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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Construction of type I blowup solutions for a higher order semilinear parabolic equation

Tej-Eddine Ghoul / Van Tien Nguyen / Hatem Zaag
Published Online: 2019-06-01 | DOI: https://doi.org/10.1515/anona-2020-0006

Abstract

We consider the higher-order semilinear parabolic equation

tu=(Δ)mu+u|u|p1,

in the whole space ℝN, where p > 1 and m ≥ 1 is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].

Keywords: Higher order parabolic equation; Blowup solution; Blowup profile; Stability

MSC 2010: Primary: 35K50; 35B40; Secondary: 35K55; 35K57

1 Introduction

We are interested in the semilinear parabolic equation

tu=Amu+u|u|p1,u(0)=u0Am(Δ)m,(1.1)

where u(t) : ℝN → ℝ with N ≤ 1, Δ stands for the standard Laplace operator in ℝN, and the exponents p and m are fixed,

p>1andmN,m1odd.

The higher-order semilinear parabolic equation (1.1) is a natural generation of the classical semilinear heat equation (m = 1). It arises in many physical applications such as theory of thin film, lubrication, convection-explosion, phase translation, or applications to structural mechanics (see the Petetier-Troy book [28] and references therein).

By standard results the local Cauchy problem for equation (1.1) can be solved in L1L thanks to the integral representation

u(t)=Km(t)u0+0tKm(ts)u(s)|u(s)|p1ds,(1.2)

where 𝓚m(t) is the fundamental solution of the linear parabolic t𝓚m = Am𝓚m, defined via the inverse Fourier transform

Km(x,t)=1(2π)NRNe|ξ|2mtı(ξx)dξ,Km(x,0)=δ(x).

From Fujita [12] (m = 1) and Galaktionov-Pohozaev [14] (m > 1), we know that

pF1+2mN,

is the critical Fujita exponent for the problem in the following sense. If p > pF, for any sufficient small initial data u0L1(ℝN) ∩ L(ℝN) the Cauchy problem (1.1) admits a global solution satisfying u(t) → 0 as t → +∞ uniformly in ℝN. If p ∈ (1, pF] and the initial data u0 ≢ 0 with ∫N u0 dx ≤ 0 for the nonlinearity u|u|p–1 replaced by |u|p or u0 ≥ 0, then the corresponding solution to problem (1.1) blows up in some finite time T > 0, namely that

limtTu(t)L(RN)=+.

Here T is called the blowup time, and a point a ∈ ℝN is called a blowup point if and only if there exists a sequence (an, tn) → (a, T) such that |u(an, tn)| → +∞ as n → +∞. A solution of (1.1) is called Type I blowup if it satisfies

c(Tt)1p1u(t)LC(Tt)1p1,(1.3)

otherwise, it is of Type II blowup. In addition, we call a blowup solution self-similar if it is of the form

u(x,t)=(Tt)1p1Φ(y),y=x(Tt)12m,(1.4)

where Φ is not identically constant. Obviously, the self-similar blowup solution is of Type I.

When m = 1, problem (1.1) reduces to the classical semilinear heat equation

tu=Δu+u|u|p1,(1.5)

which has been extensively studied in the last four decades, and no rewiew can be exhaustive. Given our interest in the construction of solutions with a prescribed blowup behavior, we only mention previous work in this direction. The first conctructive result was given by Bricmont-Kupiainen [5] who showed the existence of type I blowup solution to equation (1.5) according to the asymptotic dynamic

supxRN(Tt)1p1u(x,t)κ1+(p1)4p|x|2(Tt)|log(Tt)|1p10astT,(1.6)

for some universal positive constant κ = κ(p). Note that the authors of [5] also exhibited finite time blowup solutions that verify other asymptotic behaviors which are expected to be unstable. Note also that Bressan [3, 4] made a similar construction in the case of an exponential nonlinearity. Later, Merle-Zaag [23] suggested a modification of the argument of [5] and obtained the stability of the constructed solution verifying (1.6) under small perturbations of initial data. The stability of the asymptotic behavior (1.6) had been observed numerically by Berger-Kohn [2] (see also Nguyen [24] for other numerical analysis). In particular, Herrero-Velázquez [19] proved that the blowup dynamic (1.6) is generic in one dimensional case, and they announced the same for higher dimensional case (but never published it).

The method of [5] and [23] relies on the understanding of the spectral property of the linearized operator around an expected profile in the similarity variables setting. Roughly speaking, the linearized operator possesses a finite number of positive eigenvalues, a null eigenvalue and a negative spectrum; then they proceed in two steps:

  • Reduction of an infinite dimensional problem to a finite dimensional one in the sense that the control of the error reduces to the control of the components corresponding to the positive eigenvalues.

  • Solving the finite dimensional problem thanks to a classical topological argument based on the index theory.

This general two-step procedure has been extended to various situations such as the case of the complex Ginzgburg-Landau equation by Masmoudi-Zaag [21], Nouaili-Zaag [27] (see also Zaag [31] for an earlier work); the complex semilinear heat equation with no variational structure by Duong [7], Nouaili-Zaag [26]; non-scaling invariant semilinear heat equations by Ebde-Zaag [9], Nguyen-Zaag [25], Duong-Nguyen-Zaag [8]. We also mention the work of Tayachi-Zaag [29, 30] and Ghoul-Nguyen-Zaag [16] dealing with a nonlinear heat equation with a double source depending on the solution and its gradient in some critical setting. In [17, 18], we successfully adapted the method to construct a stable blowup solution for a non variational semilinear parabolic system.

As for the present paper, we aim at extending the above mentioned method to construct for problem (1.1) finite time blowup solutions satisfying some prescribed asymptotic behavior. Although the general idea is the same as for the classical case (1.5), we would like to emphasis that the above mentioned strategy is heavy and its implementation never being straightforward, the context and difficulties are different for each specific problem. As a step forward to better understanding the blowup dynamics for (1.1), we obtain the following result.

Theorem 1.1

(Type I blowup solutions for (1.1) with a prescibed behavior) Let m ≥ 1 be an odd integer. There exists initial data u0L(ℝN) such that the corresponding solution u(t) to equation (1.1) satisfies

  1. u(t) blows up in a finite time T = T(u0) only at the origin.

  2. (Asymptotic behavior)

    supxRN(Tt)1p1u(x,t)Φx[(Tt)|log(Tt)|]12mC|log(Tt)|12m,(1.7)

    where

    Φ(ξ)=κ(1+Bm,p|ξ|2m)1p1,(1.8)

    with

    κ=(p1)1p1,Bm,p=(1)m+12(p1)(2m)!p((4m)!2[(2m)!]2).(1.9)

  3. (Final blowup profile) There exists u* ∈ 𝓒(ℝN ∖ {0}, ℝ) such that u(x, t) → u*(x) as tT uniformly on compact subsets ofN ∖ {0}, where

    u(x)κBm,p|x|2m2m|log|x||1p1as|x|0.(1.10)

Remark 1.2

We believe that such a blowup profile (1.8) exists for all m ∈ ℕ*. We note that the constant Bm,p < 0 when m is even (see (1.9)), so the profile Φ blows up on the finite interface |ξ| = ξ* = (Bm,p)12m. This says that the case when m is even would lead to type II blowup solutions in the sense of (1.3). Although main ideas for a full justification of such a blowup behavior with m even remains the same, the proof would be very delicate and will be addressed in a separate work.

