Abstract
We consider the Dirichlet problem for the energy-critical heat equation
where \(\Omega \) is a bounded smooth domain in \(\mathbb {R}^3\). Let \(H_\gamma (x,y)\) be the regular part of the Green function of \(-\Delta -\gamma \) in \(\Omega \), where \(\gamma \in (0,\lambda _1)\) and \(\lambda _1\) is the first Dirichlet eigenvalue of \(-\Delta \). Then, given a point \(q\in \Omega \) such that \(3\gamma (q)<\lambda _1\), where
we prove the existence of a non-radial global positive and smooth solution u(x, t) which blows up in infinite time with spike in q. The solution has the asymptotic profile
where
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1 Introduction and statement of the main result
We investigate the asymptotic structure of global in time solutions u(x, t) of the energy-critical semilinear heat equation
where \(\Omega \subset \mathbb {R}^3\) is a smooth bounded domain and \(u_0\) is a smooth initial datum. The energy associated to the solution u(x, t) is
Since classical solutions of (1.1) satisfy
the energy is a Lyapunov functional for (1.1). The stationary equation on the whole space is the Yamabe problem
All positive solutions to this equation are given by the Aubin-Talenti bubbles (see [4])
where \(\mu >0,\xi \in {{\mathbb {R}}}^3\) and
Consider the Sobolev embedding \(H_0^1(\Omega )\hookrightarrow L^{p+1}(\Omega )\), which is compact for \(p\in (1,p_S)\), where \(p_S=\frac{n+2}{n-2}\), and the associated constant
The Aubin-Talenti bubbles achieve the constant \(S_{p_S}({{\mathbb {R}}}^n)\). Thus, the energy \(E(U_{\mu ,\xi })=S_{p_S}({{\mathbb {R}}}^n)\) is invariant with respect to \(\mu ,\xi \). When \(\mu \rightarrow 0\) the Aubin-Talenti bubble becomes singular. This is the reason for the loss of compactness in the Sobolev embedding for \(p=p_S\). Indeed, Struwe proved in [52] that every Palais-Smale sequence associated to the energy functional E looks like
up to subsequences, for some \(k\in \mathbb {N}\), where \(u_\infty \in H_0^1(\Omega )\) is a critical point of E and \(\mu _n^i\rightarrow 0\), \(\xi _n^i\in \Omega \). Thus, we say that the compactness is lost by ’bubbling’. When the domain is star-shaped, the Pohozaev identity constrains \(u_\infty \) to vanish.
For classical finite-energy solutions u(x, t) the problem (1.1) is well-posed in short time intervals. We refer to the monograph [48] by Quittner and Souplet for an extended review on this problem and more general semilinear parabolic equations. The aim of this paper is exhibiting classical positive finite-energy solutions u(x, t) of (1.1) which are globally defined in time and satisfy
Given any smooth function \(\varphi (x)\ge 0,\varphi \ne 0\), consider \(\alpha >0\) and \(u_\alpha (x,0){:=}\alpha \varphi (x)\) as initial datum. On one hand, if \(\alpha \) is sufficiently small, then \(u_\alpha (x,t)\) tends uniformly to zero as \(t\rightarrow \infty \). On the other hand, using the eigenfunction method of Kaplan [39], for \(\alpha \) sufficiently large \(u_\alpha (x,t)\) blows-up in finite time. Thus, the threshold number
is positive. In 1984, the first rigorous proof of the existence in \(L^1\)-weak sense of \(u_{\alpha ^*}(x,t)\) was found by Ni, Sacks and Tavantzis [47]. Du [25] and Suzuki [54] proved, that, for any unbounded sequence of times \(t_n\), \(u_{\alpha ^*}(x,t_n)\) can be decomposed as in (1.3). Thus, when constructing unbounded global solutions for the critical case, it is natural to look for an asymptotic profile as (1.2). Galaktionov and Vázquez [30] proved that, in the radial case \(\Omega =B_1(0)\) with \(\varphi \) radial non-increasing, \(u_{\alpha ^*}(x,t)\) is smooth, global and \(u=u_{\alpha ^*}\) satisfies (1.4). Thus, we naturally wonder what is the asymptotic behavior of global unbounded solutions. Most of the results about the dynamics of threshold solutions in literature concern the radial case. This particular setting allows the construction of specific solutions by means of matched expansions. In [29] Galaktionov and King proved that the threshold behavior of \(u_{\alpha ^*}\) in the radial case is
and
for some explicit constants \(\gamma _n\). Our main theorem is a non-radial extension in dimension 3. The existence of positive non-radial unbounded solutions for the Dirichlet problem in dimension \(n=4\) remains an open problem, which we will consider in a future work. The case of higher dimension \(n\ge 5\) has been already extended to the non-radial case by Cortázar et al. [11]. They found positive multi-spike global solutions which blow-up by bubbling in infinite time. Here, the term multi-spike refers to the fact that the constructed solution is unbounded in a finite number of points in \(\Omega \). Sign-changing solutions which blow-up in infinite time have been discovered by del Pino et al. [19] for \(n\ge 5\), proving stability in case \(n=5,6\).
Our solutions involve the Green function \(G_\gamma \) associated to the elliptic operator
where \(\gamma \in [0,\lambda _1)\) and \(\lambda _1\) is the principal Dirichlet eigenvalue. Namely, for all \(y\in \Omega \), \(G_\gamma \) satisfies
where \(\delta (x)\) is the Dirac delta, \(c_3 :=\alpha _3\omega _3\) and the constant \(\omega _3=4\pi \) indicates the area of the unit sphere. The Green function can be decomposed as
where \(\Gamma (x)=\alpha _3\vert x\vert ^{-1}\) and the regular part \(H_\gamma (x,y)\) is defined as the solution, for all \(y \in \Omega \), to
The diagonal \(R_\gamma (x){:=}H_\gamma (x,x)\) is called Robin function associated to \(-\Delta -\gamma \) in \(\Omega \). It turns out (see Lemma 2.1) that for any fixed \(q\in \Omega \) there exists a unique number \(\gamma (q)\in (0,\lambda _1)\) defined by
Our main theorem shows that, for any \(q\in \Omega \) such that \(3\gamma (q)<\lambda _1\), there exists a global solution to the problem (1.1) which blows-up in infinite time with spike in \(x=q\).
Theorem 1.1
Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded smooth domain. Let q be a point in \(\Omega \) such that
Then, there exist an initial condition \(u_0(x)\in C^1({\bar{\Omega }})\), smooth functions \(\xi (t),\mu (t)\) and \(\theta (x,t)\) such that the solution u(x, t) to the problem (1.1) is a positive unbounded global solution with the asymptotic profile
where \(\theta \) is a bounded function, and decays uniformly away from the point q. Moreover, the parameters \(\mu (t),\xi (t)\) are smooth functions of time and satisfy
Furthermore, thanks to the inner-outer gluing scheme, which is based only on elliptic and parabolic estimates, as in [11, 15] we get a codimension-1 stability of the solution stated by Theorem 1.1. In fact, since condition (1.6) is stable under small perturbation of \(q\in \Omega \), the stability result follows exactly as in [11, Proof of Corollary 1.1] (see Remark 7.1 in Sect. 7).
Corollary 1.1
Let u be the solution stated in Theorem 1.1 which blows up at q. Then, there exists a codimension-1 manifold \(\mathcal {M}\) in \(C^{1}({\bar{\Omega }})\) with \(u_0 \in \mathcal {M}\) and such that if \(\tilde{u}_0\in \mathcal {M}\) and it is sufficiently close to \(u_0\), then the solution \({\tilde{u}}\) to (1.1) with initial datum \({\tilde{u}}_0\) is global and blows-up in infinite time with spike in \({\tilde{q}}\) near q and profile (1.7) with \(\ln \vert \!\vert \tilde{u}(\cdot ,t)\vert \!\vert _\infty = \gamma ({\tilde{q}}) t(1+o(1))\) as \(t\rightarrow \infty \).
Condition (1.6) implies that the point q cannot be very close to boundary, since \(\gamma (q)\rightarrow \lambda _1^{-}\) as \(q\rightarrow {\partial } \Omega \) (see Lemma A.2 in Appendix A). Along the proof we need to consider Dirichlet problems of the type
for some \(f(x)\in L^p\) with \(p>2\). In order to successfully apply fixed point arguments, we need
for \(t>1\), which requires condition (1.6). Such assumption (1.6) is useful to get rid of a resonance effect, lastly due to the fact that both the Dirichlet heat kernel \(p_t^{\Omega }(x,y)\) and the parameter \(\mu (t)\) decay exponentially fast. Indeed, the long-term behavior of the Dirichlet heat kernel is
where \(\phi _1\) is the positive eigenfunction of \(-\Delta \) in \(\Omega \) with \(\vert \!\vert \phi _1\vert \!\vert _2=1\). We recall the properties of the Dirichlet heat kernel in Sect. 8. More specifically, we use assumption (1.6) in the following steps of the proof:
-
to get estimates for \(J_1,J_2\) in Lemma 2.2 and Lemma 2.3 respectively;
-
in Lemma 4.1 for solving the outer problem;
-
in Proposition 6.1 for the invertibility theory of the nonlocal operator \(\mathcal {J}\).
