) 208: 194-214. ![]() |
[17] |
Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. Math. Ann. (2016) 364: 269-292. ![]() |
[18] | P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, Duke Math. J., to appear. |
[19] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2019. doi: 10.1007/978-3-030-18222-9
![]() |
[20] |
R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture Notes in Math. Springer, 1365 (1989), 120–154. doi: 10.1007/BFb0089180
![]() |
[21] | C. Sourdis, A Liouville property for eternal solutions to a supercritical semilinear heat equation, preprint, arXiv: 1909.00498. |