Remark 1.3

The blowup solution described in Theorem 1.1 is not self-similar in the sense of (1.4). Note that in contrast to blowup solutions of the classical second order semilinear heat equation (1.5), Budd-Galaktionov-Williams [6] through numerical and asymptotic calculations conjectured that there are at least 2m2 nontrivial self-similar blowup solutions to (1.1), and that profiles having a single maximum correspond to stable (generic) self-similar blowup solutions.

Remark 1.4

The proof of Theorem 1.1 (in dimension N = 1 for simplicity) involves a detailed description of the set of initial data leading to the asymptotic dynamic (1.7). In particular, our initial data is roughly of the form (see formula (3.13) below)

u0(x)=es0Φxes0/2m+As02χ2xes0/2m,s0k=12m1dkψk(xes0/2m),

where A and s0 are large fixed constants, Φ is the profile defined by (1.8), χ is some smooth cut-off function, (d0, ⋯, d2m–1) ∈ ℝ2m are free parameters, ψk, 0 ≤ k ≤ 2m – 1 are the eigenfunctions of the linearized operator (see Proposition 2.1 for a precise definition) corresponding to the positive eigenvalue λk = 1k2m. Through a topological argument, we show that there exists a suitable choice of parameters (d0, ⋯, d2m–1) such that the solution to equation (1.1) with the initial datum u0 satisfies the conclusion of Theorem 1.1. In some sense, our constructed solution is 2(m – 1)-codimension stable in the following sense. The 2m components of the linearized solution corresponding to λk = 1k2m have the exponential growth eλks. However, the first two modes corresponding to λ0 and λ1 can be eliminated by means of the time and space translation invariance of the problem. Hence, by fixing 2(m – 1) directions ψ2, ⋯, ψ2m–1 and perturbing the remaining components (in L), we still obtain the same asymptotic dynamic (1.7) of the perturbed solution. The proof of 2(m – 1)-codimenison stability would require some Lipschitz regularity of the considered initial data set and it would be addressed separately in another work.

Remark 1.5

According to our construction, the asymptotic dynamic (1.7) lies on the center manifold generated by eigenfunctions corresponding to the null eigenvalue λ2m = 0. Our analysis can be extended to construct for equation (1.1) a finite time blowup solution having a different asymptotic dynamic from (1.7). Such solutions particularly have asymptotic dynamics laying on the stable manifold generated by eigenfunctions corresponding to the negative eigenvalue λk = 1k2m < 0 with k ≥ 2m + 1. As explained in Remark 1.4, the corresponding initial data leading to such solutions would involve k parameters with k ≤ 2m + 1 (consider N = 1), so that a topological argument is assigned in order to control the first k components corresponding to the eigenvectors ψj for 0 ≤ jk –1. Although the constructive method are similar for all cases, we decide to only deal with the case of the center manifold, since the proof is the most delicate in the sense that it requires a more refined analysis in the blowup region leading to some logarithmic correction of the blowup variable as shown in (1.7).

Strategy of the proof.

Let us sketch the main ideas of the proof of Theorem 1.1.

  • -

    Similarity variables and linearized problem. According to the scaling invariance of the problem (1.1), we introduce the similarity variables

    u(x,t)=(Tt)1p1w(y,s),y=x(Tt)12m,s=ln(Tt),(1.11)

    which leads to the new equation

    sw=Amw12myw1p1w+w|w|p1Lmwpp1w+w|w|p1,(1.12)

    where the linear operator ℒm is given by

    Lm=Am12my+Id.(1.13)

    Note that the change of variables (1.11) allows us to reduce the finite time blowup problem to a long time behavior one at the cost of an extra scaling term in the new equation (1.12). In the setting (1.11), proving the asymptotic dynamic (1.7) is equivalent to proving that

    supyRNw(y,s)Φys12m0ass+.(1.14)

    It is then natural to linearize equation (1.12) around the expected profile Φ by introducing

    q=wΦ.

    which leads to the equation of the form

    sq=(Lm+V(y,s))q+B(q)+R(y,s),(1.15)

    where B(q) is built to be quadratic, R measures the error generated by Φ and is uniformly bounded by 𝓞(s–1), and V is the potential defined as

    V(y,s)=pΦp1κp1.

    Our goal is to construct for equation (1.15) a solution q defined for all (y, s) ∈ ℝN × [s0, +∞) such that

    supyRN|q(y,s)|0ass+.

  • -

    Properties of the linearized operator. In view of equation (1.15), we see that the nonlinear quadratic and the error term are small and can be negligible in comparison with the linear term. Roughly speaking, the linear part will play an important role in the dynamic of the solution. It is essential to determine the spectrum and corresponding eigenfunctions of both ℒm and its adjoint Lm. According to [15], the spectrum of the linear operator ℒm comprises real simple eigenvalues only,

    spec(Lm)=λβ=1|β|2m,β=(β1,,βn)Nn,|β|=β1++βn,

    and the corresponding eigenfunction ψβ with |β| = n is polynomial of order n (see Proposition 2.1 below). Moreover, the family of the eigenfunctions {ψβ}β∈ℕn forms a complete subset in Lρ2(RN) where ρ is some exponentially decaying weight function.

    Depending on the asymptotic behavior of the potential V, we observe that

    • Inside the blowup region, |y|Ks12m for some K large, the effect of V is regarded as a perturbation of ℒm.

    • Outside the blowup region, |y|Ks12m, the full linear part ℒm + V behaves like ℒmpp1, which has a purely negative spectrum. Hence, the control of the solution in this region is simple.

  • -

    Decomposition of the solution and reduction to a finite dimensional problem. According to the spectrum of ℒm, we decompose

    q(y,s)=|β|=02mqβ(s)ψβ(y)+q(y,s),

    where q is the projection of q on the subspace of ℒm corresponding to strictly negative eigenvalues. Since the spectrum of the linear part of the equation satisfied by q is negative, it is controllable to zero. We would like to notice that we do not use the Feymann-Kac representation1 as for the case m = 1 treated in [23], because of its complicated implementation for higher order cases m ≥ 2. To avoid such a formula, we further decompose

    q(y,s)=|β|=2m+1Mqβ(s)ψβ+qM,(y,s),

    for some M large enough (typically M4VLy,s). A direct projection yields

    qβ=1|β|2mqβ+Os|β|+22m,|β|=2m+1,,M,

    from which we obtain the rough bound |θβ(s)|=O(s|β|+22m)for|β|=2m+1,,M. For the infinite part qM,⊥, we explore the properties of the semigroup esm and a standard Gronwall inequality to close the estimate for this part.

    The control of the null mode q2m is delicate since the potential has in some sense a critical size in our analysis. In particular, we need a careful refinement of the asymptotic behavior of V to derive the sharp ODE

    qβ=2sqβ+O1s3for|β|=2m,

    which shows a negative spectrum after changing the variable τ = ln s, hence, the rough bound |qβ(s)|=Ologss2for|β|=2m. Here the precise value of Bm,p given in (1.9) is crucial for many algebraic identities to derive this sharp ODE. At this stage, we reduce the infinite dimensional problem to a finite dimensional one in the sense that it remains to control a finite number of positive modes qβ for |β| ≤ 2m – 1. This is done through a classical topological argument based on the index theory.

    The rest of the paper is organized as follows: In Section 2 we recall basic spectral properties of the linearized operator ℒ and its adjoint ℒ*, then we perform a formal spectral analysis to derive an approximate blowup profile served for our analysis later. In Section 3 we give all arguments for the proof of Theorem 1.1 without going to technical details (the reader who is not interested in the technical details can stop at this section). Section 4 is the hearth of our analysis: it is devoted to the study of the dynamic of the linearized problem from which we reduce the problem to a finite dimensional one.