The number \(\gamma (q)\) is related to the Brezis-Nirenberg problem. Define
In the celebrated work [2], Brezis and Nirenberg proved the existence of a constant \(\mu _{\text {BN}}\in (0,\lambda _1)\) such that
Then, Druet [24] proved
Thus, when \(3\mu _{BN}(\Omega )<\lambda _1(\Omega )\) is true, condition (1.6) is satisfied in some open set \(\mathcal {O}\subset \Omega \), and Theorem 1.1 gives the desired solution with blow-up at any fixed point \(q\in \mathcal {O}\).
When we consider the radial case \(\Omega = B_1(0)\) and \(q=0\), an explicit computation gives \(\gamma (0)=\pi ^2/4\), that is consistent with (1.5). In fact, this is the minimum value for \(\gamma (q)\) since Brezis and Nirenberg computed \(\mu _{BN}(B_1)=\pi ^2/4\). By symmetry, we deduce that condition (1.6) is satisfied in the ball \(B_{d^*}\), where \(d^*=\vert q^*\vert \) and \(q^*\) is a point such that \(\gamma (q^*)=\lambda _1/3\).
Also, we can consider smooth perturbation of the ball. Let \(f: \bar{B}_1 \rightarrow {{\mathbb {R}}}^3\) a smooth map and for \(t>0\) define
For small t the domain \(\Omega _t\) is diffeomorphic to the ball. Writing \(\lambda _1\) as Rayleigh quotient and using the definition \(\mu _{BN}\) we can easily see that \(\mu (\Omega _t)=\mu (B_1)+\varepsilon (t)\) and \(\lambda _1(\Omega _t)=\lambda _1(B_1)+{\tilde{\varepsilon }}(t)\) where \(\varepsilon (t),{{\tilde{\varepsilon }}}(t)\rightarrow 0\) as \(t\rightarrow 0\). Thus, for t sufficiently small, the relation \(3\mu _{\text {BN}}(\Omega _t)<\lambda _1(\Omega _t)\) holds, and Theorem 1.1 applies to the domain \(\Omega _t\). This shows that Galaktionov-King’s radial result is stable under small perturbation of the domain.
For the unit cube \( {{\mathcal {C}}}_1\) it is known (see [58, Remark 4.3]) that \(3\mu _{\text {BN}}( {{\mathcal {C}}}_1)< \lambda _1(\mathcal {C}_1)\). Indeed, from \(B_{1/2}\subset \mathcal {C}_1\) and the strict monotonicity of \(\mu _{BN}(\Omega )\) with respect to \(\Omega \) we deduce \(\mu _{\text {BN}}\left( \mathcal {C}_1\right) < \mu _{\text {BN}}\left( B_{1/2}\right) =\pi ^2\). By separation of variables we easily compute \(\lambda _1(\mathcal {C}_1)=3\pi ^2\), thus
Hence, a slight modification of Theorem 1.1 applies: since \( {{\mathcal {C}}}_1\) is a Lipschitz domain, by the parabolic regularity theory we get a smooth solution u(x, t) in \(\Omega \times {{\mathbb {R}}}^+\) which is Lipschitz continuous in \({\bar{\Omega }} \times [t_0,\infty )\).
Let \(\Omega ^*\) be the ball with the same volume as \(\Omega \). The following estimate holds true:
The lower bound was proved in [2] by means of a symmetrization argument. Using harmonic transplantation Bandle and Flucher [1] proved the upper bound. Thus, if it happens that we know \(\min _{x\in \Omega } R_0(x)^2<4/3\) we can apply Theorem 1.1 to \(\Omega \). Wang [58] conjectured that \( {\mu _{\text {BN}}}/\lambda _1\in [1/4,4/9)\). In particular, condition \(3\mu _{BN}(\Omega )<\lambda _1(\Omega )\) could be false for "very thin rectangles" (see [58]). The range [1/4, 4/9) is supported by numerical computations made by Budd and Humphries [3].
The main differences with respect to the analogue result [11] in dimension \(n\ge 5\) are the following:
-
the main asymptotic behavior in Theorem 1.1 of the blow-up is dependent on the position of the point \(q\in \Omega .\) As far as we know, this is a completely new phenomenon;
-
since condition (1.6) is not satisfied close to the boundary, we cannot straightforward construct multi-spike solutions in the spirit of [11]. Indeed, roughly speaking, such construction requires spikes relatively far from each other and close to the boundary to suitably bound the interaction between the bubbles.
-
a nonlocal operator controls the dynamic of the parameter \(\mu (t)\). A similar operator has been treated in [15], where the domain \(\Omega ={{\mathbb {R}}}^3\) allows an explicit inversion of the Laplace transform.
The approach developed in this work is inspired by [11, 13, 15]. It is constructive and allows an accurate analysis of the asymptotic dynamics and stability. Let describe the general strategy. The first step consists in choosing a good approximated solution \(u_3\). Here the word ’good’ means that the associated error function
is sufficiently small in \(\Omega \). Part of the problem consists in understanding what smallness on S[u] is sufficient to find a perturbation \({{\tilde{\phi }}}\) such that
is an exact solution to (1.1). In Sect. 2 we start with the scaled Aubin-Talenti bubble as building block and we modify it to match the boundary at the first order. Then we realize that we need two improvements. The first one is a global correction useful to get solvability conditions for the elliptic linearized operator around the standard bubble
Such improvement produces a nonlocal operator which governs the second order term in the expansion of the scaling parameter \(\mu (t)\). This is a low-dimensional effect, lastly due to the fact that
where \(Z_{n+1}\) is the unique (up to multiples) bounded radial function belonging to the kernel of \(L[\phi ]\). Actually, the dimensional restriction in [11] was specially designed to avoid this effect and the presence of the corresponding nonlocal term. Then, by choosing \(\gamma (q)\) as in (1.6) we reduce the error close to \(x=q\); this gives the asymptotic behavior (1.8) of \(\mu (t)\) at the first order. A second correction, local in nature, removes non-radial slow-decay terms and gives the asymptotic for \(\xi \) written in (1.8). At this point we have a sufficiently good ansatz, called \(u_3\), to start the so called inner-outer gluing procedure in Sect. 3: we decompose the problem in a system of nonlinear problems, namely an inner and an outer problem which are weakly coupled thanks to the smallness of \(S[u_3]\). We solve the outer problem in §4, that is a perturbation of the standard heat equation, for suitable parameters \(\mu ,\xi \) and decaying solution \(\phi \) of the inner problem. Then, we look at the inner regime. We can find the inner solution, by fixed point argument, using the adaptation to \(n=3\) of the linear theory for the inner problem developed in [11]. This requires the solvability of orthogonality conditions which, in Sect. 5, we prove to be equivalent to a nonlocal system in the parameters \(\mu ,\xi \). We solve it in Sect. 6 using the invertibility of a nonlocal equation, which we achieve in Sect. 8 by means of a Laplace transform argument combined with asymptotic properties of the heat kernel \(p_t^\Omega (x,y)\). At this point we are ready to find the inner solution \(\phi \) in Sect. 7, which concludes the proof of Theorem 1.1.
Of course, the full problem consists in finding the exact initial datum that evolves in an infinite time blow-up solution. We find the positive initial condition
for \(t_0\) fixed sufficiently large, where the existence of \(\mu ,\xi ,\phi ,\psi \) and the constant \(e_0\) is a consequence of fixed point arguments, \(\eta _l,l,\eta _R,R\) are defined in (2.5), (2.17) and the functions \(\phi _3,J_1\) solve the problems (2.21) and (2.14). We remark that we do not know if the solution with this initial datum corresponds to a threshold solution in the sense of [47].
We conclude this introduction giving a short bibliographic overview about related problems and recent developments. The rigorous construction of blow-up solutions by bubbling, that is a solution \(u(x,t)\approx U_{\mu (t),\xi (t)}(x)\) with \(\mu \rightarrow 0\) for some special profile U, has been extensively studied in many important problems with criticality. For instance, in the harmonic map flow [13, 49, 50], in the Patlak-Keller-Segel model for chemotaxis [8, 9, 12, 31], in the energy-critical wave equation [26, 36, 40, 41] and energy critical Schrödinger map problem [45].
Concerning the Cauchy problem
infinite blow-up positive solutions have been found in dimension \(n=3\) in del Pino et al. [15], with different blow-up rates depending on the space decay of the initial datum. Recently, Wei et al. [59] detected analogue solutions in dimension \(n=4\). These works were inspired by conjectures presented in [27], where Fila and King used matched asymptotic methods to formally analyze the behavior of infinite blow-up solutions in the radial case, also conjecturing that for \(n\ge 5\) such solutions do not exist. However, adding drift terms to the equation, Wang et al. [56] have shown examples of positive initial datum which evolves in multi-spike infinite blow-up by bubbling. For \(n\ge 7\), del Pino et al. [16] proved the existence of sign-changing solutions which blow-up in infinite time in the form of tower of bubbles, that is a supersolution of Aubin-Talenti bubbles at a single point. For the analogue backward problem where \(t\in (-\infty ,0)\), ancient solutions which blow-up in infinite time have been detected by Sun et al. [53] for \(n\ge 7\).