2 A formal approach via spectral analysis

In this section we first recall basic spectral properties of the linearized operator, then present a formal approach based on a spectral analysis to derive the blowup profile given in Theorem 1.1.

2.1 Spectral properties of the linearized operator

In this subsection, we recall from [15] the basic spectral properties of the linear operator ℒm and its adjoint Lm. The case m = 1 is well known, since we can rewrite

L1=1ρ1.(ρ1)+Idwithρ1(y)=(4π)N2e|y|24,

which is a self-adjoint operator in the weighted Hilbert space L2(e–|y|2/4dy) with the domain 𝓓(ℒ) = H2(e–|y|2/4dy). It has a real discrete spectrum and the corresponding eigenfunctions are derived from Hermite polynomials.

For m ≥ 2, the operator ℒm is not symmetric and does not admit a self-adjoint extension. Then we denote Lm the formal adjoint of ℒm as

Lmf=Amf+12m.(yf)+f,(2.1)

in the sense that

Lmf,g=f,Lmgfor(f,g)D(Lm)×D(Lm).(2.2)

From [10] and [11], we know that the following elliptic equation

AmF+12m.(yF)=0inRNwithRNF(y)dy=1.(2.3)

has a unique radial solution given by the explicit formula

F(y)=(2π)N20es2msN2JN22(s|y|)ds,(2.4)

with Jν being the Bessel function. In particular, F satisfies the estimate

|F(y)|<Ded|y|νinRN,whereν=2m2m1,(2.5)

for some positive constants D and d depending on m and N.

We introduce the weight functions

ρ=ρ1,ρ(y)=ea|y|ν,(2.6)

where a = a(m, N) ∈ (0, d] is a small constant, d and ν are introduced in (2.5).

Proposition 2.1

(Spectral properties of ℒm and Lm, [15]). Let m ∈ ℕ*, we have

  1. Lm:Hρ2m(RN)Lρ2(RN) is a bounded linear operator with the spectrum

    spec(Lm)=λβ=1|β|2m,β=(β1,,βN)NN,|β|=0,1,2,.(2.7)

    The set of eigenfunctions {ψβ}|β|≥0 is complete in Lρ2(RN), where ψβ has the separable decomposition

    ψβ(y)=ψβ1(y1)ψβN(yN)withψk(ξ)=1k!j=0k2m(1)jmj!ξ2jmξk.(2.8)

  2. Lm:Hρ2m(RN)Lρ2(RN) is a bounded linear operator with the spectrum

    spec(Lm)=λβ=1|β|2m,β=(β1,,βN)NN,|β|=0,1,2,.(2.9)

    The set of eigenfunctions {ψβ}|β|0 is complete in Lρ2(RN), where ψβs are given by

    ψβ(y)=(1)|β|β!yβF(y),(2.10)

    where β!=β1!βN!,yβ=β1y1βNyN and the function F is defined by (2.4).

  3. (Orthogonality)

    ψβ,ψy=δβ,y,(2.11)

    where δβ,y is the Kronecker delta.

Remark 2.2

We note from the orthogonality (2.11) and the definition of ψk that for all polynomials Pn(y) of degree n < |y|, we have Pn,ψy=0.

For N = 1, we have

ψk(y)=akykfor0k<2m,ak=1k!,ψ2mj(y)=i=0jc2miy2miandy2mj=i=0jc2miψ2mi(y)forjN.

We end this subsection by recalling basic properties of the semigroup esm for s > 0.

Lemma 2.3

(Properties of the semigroup esm). The kernel of the semigroup esm is given by

esLm(y,x)=es[(1es)]N2mFyes2mx[2m(1es)]12m,s>0,(2.12)

where F is defined as in (2.4). The action of esm is defined by

esLmg(y)=RNesLm(y,x)g(x)dx.(2.13)

We also have the following estimates:

  1. esmgLCesgL for all gL(ℝN).

  2. esm div(g)∥LCes(1es)12mgL for all gL(ℝN).

  3. If |f(x)| ≤ η(1 + |x|M+1) for all x ∈ ℝN, then

    esLmΠM,f(y)CηeMs2m(1+|y|M+1),yRN,(2.14)

    where ΠM,⊥ is defined as in (3.10).

Proof

The formula (2.12) can be verified by a direct computation thanks to equation (2.3). The estimates (i)-(iii) are straightforward from the definitions (2.12) and (2.13).□

2.2 Approximate blowup profile

In this subsection we recall the formal approach of [13] (see also [6]) to figure out an appropriate blowup profile for our analysis later. This approach had been used in several problems involving the second order Laplacian, see for example [5], [30], [16, 17, 18]. The argument relies on the basis of the known spectral properties of the rescaled operator ℒm and its adjoint Lm given in the previous subsection. For simplicity, we consider the one dimensional case and symmetric positive solutions.

Let us introduce

w¯=wκ,

where κ=(p1)1p1 is the constant equilibrium to equation (1.12). This yields the following perturbed equation

sw¯=Lmw¯+p2κw¯2+R(w¯),(2.15)

where |R()| ≤ C||3 for || ≪ 1.

From Proposition 2.1, we know that ψn with n ≥ 2m + 1 correspond to negative eigenvalues of ℒm. Therefore, we may consider

w¯(y,s)=i=0mw¯2i(s)ψ2i(y),(2.16)

where i=0m|2i(s)| → 0 as s → +∞. Plugging this ansatz to equation (2.15), taking the scalar product with ψ2j and using Proposition 2.1, we find that for 0 ≤ jm,

w¯2j(s)=1jmw2j(s)+p2κj=0mw¯2i(s)ψ2i(y)2,ψ2j+O(i=0m|w¯2i(s)|3).(2.17)

Assuming that 2m is dominant, i.e.

for0jm1,|w¯2j(s)||w¯2m(s)|ass+,(2.18)

then system (2.17) reduces to

w¯2j(s)=1jmw2j(s)+O(|w¯2m(s)|2)for0jm1,w¯2m(s)=p2κμ¯2mw¯2m2+o(|w¯2m(s)|2),whereμ¯2m=ψ2m2,ψ2m.

Solving this system yields

w¯2m(s)=2κpμ¯2m1s+Oln2ss,i=0m1|w¯2i(s)|=O1s2ass+,

which is in agreement with the assumption (2.18).

Hence, from (2.16) and the definition of ψ2m, we derive the following asymptotic behavior

w(y,s)=κ2κpμ¯2msy2m+(1)m(2m)!(2m)!+Olnss2,(2.19)

where the convergence takes place in Lρ2(ℝ) as well as uniformly in compact sets by standard parabolic regularity.

The expansion (2.19) provides a relevant variable for blowup, namely z=ys12m, that governs the behavior in the intermediate region. In particular, we try to search formally a solution w of equation (1.12) of the form

w(y,s)=Φ(z)+(1)m+12κ(2m)!pμ¯2ms+O1s1+ϵ,(2.20)

for some ϵ > 0, with the boundary condition Φ(0) = κ.

Plugging this ansatz to equation (1.12) and comparing the leading order terms, we arrive at

z2mΦ1p1Φ+Φp=0,Φ(0)=κ.(2.21)

Solving this ODE yields

Φ(z)=κ(1+Bm,pz2m)1p1for someBm,pR.

By matching expansions (2.19) and (2.20), we find that

Bm,p=2(p1)pμ¯2m(2m)!,μ¯2m=ψ2m2,ψ2m.