As we have already mentioned, blow-up for the nonlinear heat equation
can also happen in finite time \(T<\infty \). We call it Type I blow-up if the solution satisfies
otherwise, if
we have Type II blow-up. Several works have focused on constructing finite time blow-up solutions for the Cauchy problem. Positive Type II blow-up solutions do not exist in dimension \(n\ge 7\), see Wang and Wei [57], or under radial assumptions in any dimension \(n\ge 3\), see Matano-Merle [44] and the pioneering work by Filippas-Herrero-Velázquez [28]. In dimension \(n \ge 7\), Collot et al. [10] classified the dynamics near the Aubin-Talenti bubble U in the \({\dot{H}}^1\) topology. In particular, they ruled out the Type II scenario for initial conditions \(u_0\) such that \(\vert \!\vert u_0-U\vert \!\vert _{\dot{H}^1({{\mathbb {R}}}^n)}\) is sufficiently small. The existence of positive Type II blow-up in dimensions \(n\in \{3,4,5,6\}\) is an open problem.
Type II blow-up it is still admissible for sign-changing solutions, and in fact examples have been found. Type II blow-up solutions have been constructed by Schweyer [51] in dimension 4 under radial assumption and later by del Pino et al. in the non-radial setting [20] with admissible multi-spike behavior. Also, Type II blow-up solutions have been detected in dimension \(n\in \{3,5,6\}\) in [18, 21, 33, 34] with different blow-up rates. Type II blow-up for the critical heat equation can also happen on curves contained in the boundary of special domains with axial symmetry, see [17].
There have been developments also in the nonlocal generalization of these problems. Concerning the fractional heat equation with critical exponent
Cai et al. [5] have recently constructed solutions for both the forward and backward Cauchy problem which are sing-changing tower of bubbles at the origin for \(n>6s\), and \(s\in (0,1)\). For \(n \in (4\,s,6\,s)\) and \(s\in (0,1)\) blow-up in finite time has been proved in [7], which is a fractional continuation of the local Type II blow-up cases \(n=4,s=1\) in [51] and \(n=5,s=1\) in [21]. Regarding the associated Dirichlet problem, Musso et al. provided in [46] the existence of positive multi-spike infinite-time blow-up on bounded smooth domains for \(n\in (4s,6s)\) and \(s\in (0,1)\).
2 Approximate solution and estimate of the associated error
In this section we construct an approximate solution to the problem
and we compute the associated error. Without loss of generality, we construct a solution that blows-up at \(q=0 \in \Omega \). The first approximation \(u_1\) is chosen by selecting a time-scaled version of the stationary solution to the Yamabe problem
properly adjusted to be small at the boundary \( {\partial } \Omega \). This is constructed in Sect. 2.1. In order to make the error small at the blow-up point, we need to select a precise first order for the dilation parameter \(\mu (t)\), which matches the radial asymptotic found in [29]. However, we observe in Sect. 2.2 that \(u_1\) is not close enough to an exact solution to make our perturbative scheme rigorous. In Sect. 2.3 we make a global improvement \(u_2\). Such correction involves a nonlocal operator in the lower order term of \(\mu (t)\), similar to a half-fractional Caputo derivative. The last improvement \(u_3\) is only local, and it removes slow-decaying terms in non-radial modes by selecting the first order asymptotic of the translation parameter \(\xi (t)\).
2.1 First global approximation
Our building blocks are the scaled Aubin-Talenti bubbles (1.2) which satisfy
We look for a solution of the form \(u_1(x,t) \approx U_{\mu (t),\xi (t)}(x)\). We make an ansatz for the parameters \(\mu (t),\xi (t)\). Assuming that \(\mu (t)\rightarrow 0\) and \(\xi \rightarrow 0\in \Omega \) as \(t\rightarrow \infty \), we notice that \(U_{\mu ,\xi }(x)\) is concentrating around \(x=0\) and it is uniformly small away from it. For this reason, we should have
Let \(\mu _0(t)\) the first order of \(\mu (t)\), that is
From (2.3) we define the scaled function
which should satisfy
We choose the parameter \(\mu _0(t)\) such that
for some \(\gamma \in {{\mathbb {R}}}^+\) that will be fixed later. This is equivalent to choose
for some \(b\in {{\mathbb {R}}}^+\). We can fix \(b=1\). Indeed, the equation is translation-invariant in time: we construct, for a sufficiently large initial time \(t_0\), a solution u(x, t) in \(\Omega \times [t_0,\infty )\) and we conclude that \(u_0(x,t){:=} u(x,t-t_0)\) is a solution to (2.1) in \(\Omega \times [0,\infty )\). We observe that after shifting the initial time, the main dilation parameter \(\mu _0\) becomes \(\mu _0(t-t_0)=e^{2\gamma t_0}e^{-2\gamma t}\).
With this choice (2.4) reads
Hence, for large time we should have
where \(G_\gamma (x,y)\) is the Green function for the boundary value problem
We write
where
is (a multiple of) the fundamental solution of the Laplacian in \({{\mathbb {R}}}^3\) and the regular part \(H_\gamma (x,y)\), for fixed \(y\in \Omega \), satisfies
The function \(H_\gamma (\cdot ,y)\in C^{0,1}(\Omega )\) when \(\gamma \in (0,\lambda _1)\). For later purpose, we also write
where
and \(h_\gamma (\cdot ,y)\in C^\infty (\Omega )\) solves
We also define the Robin function
In terms of the original function \(u_1\) the Eq. (2.6) reads as
We notice that far away from the origin we have
This formal analysis suggests the ansatz
2.1.1 Dilation parameter \(\mu (t)\)
We write the full dilation parameter in the form
for some \(\Lambda (t)=o(1)\) as \(t\rightarrow \infty \) to be found, where
In this notation we have
and
where \({\dot{\Lambda }}(s)\) is an integrable function in any \([t_0,\infty )\).
2.2 Error associated to \(u_1\)
The next step consists in computing the error associated to the first ansatz \(u_1\). We define the error operator
Of course, solving \(S[u]=0\) is equivalent to solve the equation in (2.1). It is well-known that all bounded solutions to the linearized operator
are linear combinations of the functions \(\{Z_i\}_{i=1}^4\) defined as
and
We define the scaled variable
Now, we compute \(S[u_1](x,t)\) for \(x\ne \xi (t)\). We have
where we used equations (2.2) and (2.9) for U and \(H_\gamma \) respectively. Using the definition of \(Z_4\), the time-derivative can be written as
Hence, the error associated to \(u_1\) is
2.3 Global improvement
The remaining part of this section concerns the improvement of the natural ansatz \(u_1\). Later in the argument we will divide the error in outer and inner part. We realize that solving the inner-outer system requires a global and a local improvement. Reading Proposition 3.1, which is the linear theory for the inner problem, we see that, to get decay in \(\phi (y,\tau )\) at distance R we need \(a>1\) in the definition of \(\vert \!\vert h\vert \!\vert _{\nu ,2+a}\). This smallness at distance R will make the inner and outer regime weakly decoupled. Our particular h will satisfy \(\vert \!\vert h\vert \!\vert _{\nu ,2+a}<\infty \) with \(a=2\), hence we will use estimate (3.17). Thus, we say that a term is slow-decay in space if it is not controlled by \((1+\vert y\vert )^{-4}\). We can find an exact perturbation with our scheme if we remove such terms. Looking at (2.13) we observe that all the terms in the first two lines are slow-decay. Using the inequality \(\mu (t)\lesssim (1+\vert y\vert )^{-1}\) we can negotiate decay in time with decay in space if needed in other terms. For the moment we can assume \({\dot{\Lambda }}, \Lambda , {\dot{\xi }},\xi \) bounded by some power of \(\mu (t)\). Later we shall specify precise norms for these parameters. Firstly, we decompose
We define
The new error reads as
Let
Plugging \(S[u_1]\) given by (2.13) into \(S[u_2]\) we get
We select \(J_1[{\dot{\Lambda }}](x,t)\) such that
and
The choice of defining \(J_1\) from the time \(t_0-1\), as well as \(\dot{\Lambda }(t)\), will become clear in Sect. 8. For the variable \(\xi (t)\) it is enough to define the extension \(\xi (t)=\xi (t_0)\) for \(t\in [t_0-1,t_0)\). With these choices the error associated to \(u_2\) reads as
2.3.1 Choice of \(\gamma \)
We observe that with this choice of \(J_2\) we remove the singular term \(\vert x-\xi \vert ^{-1}\) from (2.13). At this point, the main error at \(x=\xi (t)\) is given by the first order of the nonlinear term
which is, in general, of size \(\mu (t)^{-3/2}\). We realize that we can reduce this error by selecting \(\gamma \) such that \(R_\gamma (0)=0\). The existence and uniqueness of such number is given by the following lemma.
Lemma 2.1
There exists a unique \(\gamma =\gamma ^*(0) \in (0,{\lambda _1})\) such that \(R_{\gamma ^*}(0)=0\).
Proof
We consider the function \(R_\gamma (0)\) as a function of \(\gamma \). Lemma A.2 in [14] shows that
is smooth in \((0,\lambda _1)\) and \( {\partial } _\gamma R_\gamma (0)<0\). Lemma A.1 in Appendix A shows that \(R_\gamma (0)\rightarrow -\infty \) as \(\gamma \rightarrow \lambda _1^{-}\). By the maximum principle \(H_0(x,y)>0\) for all \(x,y\in \Omega \), hence we have \(R_0(0)>0\) and the intermediate value theorem gives the existence of
Finally the monotonicity of \(R_\gamma (0)\) implies the uniqueness of \(\gamma ^*(0)\). \(\square \)
Remark 2.2
(Regularity of \(\gamma ^*(x)\)) Let \(R(\gamma ,x) :=R_{\gamma }(x)\). Since \(R(\gamma ^*(x),x)=0\) and \( {\partial } _\gamma R(\gamma ,x)<0\) for all \(x\in \Omega \), the implicit function theorem implies that \(\gamma ^*(x)\in C^1(\Omega )\) with
Remark 2.3
(Radial case) We compute \(\gamma (0)\) in case \(\Omega =B_1(0)\). We look for a radial solution to
We define \(l_0(\vert x\vert ){:=}H_\gamma (x,0)\) for a function \(l_0:[0,1]\rightarrow {{\mathbb {R}}}\). Then \(l_0\) solves
We write \(l_0(r)=\alpha _3l(r)/r\), where l(r) solves
The solution to this problem is given by
and we conclude that
In particular, for \(r=0\) we find
Asking for \(R_\gamma (0)=0\)
and, recalling that \(\lambda _1(B_1)=\pi ^2\), the unique value in \((0,\lambda _1)\) is
as predicted in the analysis of Galaktionov and King [29].
For the sake of simplicity we continue to use \(\gamma =\gamma (0)\) to denote the selected number \(\gamma ^*(0)\). Since \(R_\gamma (x)\in C^\infty (\Omega )\) we expand
for some \(\xi ^*\in \overline{[0,\xi ]}\). Assuming \(\xi =O(\mu )\) we conclude
2.4 Local improvement and computation of the final error
In this section we make a further improvement and we obtain the final ansatz. We still need to remove from (2.13) the main order of the terms
We define the final ansatz
The function \(\eta : [0,\infty )\rightarrow [0,1]\) denotes a smooth cut-off function such that \(\eta (s)\equiv 1\) for \(s<1\) and \({{\,\textrm{supp}\,}}\eta \subset [0,2]\), and we define
where \(k_2\) is a constant such that \(B_{\frac{2}{k_2}}(0)\subset \Omega \), to ensure that \({{\,\textrm{supp}\,}}\eta _{l} \Subset \Omega \). Also we define the variable
We compute
and
We define
Thus, using (2.16),
By Taylor expanding \(h_\gamma (x,\xi )\) centered at \(x=\xi \) we have
for some \({\bar{x}}\in \overline{[\xi ,x]}\). Now, we expand the first terms at \((\xi ,\xi )=(0,0)\). By the Chain Rule we have \(\nabla _{x_1}h_\gamma (x,x)=2 \nabla _x R_\gamma (x)\). Hence, we have
for some \(\xi ^{**}\in \overline{[0,\xi ]}\). Furthermore, since \(R_\gamma (0)=0\), we have
for some \(\xi ^* \in \overline{[0,\xi ]}\). Plugging these identities in (2.18) we obtain
We write
Now, we assume the following decay for the parameters \(\xi _1,\dot{\xi }_1, \Lambda ,{\dot{\Lambda }}\):
for some positive constants \(k,l_0,l_1\) to be chosen (in §3.1.3). We write the full error
where
For any fixed \(t>t_0\), we select \(\phi _3(x,t)\) as the bounded solution to the elliptic problem
with the following orthogonality conditions on the right-hand side:
As we shall see in the proof of Lemma 2.4, the conditions (2.22) are essential to have \(\phi _3\) bounded in space and equivalent to choose \(\xi _0(t)\). The condition corresponding to the index \(i=4\) is satisfied by symmetry. When \(i=1,2,3\) the orthogonality condition (2.22) is equivalent to
Hence, we select \(\xi _{0,i}\) such that
With the condition \(\lim \nolimits _{t\rightarrow \infty }\xi _i(t)=0\) we get
Also, we define \(\mathfrak {c}{:=}(\mathfrak {c}_1,\mathfrak {c}_2,\mathfrak {c}_3)\).
Remark 4.1
(No local improvement in the radial case) In case \(\Omega =B_1(0)\), searching \(h_\gamma (r,0)\) solution to (2.12) in the radial form, we see that
hence conditions (2.22) imply \(\xi _0={0}\), as expected. Thus, the local improvement \(\phi _3\), which in fact involves only non-zero modes, is null in the radial case.
With these choices for \(\phi _3\) and \(\xi _0\) we conclude with the following expression of the error associated to the final ansatz \(u_3\):
2.5 Estimate of the inner and outer error
For later purpose, we split \(S[u_3]\) in inner and outer error. At this stage, it is important to treat the terms involving directly \({{\dot{\Lambda }}}\) as part of the outer error, since, as we shall see, a priori those are the terms with less regularity. Let
where we define the inner error
the outer error
and the radius
for some constant \(\delta >0\) which will be chosen in (2.31) to make both the errors \(S_{\text {in}}\eta _R\) and \(S_{\text {in}}(1-\eta _R)+S_{\text {out}}\) suitably small for a final contraction.
Size of \(S_{\text {in}}\eta _{R}\). We proceed with the estimate of \(S_{\text {in}}\eta _{R}\). More precisely, we need the following conditions on \(\delta ,l_0,l_1,k\):
The condition (2.27) is used to get the estimate in the linear outer problem, and it is due to the fact that both the heat kernel \(p_t^\Omega \) and the parameter \(\mu _0(t)\) have an exponential decay for t large. To make the quadratic term \(U^3 {{\tilde{\phi }}}^2\) smaller than \(S_{\text {in}}\) in the inner problem we need the upper bound in (2.28). The lower bound is necessary to get a positive Hölder exponent in the regularity of \({\dot{\Lambda }}\). The last two conditions (2.29)–(2.30) insure that \(S_{\text {in}}\) is controlled by the first term in (2.24). Thus, we fix the following values satisfying (2.27)–(2.28):
Here and in what follows, we write \(a\lesssim b\) if there exists a constant C, independent of \(t_0\), such that \(a\le Cb\). If both the inequalities \(a \lesssim b\) and \(b\lesssim a\) hold we write \(a \sim b\). Using (2.34) and (2.35) we estimate
and, since we are in the region where \(\eta _R \ne 0\), using (2.38), we obtain
Also,
Now, we estimate the last term of \(S_{\text {in}}\eta _R\) using expansion (2.19) and \(\mu /\mu _0=e^{2\Lambda }\) we get
Combining these estimates we obtain
and using the values (2.31) we get
Size of \(S_{\text {out}}\). For the first term in \(S_{\text {out}}\) we have
and using the estimates given by Lemma 2.4 on \(\phi _3,\nabla _y \phi _3\) and \( {\partial } _t \phi _3\) we get
Finally,
We conclude that
Size of \(S_{\text {in}}(1-\eta _R)\). It remains to estimate the size of \(S_{\text {in}}(1-\eta _R)\). We have
Then,
In particular we observe that this is smaller than (2.32), thanks to (2.28). Also,
and
Combining these estimates we find
We conclude that
2.6 Estimates of \(J_1,J_2\) and \(\phi _3\)
The following lemma gives an estimate of \(J_1[{\dot{\Lambda }}](x,t)\) in terms of \({\dot{\Lambda }}\). Observe that
thus, for \(t_0\) large, we will approximate \(J_1\) with \(\mathcal {J}\), that is the solution to
We define the \(L^\infty \)-weighted space
where
Lemma 2.2
(Estimate of \(J_1\)) Suppose \(2 \gamma l_1<\lambda _1-\gamma \) and
Then we have
for \(t\ge t_0\).
Since we have selected \(l_1<1\) in (2.31), condition (1.6) guarantees that \(2\gamma l_1<\lambda _1-\gamma \).
Proof
By parabolic comparison, it is enough to prove the bound for \(\mathcal {J}\) defined as the solution to (2.33). Indeed, we have
We decompose
where \(w_k\) is the k-th eigenfunction of \(-\Delta \) on \(\Omega \). Plugging the decomposition into the equation we find
In particular, we have
Using \(\vert \!\vert {\dot{\Lambda }}\vert \!\vert _{\infty ,l_1}<\infty \) and \(2\gamma l_1<\lambda _1-\gamma \) we obtain
Finally, from standard parabolic estimates, using the \(L^2\)-bound and Eq. (2.33), we get for \(t\ge t_0\)
for any \(\Omega ' \Subset \Omega \). By boundary regularity estimates this inequality can be extended to \(\Omega \) thanks to the smoothness of \( {\partial } \Omega \). \(\square \)
Lemma 2.3
(Estimate of \(J_2\)) Let \(J_2(x,t)\) be the unique solution to the problem
Suppose that \(3\gamma <\lambda _1\). Then, there exists \(t_0\) large such that
for any \(\varepsilon >0\) and for all \((x,t)\in \Omega \times [t_0,\infty )\) where \(y=(x-\xi )/\mu \).