Let us compute μ̄2m in the following. Using (2.8), we write

ψ2m(y)=1(2m)!(y2m+(1)m(2m)!),ψ4m(y)=1(4m)!y4m+(1)m(4m)!(2m)!y2m+(4m)!,ψ2m2(y)=1(2m)!(4m)!ψ4m+(1)m+1(2m)!(4m)!(2m)!2(2m)!ψ2m+c0(m)ψ0.

Using the orthogonality relation (2.11), the definition (2.10) of ψ2m and the fact that ∫ F(y)dy = 1, we compute by an integration by parts,

μ¯2m=(1)m+1(2m)!(4m)!(2m)!2(2m)!Rψ2m(y)ψ2m(y)dy=(1)m+1(2m)!(4m)![(2m)!]22R(y2m+c2m,0)y2mF(y)dy=(1)m+1(2m)(2m)!(4m)![(2m)!]22RF(y)dy=(1)m+1(2m)!(4m)![(2m)!]22.(2.22)

In conclusion, we have derived the following candidate for the blowup profile in the similarity variables:

w(y,s)φ(y,s):=κ1+Bm,py2ms1p1+Am,ps,(2.23)

where

Bm,p=2(p1)pμ¯2m(2m)!,Am,p=(1)m+12κ(2m)!pμ¯2m.(2.24)

Since we want the profile φ bounded, this requests Bm,p > 0 or μ̄2m > 0, which only happens for m odd.

3 Proof of Theorem 1.1 without technical details

In this section we give all arguments of the proof of Theorem 1.1. We only deal with the one dimensional case N = 1 for simplicity since the analysis for the higher dimensional cases N ≥ 2 is exactly the same up to some complicated calculation of the projection of (1.15) on the eigenspaces of ℒm. The proof of Theorem 1.1 is completed in three parts:

  • In the first part we formulate the problem by linearizing the rescaled equation (1.12) around the approximate profile φ given by (2.23). We also introduce a shrinking set in which the constructed solution of the linearized equation is trapped.

  • In the second part we exhibit an explicit formula of the initial data and show that the corresponding solution belongs to the shrinking set. In particular, the reader can find how to reduce the problem to a finite dimensional one (all technical details will be left to the next section) and the use of a topological argument based on index theory to conclude.

  • The last part is devoted to the proof of items (i) and (iii) of Theorem 1.1. Although the argument of the proof is almost the same as for the classical case m = 1, we would like to sketch the main ideas for the reader convenience.

3.1 Formulation of the problem

According to the formal analysis given in Section 2.2, we introduce

q(y,s)=w(y,s)φ(y,s),(3.1)

and write from (1.12) the equation driving by q,

sq=(Lm+V(y,s))q+B(q)+R(y,s),(3.2)

where ℒm is the linearized operator defined by (1.13) and

V(y,s)=pφp1κp1,(3.3)

B(q)=(q+φ)|q+φ|p1φppφp1q,(3.4)

R(y,s)=sφ+Amφ12myφφp1+φp.(3.5)

Let χ0C0(R+) be a cut-off function with supp(χ0) ⊂ [0, 2] and χ0 ≡ 1 on [0, 1]. We introduce

χ(y,s)=χ0|y|Ks12mwithK1fixed,(3.6)

and define

qe(y,s)=q(y,s)(1χ(y,s)).(3.7)

We decompose

q(y,s)=k=0Mqk(s)ψk(y)+qM,(y,s),(3.8)

where

  • qk = Πk(q) is the projection of q on the eigenmode corresponding to eigenvalue λk=1k2m, defined as

    qk(s)=q,ψkwithψkbeing introduced in(2.10).(3.9)

  • qM,⊥ = ΠM,⊥(q) is called the infinite dimensional part of q, where ΠM,⊥(q) is the projection of q on the eigensubspace where the spectrum of ℒm is lower than 1M2m. Note that we have the orthogonality

    qM,,ψk=0forkM.(3.10)

  • M is typically a large constant

    M=4mNwithNN,NVLy,s.(3.11)

    which allows us to successfully apply a standard Gronwall’s inequality to the control of the infinite dimensional part qM,⊥.

We aim at constructing for equation (1.15) a global in time solution q such that

q(s)L(RN)0ass+.(3.12)

According to the decomposition (3.8), it is enough to show that there exists a solution q belonging to the following set.

Definition 3.1

(Shrinking set to trap solutions). For each A > 0, for each s > 0, we denote 𝓥A(s) the set of all functions q(y, s) in L(ℝ) such that

|qk(s)|As2for0k2m1,|q2m(s)|A2logss2,|qk(s)|Aksk+12mfor2m+1kM,qe(s)L(R)AM+2s12m,yR,|qM,(y,s)|AM+1sM+22m(|y|M+1+1),

where qk, qM,⊥ and qe are defined as in the decomposition (3.8) and (3.7).

Remark 3.2

By definition, we see that if q(s) ∈ 𝓥A(s), then the following bound holds

|q(y,s)|CAM+1k=0M+1(|y|k+1)sk+12m1{|y|2Ks12m}+qeL(RN)CAM+2s12mfor allyR.

Hence, our goal (3.12) reduces to constructing for equation (1.15) a solution q(s) belonging to the shrinking set 𝓥A(s) for all s ∈ [s0, +∞).

3.2 Existence of solutions trapped in 𝓥A

In this step we aim at proving that there actually exists initial data ϕ(y) = q(y, s0) such that the corresponding solution q(y, s) to (1.15) belongs to the shrinking set 𝓥A(s). Given A ≥ 1 and s0e, we consider initial data of the form

ϕA,s0,d(y)=As02χ(2y,s0)k=02m1dkψk(y),(3.13)

where d = (d0, d1 ⋯, d2m–1) ∈ ℝ2m are real parameters to be fixed later, χ is introduced in (3.6). In particular, the initial data (3.13) belongs to 𝓥A(s0) as shown in the following proposition.

Proposition 3.3

(Properties of initial data (3.13)). For each A ≫ 1, there exist s0 = s0(A) ≫ 1 and a cuboid 𝓓s0 ⊂ [–A, A]2m such that for all (d0, ⋯, d2m–1) ∈ 𝓓s0, the following properties hold:

  1. The initial data ϕA,s0,d defined in (3.13) belongs to 𝓥A(s0) with strict inequalities except for the positive modes ϕk with 0 ≤ k ≤ 2m – 1, where ϕk = Πk(ϕA,s0,d).

  2. The map Γ : 𝓓s0 → ℝ2m, defined as Γ(d0, ⋯, d2m–1) = (ϕ0, ⋯, ϕ2m–1), is linear, one to one from 𝓓s0 to [As02,As02]2m, and maps 𝓓s0 into ([As02,As02]2m). Moreover, the degree of Γ on the boundary is different from zero.

Proof

The proof directly follows from the definition of the projection Πk(ϕA,s0,d) and it is straightforward, so we omit it here.□

Starting with initial data ϕA,s0,d belonging to 𝓥A(s0), we claim that we can fine-tune the parameters (d0, ⋯, d2m–1) ∈ 𝓓s0 such that the corresponding solution q(s) to equation (1.15) stays in 𝓥A(s) for all ss0. More precisely, we claim the following.

Proposition 3.4

(Existence of solutions trapped in 𝓥A(s)). There exist A ≫ 1, s0 = s0(A) ≫ 1 and (d0, ⋯, d2m–1) ∈ 𝓓s0 such that if q(s) is the solution to equation (1.15) with initial data (3.13), then q(s) ∈ 𝓥A(s) for all ss0(A).

Proof

From the local Cauchy problem of (1.1) in L(ℝ), we see that for each initial data ϕA,s0,d ∈ 𝓥A(s0), equation (1.15) has a unique solution q(s) ∈ 𝓥A(s) for all s ∈ [s0, s*) with s* = s*(d). If s* = +∞ for some d = (d0, ⋯, d2m–1) ∈ 𝓓s0, we are done. Otherwise, we proceed by contradiction and assume that s*(d) < +∞ for all d ∈ 𝓓s0. By continuity and the definition of s*, we remark that q(s*) ∈ 𝓥A(s*). We claim the following.

Proposition 3.5

(Finite dimensional reduction). There exist A ≫ 1, s0 = s0(A) ≫ 1 and (d0, ⋯, d2m–1) ∈ 𝓓s0 such that the following properties hold: If q(s) is the solution to equation (1.15) with initial data (3.13) and q(s) ∈ 𝓥A(s) for all s ∈ [s0, s1] and q(s1) ∈ 𝓥A(s1), then

  1. (Reduction to a finite dimensional problem) (q0, ⋯, q2m–1)(s1) ∈ [As12,As12]2m.

  2. (Transversality) There exists μ0 > 0 such that q(s1 + μ) ∉ 𝓥A(s1 + μ) for all μ ∈ (0, μ0).

Let us postpone the proof of Proposition 3.5 to the next section and continue our argument. From (i) of Proposition 3.5, we obtain (q0, ⋯, q2m–1)(s*) ∈ [As2,As2]2m, and the following mapping is well defined

Θ:Ds0[1,1]2m(d0,,d2m1)s2A(q0,,q2m1)(s).

From the transversality given in item (ii) of Proposition 3.5, (q0, ⋯, q2m–1) actually crosses its boundary at s = s*, resulting in the continuity of s* and Θ. Applying again the transversality, we see that if (d0, ⋯, d2m–1) ∈ 𝓓s0, then q(s) leaves 𝓥A(s) at s = s0, thus, s* = s0 and Θ| 𝓓s0 = Γ, the map defined in item (ii) of Proposition 3.3. Using that item, we see that the degree of Θ is not zero. Since Θ is continuous, this is a contradiction. This concludes the proof of Proposition 3.4, assuming that Proposition 3.5 holds.□

3.3 Conclusion of Theorem 1.1

From Proposition 3.4, there exists a solution q to equation (1.15) such that q(s) ∈ 𝓥A(s) for all ss0. From Remark 3.2, we deduce that q(s)L(R)C(A)s12m for all ss0. The conclusion of item (ii) then follows from (1.11) and (3.1). Item (i) of Theorem 1.1 is just a direct consequence of items (ii) and (iii). Because the proof of item (iii) is similar to the classical case m = 1, we only sketch the main ideas for the reader’s convenience. The existence of the final blowup profile u* ∈ 𝓒(ℝ ∖ {0}) follows from the technique of Merle [22]. Here we focus on a precise description of the final blowup profile u* in a neighborhood of the singularity. To do so, we follow the technique of Herrero-Velázquez [20] (see also Bebernes-Bricher [1], Zaag [31] for a similar approach) by introducing the auxiliary function

h(ξ,τ;x0)=(Tt0(x0))1p1u(x,t),(3.14)

where

ξ=xx0[Tt0(x0)]12m,τ=tt0(x0)Tt0(x0),

and t0(x0) is uniquely determined by

|x0|=K[(Tt0(x0))|log(Tt0(x0))|]12m,K1fixed.(3.15)

We note that h(ξ, τ; x0) is also a solution to (1.1) because of the invariance of (1.1) under dilations. From (1.7), we have

sup|ξ||log(Tt0(x0))|14mh(ξ,0,x0)Φ(K)C(Tt0(x0))12m0as|x0|0.

Let ĥK(τ) be the solution to (1.1) with the constant initial datum Φ(K), defined as

h^K(τ)=κ1τ+Bm,pK2m1p1,τ[0,1).

By the continuity with respect to initial data for equation (1.1), one can show that

sup|ξ||log(Tt0(x0))|14m,0τ<1h(ξ,τ;x0)h^K(τ)ϵ(x0)0,as|x0|0.

Passing to the limit τ → 1 yields

u(x0)=(Tt0(x0))1p1limτ1h(0,τ;x0)(Tt0(x0))1p1h^K(1).

From the definition (3.15), we compute

|log(Tt0(x0))|2m|log|x0||,Tt0(x0)|x0|2m2mK2m|log|x0||as|x0|0,

from which we obtain the asymptotic behavior

u(x0)κBm,p|x0|2m2m|log|x0||1p1as|x0|0.

This concludes the proof of Theorem 1.1, assuming that Proposition 3.5 holds.

4 Reduction to a finite dimensional problem

This section is the central part in our analysis where we give all details of the proof of Proposition 3.5, completing hence the proof of Theorem 1.1. The essential idea is to project equation (1.15) onto different components according to the decomposition (3.8). In particular, we claim that Proposition 3.5 is a direct consequence of the following.

Proposition 4.1

(Dynamics of equation (1.15)). For all A ≫ 1, there exists s0 = s0(A) ≫ 1 such that if q(s) ∈ 𝓥A(s) for all s ∈ [τ, τ1] with τ1τs0, then the following estimates hold for all s ∈ [τ, τ1]:

  1. (Control of the finite dimensional part)

    qk(s)1k2mqkCs2for0k2m1.(4.1)

    q2m(s)+2sq2mCA3s3,(4.2)

    |qj(s)|Cej2m1(sτ)|qj(τ)|+CAj1sj+12mfor2m+1jM.(4.3)

  2. (Control of the infinite dimensional and outer part)

    qM,(y,s)1+|y|M+1L(R)CeM(sτ)2mqM,(y,τ)1+|y|M+1L(R)+CAMsM+22.(4.4)

    qe(s)L(R)Ce(sτ)2(p1)qe(τ)L(R)+CAM+1s12m(1+sτ).(4.5)

Remark 4.2

Note that the sharp ODE (4.2) comes from the precise choice of the constant Bm,p appearing in the profile Φ defined in (1.8). Other choices would give the error of size O1s2 which is too large to close the estimate for q2m.

Let us postpone the proof of Proposition 4.1 and proceed with the proof of Proposition 3.5.

Proof of Proposition 3.5 assuming Proposition 4.1

Since the argument of the proof is similar to what was done in [23], we only sketch the proof for the reader’s convenience. We recall from the assumption that

q(s)VA(s)for alls[s0,s1]andq(s1)VA(s1).(4.6)

Thus, part (i) of Proposition 3.5 will be proved if we show that all the bounds given in Definition 3.1 can be improved, except for the first 2m components qk with 0 ≤ k ≤ 2m – 1, in the sense that for all s ∈ [s0, s1]:

|q2m(s)|<A2logss2,|qk(s)|<Aksk+12mfor2m+1kM,(4.7)

qM,(y,s)1+|y|M+1L<AM+1sM+22m,qe(s)L<AM+2s12m.(4.8)

We first deal with q2m. We argue by contradiction by assuming that there is s* ∈ [s0, s1] such that for all s ∈ [s0, s*),

|q2m(s)|<A2logss2,|q2m(s)|=A2logss2,

(note that the existence of s* is guaranteed by item (i) of Proposition 3.3). Considering q2m(s*) > 0 and using minimality (the case q2m(s*) < 0 is similar), we have on the one hand,

q2m(s)A2ddslogss2=A2s32A2logss3,

which holds thanks to (4.6). On the other hand, we obtain from the ODE (4.2) that

q2m(s)2A2logss3+CAs3,

and the contradiction follows if A ≥ 2C + 1.