Proof
Firstly, we observe that
Also, by Taylor expanding the function \(\theta _\gamma \) in (2.11) near the origin, we see that
where \(\epsilon >0\) can be taken arbitrarily small. Thus, by parabolic comparison, it is enough to find a supersolution to the problem
Let \(v(x,t){:=} \mu (t)^{-1} u(x,t)\), which satisfies
We look for a supersolution \({{\bar{v}}}\) of the form
We need
with \(v_1\ge 0\) on \( {\partial } \Omega \times [t_0,\infty )\) and \(v_0(y(x,t_0))\ge 0\) for \(x\in \Omega \). Without loss of generality let \(\Omega \subset B_1\). Consider the positive radial solution \(v_0(\vert y\vert ,t)\) to
given by the formula of variation of parameters
From this formula we obtain the following estimates in \((x,t)\in \Omega \times [t_0,\infty )\):
Thus, if \(\vert x-\xi \vert <C_0\), for \(C_0\) sufficiently small, then
Then, let \(v_1\) be the solution to
with
In the right-hand side we have
Since \(3\gamma -{\dot{\Lambda }}(t)<\lambda _1\) provided that \(t_0\) is sufficiently large, the comparison principle applies and we get \(\vert v_0\vert \lesssim \mu ^{1-\varepsilon }\). Thus, we verified inequality (2.36). Also, we have \(v=v_1\ge 0\) on \( {\partial } \Omega \times [t_0,\infty )\) and \(\eta v_0(y(x,t_0))\ge 0\). Thus, v is a supersolution, and going back to the original function \(u=\mu v\) we get estimate (2.35) for \(J_2\). \(\square \)
Lemma 2.4
(Estimate on \(\phi _3\)) Let \(\mathcal {M}[\xi _0,\mu _0]\) be defined as in (2.20). If the orthogonality conditions (2.22) on \(\mathcal {M}[\xi _0,\mu _0]\) hold, then there exists a bounded solution to the problem
We have the following estimates on \(\phi _3\) and its derivatives:
where f is a smooth bounded function.
Proof
From the explicit form of the function \(\mathcal {M}\) given in (2.20) we estimate its size by
and we observe that \(\mathcal {M}\) has only modes \(i=1,2,3\). Thus, we decompose \(\phi _3\) in such modes:
Similarly, we define
The formula of variation of constants gives
where
and
Since
and
we deduce
Also, by the orthogonality conditions (2.22) we have
With these estimates we conclude
Similarly, taking the space and time derivatives of Eq. (2.37), we deduce the bounds on \(\nabla _y \phi _3\) and \( {\partial } _t \phi _3\). \(\square \)
We conclude this section with summarizing the estimates of the error \(S[u_3]\).
Lemma 2.5
Let \(3\gamma <\lambda _1\), \(\mu =\mu _0 e^{2\Lambda }\) and \(\xi =\xi _0+\xi _1\), where \(\mu _0,\xi _0\) are given by (2.5) and (2.23) respectively. Assume
for positive constant \(\delta ,l_0,l_1,k\) satisfying (2.27),(2.28), (2.29) and (2.30). Then, setting \(x=\mu y+\xi \), for \(t_0\) sufficiently large the following estimate on the error function \(S[u_3]\) holds:
where
3 The inner-outer scheme
We recall that our final purpose is to find an unbounded global in time solution u to (2.1) of the form
for a small perturbation \({{\tilde{\phi }}}\). The latter is constructed by means of the inner-outer gluing method. This consists in looking for a perturbation of the form
where
and \(\eta (s)\) is a cut-off function with \({{\,\textrm{supp}\,}}\eta \subset [0,2]\) and \(\eta \equiv 1\) in [0, 1]. We have already chosen \(R=R(t)\) in (2.26). In terms of \({{\tilde{\phi }}}\) the equation reads as
where
Hence the problem for \({{\tilde{\phi }}}\) is
Now, the main idea is to split the problem for \({{\tilde{\phi }}}\) in a system for \((\psi ,\phi )\), localizing the inner regime. We divide the error in
where \(S_{\text {in}}, S_{\text {out}}\) are defined in (2.24) and (2.25) respectively. Considering \({{\tilde{\phi }}}\) as in (3.2) we compute
and
where \(z{:=}y/R\). We split
Hence, the full equation becomes
We divide the full problem in a system. Firstly, we look for a solution \(\psi \) to
Thus, after dividing by \(\mu _0^{1/2}\), \(\psi \) solves the outer problem
Then, \(\phi \) has to solve the problem
Equivalently, multiplying by \(\mu ^{5/2}\), \(\phi \) solves
where \(B_0\) is the linear operator
3.1 General strategy for solving the inner-outer system
We now describe the method we use to solve system (3.4)–(3.5). Firstly, for fixed parameters \(\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }}\) and inner function \(\phi \) in suitable weighted spaces, we solve problem (3.4) in \(\psi =\psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi ]\). This is done in Sect. 4. We insert such \(\psi \) in the inner problem. At this point we need to find \(\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }}\) and \(\phi \). We make the change of variable \(t(\tau )\) defined by the ODE
which explicitly gives
Expressing Eq. (3.5) in the new variables \((y,\tau )\) we get the inner problem
where
Let \(Z_0\) be the positive radially symmetric bounded eigenfunction associated to the only negative eigenvalue \(\lambda _0\) of the problem
It is known that \(\lambda _0\) is simple and
We solve (3.10) with a multiple of \(Z_0(y)\) as initial datum, namely
for some constant \(e_0=e_0[H]\) to be found. Formally, this initial datum (3.12) allows \(\phi \) to remain small along its trajectory. Indeed, multiplying (3.5) by \(Z_0\) and integrating we obtain
where
The general solution p(t) is given by
This shows that in order to get a decaying solution p(t) (and hence \(\phi (y,t)\)), the following initial conditions should hold:
This argument formally suggests that, to avoid the instability caused by \(Z_0\), the small initial value \(\phi (y,t_0)\) needs to be constrained along \(Z_0\).
Another important observation is that, in order to solve the problem (3.10)–(3.12) we need to constrain the right-hand side H to be orthogonal to \(\{Z_i\}_{i=1}^4\). Namely we need
Indeed, the elliptic kernel generated by \(\{Z_i\}_{i=1}^4\) is a subset of the kernel of the parabolic operator
Hence, we expect to have solvability of the inhomogeneous problem (3.10) with suitable space-time decay if the orthogonality conditions (3.13) are satisfied.
As we shall see in Sect. 5, condition (3.13) with index \(i=4\) is equivalent to a nonlocal problem in \(\Lambda \), for fixed \(\phi ,\xi \). Such operator is similar to an half-derivative in the sense of Caputo [6], and we develop an invertibility theory in Sect. 8. In Sect. 5 we solve (3.13) by fixed-point argument and hence we find \(\Lambda ,\xi \). A main ingredient of the full proof is the linear theory for the inner problem developed in [11] and adapted in dimension 3 in [15].
3.1.1 Statement of the linear estimate for the inner problem
We recall the result on the linear theory in dimension 3, proved in [15]. To state the result, we decompose a general function \(h(\cdot ,\tau )\in L^2(B_{2R})\) for any \(\tau \in [\tau _0,\infty )\) in spherical modes. Let \(\{\vartheta _m\}_{m=0}^\infty \) the orthonormal basis of \(L^2(S^2)\) made up of spherical harmonics, namely the eigenfunctions of the problem
where \(0=\lambda _0<\lambda _1=\lambda _2=\lambda _3=2<\lambda _4\le \cdots \). We decompose h into the form
Furthermore, we write \( h=h^0+h^1+h^\perp \) where
We solve the inner problem (3.15) for functions h in the space \(X_{\nu ,2+a}\) defined by
where
Proposition 3.1
Let \(\nu ,a\) be positive constants. Then for all sufficiently large \(R>0\) and any \(h(y,\tau )\) with \(\vert \!\vert h\vert \!\vert _{\nu ,2+a}<\infty \) such that
there exist \(\phi [h]\) and \(e_0[h]\) which solves
They define linear operators of h that satisfy the estimates
and
where
As we said in Sect. 2.3, in order to make the system for \((\phi ,\psi )\) weakly coupled, \(\phi \) needs to be small at distance \(y\sim R\). For this reason, we need to take \(a>1\) in the statement of Proposition 3.1. This makes clear why we need to improve ansatz \(u_1\) to \(u_3\) in Sect. 2. Since in our problem \(h=H\) as in (3.10) decays as
where \(\tau \) is given in (3.7), we apply estimate (3.16) with constants
in the simplified form
and observe that
We look for \(\phi \) in the space of functions
where
for \(\varepsilon >0\) fixed small (as in Sect. 3.1.3).
We notice that, by standard parabolic estimates, from (3.17) we also get the bound on the Hölder seminorms in (3.18), thus
3.1.2 Spaces for the parameters
We introduce weighted Hölder spaces for the parameters \(\Lambda ,\xi \). Let
where
and
We look for \(\Lambda \) such that
for some positive constant \(\varepsilon ,\delta _0,\delta _1,l_0,l_1\) to be chosen (see Sect. 3.1.3). We also define \(X_{\sharp ,c,\sigma }{:=}X_{\sharp ,c,c,\sigma }\) and
We consider \(\xi _1\) such that
for some \(k>0\) (see Sect. 3.1.3). The positive constants \(\mathfrak {b}_{1},\mathfrak {b}_{2}\) will be selected as small as needed.
3.1.3 Choice of constants
Here we select the constants
which are sufficient to find the perturbation \({{\tilde{\phi }}}\) in (3.1) by the inner-outer gluing scheme. Firstly, we indicate where the constants appear in the scheme:
-
\(l_0,l_1,\delta _0,\delta _1,\varepsilon \) appear in the definition (3.20);
-
k is used in the norm (3.21) for \(\xi \);
-
\(\delta \) appears in \(R(t)=\mu ^{-\delta }\), that is the radius of the inner regime;
-
\(\alpha ,\beta \) is used in the norms for the outer problem, see (4.6) and (4.14);
-
\(\sigma >0\) appears in the choice of \(\beta =l_1+\delta +\sigma \) in the outer problem;
-
\(\kappa >0\) is the constant appearing in Proposition 4.1.
We fix the following values:
-
\(\delta =\frac{2}{9}\);
-
\(l_1=k=\frac{2}{3}\);
-
\(l_0=l_1+\frac{\delta }{2}=\frac{7}{9}\);
-
\(\sigma =2\alpha =\varepsilon =\frac{1}{100}\);
-
\(\delta _1=l_1+\delta -\sigma -(1-\delta )(1+\alpha /2)(1+2\varepsilon )\);
-
\(\delta _0 = l_1 + \delta - \sigma - (1-\delta )(1+\alpha /2)2\varepsilon \);
-
\(\beta =\frac{1}{2}+l_1+\delta -\sigma \);
-
\(\kappa = \gamma (\sigma -\alpha \delta )\)
These choices are dictated by the following constraints, based on the estimate of the approximate solution, the linear theory for inner (Proposition 3.1) and outer problem (Lemma 4.1), the characterization of the orthogonality conditions (6.1) and the estimates in Proposition 6.1:
-
\(l_1+\delta <1\) to make \(\beta <3/2\) and apply the outer linear estimate (4.8);
-
we need
$$\begin{aligned}\delta \in \left( \frac{1-l_1}{2}+{\hat{\varepsilon }},\frac{1+l_1}{6}\right) , \quad \text {where} \, {\hat{\varepsilon }}=\frac{(1+\alpha /2)(1+2\varepsilon )-l_1+\sigma }{1+(1+\alpha /2)(1+2\varepsilon )}-\frac{1-l_1}{2}.\end{aligned}$$Up to choosing \(\sigma>\alpha >0\) and \(\varepsilon >0\) small enough, these range is equivalent to (2.27), which, together with the previous restriction, impose a range for \(\delta \) and \(l_1\) leading (for instance) to the choice (2.31);
-
\(l_0\ge l_1\) and \(k+1\ge 2\delta +l_1\) to get \(\mu ^{5/2}S_{\text {in}}\) controlled by the term \(\mu (t) 5U(y)^4 J(x,t)\);
-
\(\sigma>\alpha \delta >0\), \(\varepsilon >0\) and \(\kappa \in (0,2\gamma (\sigma -\alpha \delta ))\). This allows to estimate the \(R^{\alpha }\log R\lesssim e^{-\kappa t}\mu ^{-\sigma }\) when we need to control the term \(\mu ^{-1} \phi \Delta _x \eta _R\) in the outer error;
-
\(k=l_1\). From (5.6) we need \(\vert \xi _1\vert +\vert *\vert {{\dot{\xi }}_1} \lesssim \mu ^{1+l_1}\), thus the choice of k, which is consistent with (2.30);
-
in the outer problem we obtain \(\vert \psi (x,t)\vert \lesssim \frac{\mu ^{l_1-\sigma }R^{-1}}{1+\vert y\vert ^\alpha }\). The nonlocal equation (5.2) and the estimate (2.29) asks for \(\vert \Lambda (t)\vert \lesssim \vert \psi (\xi (t),t)\vert \). Thus, this leads to the a choice of \(l_0\in [l_1,l_1+\delta -\sigma ]\);
-
from estimate (6.3), Eq. (5.2) and the bound on the \(\varepsilon \)-Hölder seminorm of \(\psi \) we get
$$\begin{aligned}{}[\Lambda ]_{0,\frac{1}{2}+\varepsilon ,[t,t+1]}\lesssim [\psi (\xi (\cdot ),\cdot )]_{0,\varepsilon ,[t,t+1]}\lesssim \frac{\mu ^{l_1+\delta -\sigma }}{(\mu R)^{(1+\frac{\alpha }{2})2\varepsilon }}= \mu ^{\delta _0}, \end{aligned}$$which gives \(\delta _0\);
-
similarly, from (4.16) the Hölder estimate on the outer solution gives
$$\begin{aligned}_{0,\frac{1}{2}+\varepsilon ,[t,t+1]}\lesssim \mu ^{l_1+\delta -\sigma }(\mu R)^{-(1+\frac{\alpha }{2})(1+2\varepsilon )},\end{aligned}$$and by Eq. (5.2) and estimate (6.5) we need
$$\begin{aligned}_{0,\varepsilon ,[t,t+1]}\lesssim [\psi (0,\cdot )]_{0,\frac{1}{2}+\varepsilon ,[t,t+1]}.\end{aligned}$$This leads to the choice of \(\delta _1\);
-
after choosing \(\sigma =2\alpha >0\) small so that \(\delta >\sigma \), the constant \(\varepsilon \) is chosen small enough to make \(\delta _1\) positive (any choice of \(\alpha \in (0,\delta /2)\) and \(\varepsilon \) such that \(\delta _1=\delta _1(\alpha ,\varepsilon )>0\) is sufficient).
4 Solving the outer problem
We devote this section to solve the outer problem (3.4)
where \(\psi _0(x)\) is any suitable small initial condition,
and potential
which, by the definition of \(u_3\), using again the bounds on \(H_\gamma ,J,\phi _3\) and the support of \((1-\eta _R)\), satisfies
Let
Then, the problem for \(\psi _1\) becomes
where
In particular, in the proof of Proposition 4.1 we prove that for any \(\alpha >0\)
Also, using the definition of \(u_3\), in (4.17) we prove
Firstly, we consider the linear version of (4.3). Let
for some \(\beta>0,\alpha >0\), where \(\vert \!\vert F\vert \!\vert _{\beta -2,\alpha +2}\) is the best constant for such inequality. Also, for \(\delta \in (0,1/2)\) and \(\sigma \in (0,1)\) we define the Hölder norms
Lemma 4.1
Let F such that \(\vert \!\vert F\vert \!\vert _{\beta -2,\alpha +2}<\infty \) for some constants \(\beta <3/2 \) and \(\alpha \in (0,1)\). Furthermore, assume that \(\vert \!\vert e^{as}g(s)\vert \!\vert _{L^{\infty }( {\partial } \Omega \times (t_0,\infty ))}<\infty \) for some \(a>0\) and \(\vert \!\vert h\vert \!\vert _{L^\infty (\Omega )}<\infty \). Let \(\psi _1[F,g,h]\) be the unique solution to
Then, for \(b \in (0,\lambda _1)\) and \({\tilde{a}} \in (0,\min \{a,\lambda _1-\varepsilon \}]\) for \(\varepsilon >0\) arbitrary small, we have
for all \(x=\mu y+\xi \in \Omega \) and \(t>t_0\). Furthermore, the following local estimate on the gradient holds:
where \(R\le \delta \mu ^{-1}\) for sufficiently small \(\delta >0\). Also, one has
Proof
To prove the result is enough to find a supersolution to the problem
We use the notation \(\psi _2=\psi _2[F,g,h]\). Indeed, suppose that \(\bar{\psi }_2\) is a supersolution to this problem. By (4.2) we have
and hence \(\vert \!\vert V {\bar{\psi }}_2\vert \!\vert _{\beta -2,2+\alpha }<R(t_0)^{-2}\) for \(t_0\) sufficiently large. Thus, we find that a large multiple of \({\bar{\psi }}_2\) is a supersolution of (4.7). Firstly, let \(F,g\equiv 0\) and consider \(\psi _2[0,0,h]\). Let \(v_0(x)\) be the solution to
for \(b \in (0,\lambda _1)\) and define
We claim that \({\bar{\psi }}_2\) is a supersolution for \(\psi _2[0,0,h]\). Indeed, we have
where the last inequality is a consequence of the maximum principle applied to \(v_0\). Secondly, we look for a supersolution to \(\psi _2[0,g,0]\). Let \(v_1(x)\) to be the solution to
where \({\tilde{a}} \in (0,\min \{a,\lambda _1-\varepsilon \}] \) and consider
We verify that
where we used \({\tilde{a}}\le a\) to get the second inequality and \({\tilde{a}}<\lambda _1\) to get the third one by the maximum principle. It remains to find a supersolution for \(\psi _2[F,0,0]\). Let \(\psi _2[F,0,0]=e^{-c(t-t_0)}\psi _3\), where \(c=2\gamma \beta \) so that
We find a bounded \({\bar{\psi }}_3\) supersolution in case \(c<\lambda _1\), that is \(3\gamma <\lambda _1\). Consider
We need
with \({\bar{\psi }}_3(x,t)\ge 0\) on \( {\partial } {\Omega \times [t_0,\infty )}\) and initial datum \(\bar{\psi }_3(x,t_0)\ge 0\). Suppose without loss of generality that \(\Omega \subset B_1\) and take \(\uppsi _0\) as the solution to
From the variation of parameters formula
we find
and
Also, if \(\vert x-\xi \vert <d\) for d fixed sufficiently small, we obtain
Now, we take \(\uppsi _1\) as the solution to
We estimate the right-hand side by