As for qj with 2m + 1 ≤ jM and qM,⊥ and qe, we argue as follows: Let λ = log A and assume that s0λ such that for all τs0 and s ∈ [τ, τ + λ] we have

τsτ+λτ+s02τ,hence,1τ1s.

We distinguish into two cases:

  • -

    For ss0λ, we use estimates (4.3), (4.4) and (4.5) with τ = s0 together with Proposition 3.3 to find that

    |qj(s)|C(1+Aj1)sj+12m<Ajsj+12mfor2m+1jM,qM,(y,s)1+|y|M+1LC(1+AM)sM+22m<AM+1sM+22m,qe(s)LC(1+AM+1logA)s12m<AM+2s12m,

    for A large enough. Hence, estimates (4.7) and (4.8) hold for ss0λ.

  • -

    For ss0 > λ, we use estimates (4.3), (4.4) and (4.5) with τ = sλ > s0 together with (4.6) to write (remember that sλ/2)

    |qj(s)|ej2m1λAj(s/2)j+12m+CAj1sj+12m<Ajsj+12mfor2m+1jM,qM,(y,s)1+|y|M+1LCeMλ4mAM+1(s/2)M+22m+CAMsM+22<AM+1sM+22m,qe(s)LCeλ2(p1)AM+2(s/2)12m+CAM+1(1+logA)s12m<AM+2s12m.

    Therefore, estimates (4.7) and (4.8) hold for all s ∈ [s0, s1], hence, the conclusion of part (i) of Proposition 3.5 follows.

Part (ii) of Proposition 3.5 is a direct consequence of the dynamics of qk given in (4.1) and (4.6). Indeed, from part (i) of Proposition 4.1, we know that qk(s1) = ϵAs12 for some k ∈ {0, ⋯, 2m – 1} and ϵ = ±1. Using estimate (4.1) yields

ϵqk(s1)1k2mϵqk(s1)Cs12(1k2m)AC1s12.

Thus, for 0 ≤ kk2m and A large enough, we have ϵqk(s1)>0. Therefore, qk is traversal outgoing to the boundary curve sϵAs–2 at s = s1. This concludes the proof of Proposition 3.5, assuming that Proposition 4.1 holds.□

We now give the proof of Proposition 4.1 to complete the proof of Proposition 3.5. We divide the proof into two subsections according to the two parts of Proposition 4.1.

4.1 Control of the finite dimensional part

We prove item (i) of Proposition 4.1 in this part. A direct projection of equation (1.15) on the eigenfunction ψk for 0 ≤ kM yields

qk(s)1k2mqk(s)=Πk(Vq+B(q)+R).(4.9)

Estimate of Πk(Vq).

We claim the following.

Lemma 4.3

(Expansion of V). The potential V defined by (3.3) satisfies the estimate

|V(y,s)|C(1+|y|2m)s,yR,s1,(4.10)

and admits the following uniform expansion for all n ∈ ℕ*,

V(y,s)=j=1n1sjVj(y)+O1+|y|2m(n+1)sn+1,|y|s12m,s>1,(4.11)

where Vjs are even polynomial of degree 2mj. More precisely, we have

V1(y)=2μ¯2mψ2mwithμ¯2m=ψ2m2,ψ2m.(4.12)

Proof

Estimate (4.10) is trivial. Estimate (4.11) follows from a Taylor expansion in the variable z=|y|2ms. Formula (4.12) comes from the definitions (2.23) and (2.8) of φ and ψ2m.□

From Lemma 4.3, we obtain the following estimate for Πk(Vq).

Lemma 4.4

(Estimate of Πk(Vq)). Under the assumption of Proposition 4.1, we have

|Πk(Vq)|Cs2for0k2m1,Π2m(Vq)+2sq2mCAs3,|Πk(Vq)|CAk2sk+12mfor2m+1kM.

Proof

By Definition 3.1, we have

|q(y,s)|CAM+1s1+1m(1+|y|M+1)for allyR.(4.13)

Using this, the estimate (4.10) and noting from the definition (2.10) and (2.5) that ψk is exponentially decaying, we obtain for 0 ≤ k ≤ 2m – 1,

Πk(Vq)CAM+1s2+1mR(1+|y|M+1+2m)ψk(y)dyCs2.

For 2m + 1 ≤ kM, we write from the decomposition (3.8),

Πk(Vq)=i=0Mqi(s)RV(y,s)ψi(y)ψk(y)dy+RV(y,s)qM,(y,s)ψk(y)dy.

Using (4.10) and the bound of qM,⊥ given in Definition 3.1 yields

RV(y,s)qM,(y,s)ψk(y)dyCAM+1s1+M+12mR(1+|y|M+1+2m)ψk(y)dyCAM+1s1+M+12m.

From the orthogonality (2.11), we note that Pn,ψk=0 for all polynomial Pn of degree nk – 1. We then use (4.11) and (4.13) to estimate

i=0Mqi(s)RV(y,s)ψi(y)ψk(y)dy=i=0Mj=1nqi(s)sj|y|s12mVj(y)ψi(y)ψk(y)dy+OAM+1sn+1+1m|y|s12m(1+|y|2m(n+1)+M+1)|ψk(y)|dy+OAM+1s2+1m|y|s12m(1+|y|2m+M+1)|ψk(y)|dy.

The last term is bounded by 𝓞(ecs) because of the exponential decay of ψk. By taking n ∈ ℕ* such that n + 1 + 1mk+22m, the second term is bounded by OAM+1sk+22m. Using the fact that Vj ψi is polynomial of degree 2mj + i, we see that Vjψi,ψk = 0 for 2mj + ik - 1. Combining this with the bounds of qi given in Definition 3.1 of 𝓥A, we obtain the rough estimate

i=0Mj=1nqi(s)sj|y|s12mVj(y)ψi(y)ψk(y)dyi=0Mj=1,2jm+iknCAisj+i+12mCAk2sk+12m+i=k1MCAMs1+i+12mCAk2sk+12m.

As for k = 2m, we need to use the precise definition (4.12) of V1 and process similarly as for k ≥ 2m + 1. Indeed, from decomposition (3.8) and expansion (4.11) with n = 2, we have

Π2m(Vq)q2ms|y|s12mV1(y)ψ2mψ2mdy=i=0,i2mMqi(s)s|y|s12mV1ψiψ2m+OAM+1s3+1m.

Using the definition (4.12) of V1, the bound of qi given in Definition 3.1 and the fact coming from the definition (2.8) of ψβ and the orthogonality (2.11) that

Rψ2mψiψ2mdy=R(c0ψ2m+i+c1ψi+c2ψi2m)ψ2m=0for2m+1i4m1,

we arrive at

Π2m(Vq)+2sq2mi=02m1CAs3+4mMCAMs1+i+12m+CAM+1s3+1mCAs3.

This concludes the proof of Lemma 4.4. □

We now turn to the estimate of the main contribution of the nonlinear term B(q) under the projection Πk. We begin with the following expansion.

Lemma 4.5

(Expansion of B(q)). For all |q| < 1, ss0 ≫ 1 and |y| ≤ s12m, the function B(q) defined by (3.4) admits the uniform expansion

B(q)j=2M+1qjl=0M1slBj,l(ys12m)+B~j,l(y,s)C|q|M+2+CsM+1,(4.14)

where Bj,l(z)’s are even polynomials of degree l and

|B~j,l(y,s)|C(1+|y|M+1)sM+12m.

Furthermore, we have the estimate for all y ∈ ℝ and s ≥ 1,

|B(q)|C|q|p¯withp¯=min{2,p}.(4.15)

Proof

We first note from the definition (2.23) of φ that there exist two positive constants c0, C0 such that c0φ(y, s) ≤ C0 for |y| ≤ s12m. Thus, a Taylor expansion of B(q, φ) in terms of q yields

B(q)j=2M+1Bj(φ)qjC|q|M+1,

where Bj(φ) admits the following expansion in terms of 1s,

Bj(φ)l=0M1slBj,l(Φ)CsM+1.

Now, a Taylor expansion of Bj,l(Φ) in terms of the variable z=ys12m yields the desired result. This concludes the proof of Lemma 4.5. □

With Lemma 4.5 at hand, we estimate the main contribution of B(q) under the projection Πk. We claim the following.

Lemma 4.6

(Estimate for Πk(B(q))). Under the assumption of Proposition 4.1, we have

Πk(B(q))Cs2for0k2m1,Π2m(B(q))Cs3,Πk(B(q))CAksk+22mfor2m+1kM.

Proof

From (4.14), the exponential decay of ψk and (4.13), we have

Πk(B(q))=j=2M+1l=0M1sl|y|s12mqjBj,lψkdy+OAM+1s2(1+1/m)+M+12m+O(ecs).

From the decomposition (3.8), we write

qj=(i=0Mqiψi+qM,)j(q<+qM,)jforj2.

From Definition 3.1 of 𝓥A and the uniform boundedness of q, we obtain the bound for j ≥ 2,

|qjq<j|C(|qM,|j+|q<|j1|qM,|)CAj(M+1)sM+4+2m2m(1+|y|j(M+1)).

Therefore,

Πk(B(q))=j=2M+1l=0M1sl|y|s12mq<jBj,lψkdy+OAj(M+1)sM+4+2m2m.

We only handle the case k = 2m, which is the most delicate. The case k ≠ 2m can be processed similarly and we omit it. From (4.13), we have

j=3Ml=0M1sl|y|s12m|q<jBj,lψ2m|dy+l=1M1sl|y|s12m|q<2Bj,lψ2m|dyCAj(M+1)s3+1mCs3.

It remains to control the term I:=|y|s12mq<2ψ2mdy. By the definition of q<, we expand

q<2=i=02mqiψi2+2i=02mqiψii=2m+1Mqiψi+i=2m+1Mqiψi2=I1+I2+I3.

From Definition 3.1, we have |I1|CA4log2s4(1+|y|4m)and|I2|CAM+4logss3+1/m(1+|y|M+2m). Hence,

|y|s12m|(I1+I2)ψ2mdy|CA4log2s4+CAM+4logss3+1/m1s3.

Note from the orthogonality relation (2.11) that if i′ + l′ ≠ J2m for J ∈ ℕ*, we have

ψiψl,ψ2m=i=0i2mci,iy2imyil=0l2mcl,ly2lmyl,ψ2m=j=0i2m+l2mc^i+l,jyi+l2jm,ψ2m=j=0i2m+l2mc~i+l,jψi+l2jm,ψ2m=0.

Therefore, we estimate

|y|s12mI3ψ2mdy=i,l=2m+1Mci,lqi(s)ql(s)Rψiψlψ2mdy+O(ecs)=i,l=2m+1,i+l=J2mMci,lqi(s)ql(s)Rψiψlψ2mdy+O(ecs).

for J ∈ ℕ*. Since i′ + l′ ≥ 4m + 2, it follows that J ≥ 3. From Definition 3.1, we have the bound |qiql|Ai+lsi+l+22mAi+ls3+1m1s3. Therefore, |y|s12mI3ψ2mdy1s3. This concludes the proof of Lemma 4.6. □

We now deal with the generated error term R. We begin with the following expansion.

Lemma 4.7

(Expansion of R). The function R defined by (3.5) satisfies

R(s)LCs,

and admits the following expansion: for all n ∈ ℕ*, |y| ≤ s12m and s ≥ 1:

R(y,s)j=1n11sj+1Rj(y)C(1+|y|2mn)sn+1,(4.16)

where Rjs are polynomials of degree 2mj of the form Rj(y)=i=0jdiy2mi. Moreover, the coefficient of degree 2m of R1 is identically zero, hence, R1,ψ2m.

Proof

Let z=ys12m and note that Φ(z) satisfies equation (2.21). Therefore, we rewrite (3.5) as follows:

R(y,s)=z2msΦ+Am,ps2+1sAmΦAm,p(p1)s+QΦ+Am,psQ(Φ),(4.17)

where Q(h) = hp. Since c0Φ(z) ≤ C0 for all |z| ≤ 1 with c0, C0 some positive constants, we have the following expansion of Q,

QΦ+Am,psQ(Φ)j=1J1sjQj(Φ)CsJ+1forJN.

Then, we expand Qj and all the remaining terms in (4.17) in power series of Z = z2m to obtain the desired result. Note that the coefficient of 1s in the expansion of R (after an elementary computation) is given by

Am,p(2m)!κBm,pp1=0,

where we used (2.24). Moreover, R1(y) = C1 y2m + C2, where C2 = C2(p, m) and

C1=κBm,pp1(4m)!(2m)!pBm,p2(p1)pAm,pκ1=0.

Again, the precise values of Bm,p and Am,p given in (2.24) are crucial in deriving that C1 is identically zero. Therefore, the orthogonality relation (2.11) yields R1,ψ2m=0. This concludes the proof of Lemma 4.7. □

As a direct consequence of Lemma 4.7, we have the following.

Lemma 4.8

(Estimate of Πk(R)). Under the assumption of Proposition 4.1, we have

Πk(R)=O(sM)for(kmod2m)0,|Π0(R)|Cs2,|Π2m(R)|Cs3,|Π2mi(R)|Csi+1foriN.

Proof

The proof directly follows from the expansion (4.16) and the fact that

y2mj,ψk=i=0jaiψ2mi,ψk=0for(kmod2m)0.

For k ∈ ℕ with (k mod 2m) ≠ 0, we use the expansion (4.16) with n = M − 1 (we can replace M by any positive integer L ≫ 1) and write

|Πk(R)|=RR(y,s)ψk(y)dy=|y|s12mR(y,s)ψk(y)dy+|y|s12mR(y,s)ψk(y)dyj=1M2|y|s12m|Rj(y)|sj+1|ψk(y)|dy+CsMR(1+|y|2m(M1))|ψk(y)|dy+Cs|y|s12m|ψk(y)|dyCsM+CecsCsM.

For (k mod 2m) ≠ 0, i.e. k = 2m i for some i ∈ ℕ, we use (4.16) with n = i + 1 to get the conclusion. This ends the proof of Lemma 4.8. □

A collection of all estimates given in Lemmas 4.4, 4.6, 4.8 and equation (4.9) yields the conclusion of part (i) of Proposition 4.1.

4.2 Control of the infinite dimensional and the outer part

We prove item (ii) of Proposition in this part. We first deal with the infinite dimensional part qM,⊥, then the outer part qe.

Control of qM,⊥:

Applying ΠM,⊥ to equation (1.15) and using the fact that ΠM,⊥ψn = 0 for all nM (see (3.10)), we obtain

sqM,=LmqM,+ΠM,Vq+B(q)+R.(4.18)

By definition, we have ΠM,⊥(yk) = 0 for all kM. Using the uniform bound R(s)LCs and the expansion (4.16) with n=M2m (see (3.11) for the definition of M) and noting that ΠM,⊥(Rj) = 0 for jM2m, we obtain

ΠM,(R)C(1+|y|M+2m)sM2m+21|y|s12m+|R(y,s)|1|y|s12mC(1+|y|M+1)sM+12m+1.(4.19)

As for the estimate of ΠM,⊥(Vq), we claim the following.