Hence, by comparison principle using \(c<\lambda _1\) we obtain a solution \(\vert *\vert {{\bar{\psi }}_3}\lesssim \mu ^{\alpha }\). Thus, inequality (4.11) is satisfied. Also, \({\bar{\psi }}_3=0\) on \( {\partial } {\Omega \times [t_0,\infty )}\) and \(\psi _3(x,t_0)=\eta \uppsi _0(x,t_0)\ge 0\). We conclude that \({\bar{\psi }}_3\) is a supersolution and the bound (4.8) is proven. Now, we prove the gradient estimate (4.9). Let
and \({\dot{\tau }}(t)=(R(t)\mu (t))^{-2}\), that gives \(\tau (t)\sim \mu ^{-2}\). We can take \(\tau (t_0)=2\). The equation for \({{\tilde{\psi }}}\) becomes
where \({\tilde{F}}(z,\tau (t))=(R\mu )^2 F(\mu R z+\xi , \tau (t))\), and the coefficients
are uniformly bounded. Since \(\vert \!\vert F\vert \!\vert _{\beta -2,\alpha +2}<\infty \), we have
We have already proved the \(L^\infty \)-bound
We apply standard local parabolic estimates for the gradient: let \(\sigma \in (0,1)\) and \(\tau _1\ge \tau (t_0)+2\), then
In the original variables, for any \(t\ge t_0+2\) we find
By similar parabolic estimates using \(\vert \!\vert \nabla _x\psi _0\vert \!\vert _\infty <\infty \) we can extend estimate (4.13) up to \(t=t_0\), thus, the proof of (4.9) is complete. Now, we prove estimate (4.10). We consider the Hölder seminorms. We perform the change of variable
where \(z :=(x-\xi )/({R\mu _0})^{1+\frac{\alpha }{2}}\) and \(\tau \) satisfies
that is
The equation for \({\hat{\psi }}\) is
where
Then, applying local parabolic estimates on \({{\tilde{\psi }}}\), we get
where \(\tau _i=\tau (t_i)\) and \(z=z(x,t_i)\) for \(i=1,2\). Similarly, using the Hölder coefficient \((2\varepsilon ,\varepsilon )\), we get
\(\square \)
We introduce the following weighted norms for \(\psi \):
where
and define the space of functions
Now, we are ready to solve the outer problem (3.4) for \(\phi \) such that
for parameters satisfying (3.20) and (3.21).
Proposition 4.1
Assume that \(\Lambda ,\xi _1,\phi \) satisfy (3.20), (3.21) and (4.15) respectively. Also, suppose \(\psi _0 \in C^2(\bar{\Omega })\) such that
for some \(\kappa \in (0,2\gamma (\sigma -\alpha \delta ))\). Then, there exists \(t_0\) large so that problem (3.4) has a unique solution \(\psi =\Psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi ]\) and given \(\alpha >0\), there exists \(C_{**}\) such that
where \(C_{**}=C_{**}({{\textbf {b}}},\mathfrak {b}_1,\mathfrak {b}_2)\) and \({{\textbf {b}}},\mathfrak {b}_1,\mathfrak {b}_2\) are the constants in (4.15), (3.20) and (3.21) respectively.
Proof
Let \(T_1\) the linear operator, defined by Lemma 4.1, such that, given \(\beta <3/2,\alpha >0\) and functions f, g, h with bounded norms \(\vert \!\vert f\vert \!\vert _{\beta -2,\alpha +2},\vert \!\vert e^{as}g\vert \!\vert _{\infty },\vert \!\vert h\vert \!\vert _\infty \) respectively, T[f, g, h] is the solution to (4.7).
Let
where we define \(\psi _B{:=}T(0,-u_3,\mu _0(t_0)^{1/2}\psi _0)\). From the definition of \(u_3(x,t)\) we expand for \(x\in {\partial } \Omega \) and \(t_0\) large, to get
for a smooth bounded function \(f_B(x,t)\) on \( {\partial } \Omega \times [t_0,\infty )\). Hence, Lemma 4.1 gives the bound
for any \(b<\lambda _1\) and \(a<\min \{5\gamma ,\lambda _1-\varepsilon \}\) for any \(\varepsilon >0\).
Now, we apply the fixed point theorem to find \(\psi _A\) such that \(\psi \) satisfies (4.16). We obtain a solution \(\psi \) if \(\psi _A\) satisfies
where \(F(x,t)=\mu _0(t)^{1/2}f(x,t)\) and f is given by (4.1). We look for \(\psi _A\) in
where M is a fixed large constant, independent of t and \(t_0\). We prove that \(\mathcal {A}(\psi _A) \in \mathcal {B}\) for any \(\psi _A \in \mathcal {B}\). Firstly, we estimate the \(L^\infty \) norm of \(F(\psi _A)\). From (4.14) we apply Lemma 4.1 with \(\beta =1/2+l_1+\delta <3/2\). We recall that \(F=\mu _0^{1/2}f\) where
We have \(\eta ''(y/R)\ne 0\) and \(\eta '(y/R)\ne 0\) only if \(\vert y\vert \sim R\), hence we estimate
Using the bound on the gradient given in the definition of \(\vert \!\vert \phi \vert \!\vert _{*}\) we obtain
Similarly, also using the bounds on \({\dot{\Lambda }},{\dot{\xi }}\) we have
Also, since \(\delta <1/3\), we have
Furthermore, using Lemma 2.5 we estimate
and
Finally, since \(\vert \!\vert \mu _0^{-1/2}\psi _A\vert \!\vert _{**}\) is bounded we get
Summing up these estimates we conclude that
Hence, we have
and Lemma 4.1 gives
Since \(F\in L^\infty (\Omega \times [t_0,\infty ))\), classic parabolic estimates give \(\psi \in C^{1+{\hat{\sigma }},\frac{1+\hat{\sigma }}{2}}(\Omega \times [t_0,\infty ))\) for any \({\hat{\sigma }}<1\) and from Lemma 4.1 we get
for sufficiently large M (independent of \(t_0\)). This proves \(\mathcal {A}(\psi _A)\in \mathcal {B}\). Now, we claim that the map \(\mathcal {A}(\psi )\) is a contraction, that is: there exists \({{\textbf {c}}}<1\) such that, for any \(\psi _A^{(1)},\psi _A^{(2)} \in \mathcal {B}\),
Since \(\psi \) appears in \(F(\psi )\) only in the nonlinear term \(\mathcal {N}\), we get
From definition (3.3) we write
We estimate
and
Finally, using \(\beta =1/2+l_1+\delta -\sigma <3/2\) we apply \(T[\cdot ,0,0]\) to \(F(\psi )\) we obtain
Arguing as in (4.20), from (4.21) and standard parabolic estimates we obtain
with \({{\textbf {c}}}<1\) if \(t_0\) is taken sufficiently large. Applying the Banach fixed point theorem we get existence and uniqueness of \(\psi _A\) and hence of \(\psi =\mu _0^{-1/2}(\psi _A+\psi _B)\) with estimate (4.16) that is a consequence of estimates (4.8)–(4.10). \(\square \)
Remark 6.1
(Continuity with respect to the initial condition \(\psi _0\)) Given an initial datum \(\psi _0\) Proposition 4.1 defines a solution \(\psi =\Psi [\psi _0]\) to (3.4), from a small neighborhood of 0 in the \(L^\infty (\Omega )\) space with the \(C^1\)-norm \(\vert \!\vert \psi \vert \!\vert _\infty +\vert \!\vert \nabla \psi _0\vert \!\vert _\infty \) into the Banach space \(L^\infty \) with norm \(\vert \!\vert \psi \vert \!\vert _{**}\) defined in (4.14). In fact, from the proof of Proposition 4.1 and the implicit function theorem, \(\psi _0 \mapsto \Psi [\psi _0]\) is a diffeomorphism and hence
for some positive constant c.
The function \(\psi =\Psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi ]\) depends continuously on the parameters \(\Lambda ,\dot{\Lambda },\xi ,{\dot{\xi }},\phi \). To see this we argue similarly to [15, Proposition 4.3]. For example, fix \(\dot{\Lambda },\xi ,{\dot{\xi }},\phi \) and consider
for \(\Lambda _1,\Lambda _2\) satisfying (3.20). Then \({\bar{\psi }}\) solves
One can easily check each term in F and obtain
with \({{\textbf {c}}}<1\) if \(t_0\) is large enough. Also, using (4.2) we find that
Then, arguing as in the proof of (4.8), a multiple of \(\vert \!\vert \Lambda _1-\Lambda _2\vert \!\vert _{\sharp ,l_0,\delta _0,\frac{1}{2}+\varepsilon } \uppsi \), where \(\uppsi \) is the supersolution constructed in Lemma 4.1, is a supersolution for \({\bar{\psi }}\). Similarly, one obtain analogue estimates fixing \(\xi ,{\dot{\Lambda }}, {\dot{\xi }}\). Let us consider all the parameters fixed. We define \(\bar{\psi }:=\psi [\Lambda ,{\dot{\Lambda }},\xi ,\dot{\xi },\phi _1]-\psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi _2]\), which satisfies the equation
For instance, we estimate
with \({{\textbf {c}}}<1\) when \(t_0\) is fixed large enough, and arguing as in (4.18)-(4.19), we obtain similar estimate on the other terms of \(F[\phi _1]-F[\phi _2]\). Having the \(L^\infty \)-bound, the estimate for the gradient and the Hölder norms of \({\bar{\psi }}\) follow as in the proof of Lemma 4.1. We summarize the continuity of \(\psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi ]\) with respect to the parameters in the following Proposition.