Lemma 4.9

(Estimate of ΠM,⊥(Vq)). Under the assumption of Proposition 4.1, we have

ΠM,(Vq)1+|y|M+1L(R)V(s)L(R)+CsqM,1+|y|M+1L(R)+CAMsM+22m.(4.20)

Proof

By definition, we write

ΠM,(Vq)=VqM,ΠM,<(VqM,)+ΠM,(VqM,<),

where qM,< = ΠM,<(q) = (Id − ΠM,⊥)(q). Using estimate (4.10), we derive

VqM,1+|y|M+1L+ΠM,<(VqM,)1+|y|M+1LV(s)L(R)+1sqM,1+|y|M+1L.

As for the control of ΠM,(VqM,<)=iMΠM,(qiVψi), we argue as follows:

  • -

    If Mi = 0 mod 2m, we take n=Mi2min the expansion (4.11) and write

    ΠM,(Vqiψi)=j=1nqisjΠM,(Vjψi)+ΠM,(V~nqiψi),

    where |V~n(y,s)|C(1+|y|2m(n+1))sn+1. Using the fact that ΠM,⊥(yn) = 0 for nM, we have j=1nΠM,(Vjqi)=0. Hence,

    ΠM,(Vqiψi)C|qi|sMi2m+1CAMsM+22m.

  • -

    If Mi = l mod 2m for some l ∈ (1, 2m), we take n=Mil2m in the expansion (4.11) to deduce that

    ΠM,(Vqiψi)C|qi|sMil2m+1CAMsM+22m.

    This concludes the proof of Lemma 4.9. □

Concerning the control of ΠM,⊥(B(q)), we claim the following.

Lemma 4.10

(Estimate of ΠM,⊥(B(q))). Under the assumption of Proposition 4.1, we have

ΠM,(B(q))1+|y|M+1L(R)CA(M+2)2sM+1+p¯2mwithp¯=min{2,p}.(4.21)

Proof

Let BM be defined by

BM(y,s)=ΠM,<j=2M+1qjl=0M1slBj,l(ys12m),(4.22)

where ΠM,< = Id − ΠM,⊥ and Bj,l’s are introduced in Lemma 4.5. Then we claim the following: for all y ∈ ℝ and s ≥ 1,

B(q)BMCA(M+2)2sM+1+p¯2m(1+|y|M+1).(4.23)

The estimate (4.21) simply follows from (4.23) since ΠM,⊥(BM) = 0. Let us prove (4.23). In the region |y|s12m, we use (4.15) and the bounds given in Definition 3.1 to estimate

|B(q)|CAp¯(M+2)sp¯2mCAp¯(M+2)sM+1+p¯2m(1+|y|M+1)for|y|s12m.

In the region |y|s12m, we write from Lemma 4.5 the expansion

B(q)ΠM,<j=2M+1qjl=0M1slBj,l(ys12m)ΠM,j=2M+1qjl=0M1slBj,l(ys12m)+j=2M+1qjl=0M1slB~j,l(y,s)+|q|M+2+CsM+1.

A similar argument as in the proof of Lemma 4.6 shows that the coefficient of degree kM + 1 of the polynomial j=0M+1qjl=0M1slBj,l(ys12m) is bounded by Aksk+22m, hence,

ΠM,j=2M+1qjl=0M1slBj,l(ys12m)CA2(M+1)sM+32m(1+|y|M+1),|y|s12m.

From (4.13) and Remark 3.2, we obtain

j=2M+1qjl=0M1slB~j,l(y,s)CA(M+2)(M+1)s1m+M+12m(1+|y|M+1)CA(M+2)2sM+1+p¯2m(1+|y|M+1),

and

|q|M+2CAM+2s12mM+1AM+2s1+1m(1+|y|m+1)CA(M+2)2sM+1+p¯2m(1+|y|M+1).

This completes the proof of (4.23) as well as the proof of Lemma 4.10. □

Plugging (4.19), (4.20) and (4.21) into equation (4.18) yields

sqM,=LmqM,+GM,,

where GM,⊥ satisfies for s large enough,

GM,(y,s)1+|y|M+1L(R)V(s)L(R)qM,(y,s)1+|y|M+1L(R)+CAMsM+22m.

Using the semigroup representation by 𝔏m, we write for all s ∈ [τ, τ1],

qM,(s)=e(sτ)LmqM,(τ)+τse(ss)LmGM,(s)ds,

where es𝔏m is defined in Lemma 2.3. Letting λ(s)=qM,(s)1+|y|M+1L(R) and using (iii) of Lemma (2.3), we have

λ(s)eM2m(sτ)λ(τ)+τseM2m(ss)GM,(s)1+|y|M+1L(R)dseM2m(sτ)λ(τ)+τseM2m(ss)V(s)L(R)λ(s)+CAMsM+22mds.

Since we have fixed M large such that VLy,sM4m (see (3.11)), a standard Gronwall’s argument applied to the function eMs2mλ(s) yields

eMs2mλ(s)eM(sτ)4meMτ2mλ(τ)+CeMs2mAMsM+22,

from which we conclude the proof of (4.4).

Control of qe.

We write from (1.15) the equation satisfied by qe = (1 − χ(y, s))q,

sqe=Lmpp1qe+(1χ)(Q+R)+E1+E2,

where R is defined by (3.5),

Q=|q+φ|p1(q+φ)φp,E1=q(sχ+Amχ+12myχ),E2=Am(χq)+qAmχχAmq=j=12m1cjj(q2mjχ).

Using the semigroup representation by 𝔏m and items (i)-(ii) of Lemma 2.3, we write for all s ∈ [τ, τ1],

qe(s)Lesτp1qe(τ)L+τsessp1(1χ)(Q(s)+R(s))+E1(s)Lds+Cj=12m1τsessp1(1e(ss))j2mq(s)2mjχ(s)Lds.

For K large enough, we have

(1χ(y,s)Q(s))LCφ(s)Lp1qe(s)L12(p1)qe(s)L.

Recall from Lemma 4.7 that R(s)LCs. As for E1, we use the definition of χ given in (3.6) and the bounds given in Definition 3.1 to obtain the estimate

E1(s)LCq(s)L(Ks12m|y|2Ks12m)CAM+1s12m.

Since 2mjχ(s)LCs2mj2m, using the fact that ∇2mj χ is compactly supported, we estimate

j=12m1q(s)2mjχ(s)LCAM+1s1m.

A collection of these above estimates yields

qe(s)Lesτp1qe(τ)L+τsessp112(p1)qe(s)L+CAM+1s12m+CAM+1[s1e(ss)]1mds.

Applying the standard Gronwall’s inequality to the function esp1qe(s)L yields

esp1qe(s)Lesτ2(p1)eτp1λ(τ)+CAM+1s12m(sτ+1),

from which the estimate (4.5) follows. This completes the proof of Proposition 4.1. □

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading and suggestions to improve the presentation of the paper.

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Footnotes

  • 1

    In [5] and [23], the kernel 𝓚 of the heat semigroup associated to the linear operator ℒ1 + V is defined through the Feymann-Kac representationK(s,σ,y,x)=e(sσ)L1(y,x)dμy,xsσ(ω)e0sσV(ω(τ),σ+τ)dτ,where et1(y, x) is given by Mehler’s formula and dμy,xt is the oscillator measure on the continuous path: ω : [0, t] → ℝN with ω(0) = x and ω(t) = y. 

About the article

Received: 2018-05-17

Accepted: 2018-12-13

Published Online: 2019-06-01

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 388–412, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0006.

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© 2020 T. Ghoul et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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