Proposition 4.2
Under the same assumption of Proposition 4.1, the function \(\psi =\Psi [\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi ]\) is continuous with respect to the parameters \(\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\phi \). Moreover the following estimate holds:
where \({{\textbf {c}}}<1\) provided that \(t_0\) is sufficiently large and the constants \(\mathfrak {b}_1,\mathfrak {b}_2\) in (3.20), (3.21) are sufficiently small.
5 Characterization of the orthogonality conditions (3.10)
Given the function \(\psi =\Psi [\Lambda ,{\dot{\Lambda }},\xi ,\dot{\xi },\phi ]\) provided by Proposition 4.1, we plug it in the inner problem for \(\phi \). From the linear theory stated in Proposition 3.1, the inner problem (3.10) with initial datum (3.15) can be solved if the orthogonality conditions
are satisfied. The aim of this section is to characterize this set of conditions as an nonlocal system in \(\Lambda ,\xi \) for fixed \(\phi \in X_*\). The next lemma shows that the orthogonality condition with index \(i=4\) is equivalent to a nonlocal equation in the variable \( \Lambda \), for fixed \(\phi ,\xi \).
Lemma 5.1
Assume that \(\Lambda ,\xi ,\phi \) satisfy (3.20), (3.21) and (3.18) respectively. Let \(\psi =\Psi [\Lambda ,\dot{\Lambda },\xi ,{\dot{\xi }},\phi ]\) be the solution to problem (3.4) given by Proposition 4.1. Then, the condition (5.1) with index \(i=4\) is equivalent to
where \(\mathcal {J}\) is the solution to
The function a is smooth, decays as \(t\rightarrow 0\) and \(a[0,0]\equiv 0\). Then, for \(\kappa \in (0,2\gamma (\sigma -\alpha \delta ))\), the following estimates on g and G hold:
and
Furthermore, we have
with constant \({{\textbf {c}}}<1\) provided that \(t_0\) is fixed sufficiently large and \(\mathfrak {b}_i\) small for \(i=1,2\).
Proof
We recall that
Hence, (3.13) with index \(i=4\) becomes
We follow the analogue [15, Lemma 5.1] to estimate the terms \(i_j(t)\). Firstly, we analyze \(i_1\). We have
where we used that the integral of \(Z_4(y)U(y)^4 {\partial } _{y_i} U(y)\) on \(B_{2R}(0)\) is null by symmetry for \(i=1,2,3\). Also,
The main term in the left-hand side of (5.2) is given by
where \(c_1(1+O(R^{-2}))=\int _{B_{2R}} 5U^4 Z_4 \,dy\). To analyze the terms \(a_{12}\), we decompose \(w[{\dot{\Lambda }}](x,t)=J[\dot{\Lambda }](x,t)-\mathcal {J}[{\dot{\Lambda }}](x,t)\) as a sum of a solution in \({{\mathbb {R}}}^3\) and a smooth one in \(\Omega \) with more decay. Then, using the Duhamel’s formula in \({{\mathbb {R}}}^3\) as in [15, Proof of (7.5)] we deduce
We analyze \(a_{13}\) by splitting J as a sum of a solution to the same equation in \({{\mathbb {R}}}^3\) and smooth remainder in \(\Omega \) with more decay, and, proceeding as in [15, Proof of (5.10)], again by Duhamel’s formula in \({{\mathbb {R}}}^3\) we obtain
for some \(\sigma \in (0,1)\) and bounded smooth function \(\theta \), and \(\Pi [{\dot{\Lambda }},\xi ]\) satisfying the estimate above for \(a_{12}\). After integration, \(a_{13}[{\dot{\Lambda }},\xi ](t)\) satisfies (5.5). Taking into account the behavior of \(J_1,J_2\) and \(\phi _3\) given in (2.34), (2.35) and (2.38) respectively, we have
for some constant \(s\in (0,1)\) and bounded smooth function \(Q[\Lambda ,\xi ](y,t)\) satisfying (5.5).
Finally, Taylor expanding \(h_\gamma (x,\xi )\) at \(x=\xi \), we get
for some \(y^* \in \overline{[0,y]}\) and a smooth bounded function \(\Pi _2(t)\). The term \(\mu ^{-1/2}\mu _0^{-1/2}i_1(t)=\sum _{i=1}^{3}a_i(t)\) gives the left-hand side in (5.2). Now, we look at \(i_2\). We decompose
The term
is independent of parameters and, as a consequence of Proposition 4.1, satisfies the estimate
Applying the mean value theorem to \(\psi \) and using the gradient estimate we deduce the same bound for \(b_2\). This gives the main term \(b_1(t)+b_2(t)=g(t)\) in the right-hand side of (5.2). We analyze \(b_3(t)\). By Proposition 4.2 applied to
we obtain
Also, again as a consequence of the Lipschitz estimates in \(\psi \) we have for example
for any \({\dot{\Lambda }}_1,{\dot{\Lambda }}_2 \in X_{\sharp ,l_1,\delta _1,\varepsilon }\) and fixed \(\Lambda ,\xi ,\dot{\Lambda },{\dot{\xi }}\) in the respective spaces. We analyze \(i_3\). We recall that
which is linear in \(\phi ,\psi \) and satisfies
It follows that
Then, the Hölder bounds on \(\psi \) and \(\phi \) in the respective norms give estimate (5.3) for \(i_3\), and using Proposition 4.2 we also get the Lipschitz property (5.3) for \(i_3\). Finally, we have
and (5.3)–(5.3) for \(i_4\) follows arguing as for \(i_3\). Summing up the estimates we obtain \(G[\Lambda ,\dot{\Lambda },\xi ,{\dot{\xi }},\phi ](t)=b_3+i_3+i_4\) as in (5.2) with the properties (5.3) and (5.4). \(\square \)
Now, we characterize the conditions
This characterization is given in the following lemma, whose proof, similar to the one of Lemma 5.1, is omitted.
Lemma 5.2
The relation (5.1) for \(i=1,2,3\) is equivalent to
for smooth bounded function \(\Theta \) which satisfies
Furthermore, we have
with constant \({{\textbf {c}}}<1\) provided that \(t_0\) is fixed sufficiently large and \(\mathfrak {b}_i\) small for \(i=1,2\).
6 Choice of parameters \(\Lambda ,\xi \)
In the previous section we have proved that if \(\phi \in X_*\) and \(\Lambda ,\xi \) satisfy (3.20) and (3.21) then the system of orthogonality conditions
is equivalent to the nonlocal system in \([t_0,\infty )\)
with g, G, a as in Lemma 5.1 and \(\Theta _i\) as in Lemma 5.2. Next, we verify that this system is solvable for \(\Lambda ,\xi \) satisfying (3.20),(3.21) respectively. This relies on the following proposition, proved in Sect. 8, about the solvability of the nonlocal operator \(\mathcal {J}[{\dot{\Lambda }}](0,t)=g(t)\) for g as in (5.2).
Proposition 6.1
Let \(h:[t_0,\infty )\rightarrow {{\mathbb {R}}}\) a function satisfying \(\vert \!\vert h\vert \!\vert _{\sharp ,c_1,c_2,\varepsilon }<\infty \) for some constants \(\varepsilon >0\) and \(c_1,c_2\) such that
Then there exists a function \(\Lambda \in C^{\frac{1}{2}+\varepsilon }(t_0-1,\infty )\) satisfying
where \(\mathcal {J}[{\dot{\Lambda }}]\) satisfies (2.33), and there exists a constant \(C_1\) such that
Moreover, if \(\vert \!\vert h\vert \!\vert _{\sharp ,c_1,c_2,\frac{1}{2}+\varepsilon } <\infty \) then \(\Lambda \in C^{1,\varepsilon }(t_0-1,\infty )\) and there exists a constant \(C_2\) such that
Thus, the linear operators
and
are well-defined and continuous.
We are ready to solve the system (6.1) in \(\Lambda ,\xi \) for fixed \(\phi \in X_*\).
Proposition 6.2
Suppose that \(\phi \) satisfies (4.15). Then, there exist \(\Lambda =\Lambda [\phi ](t)\) and \(\xi =\xi [\phi ](t)\) to the nonlinear nonlocal system (6.1) which satisfy (3.20) and (3.21) respectively. Moreover, they satisfy
with constant \({{\textbf {c}}}<1\) provided that \(t_0\) is fixed sufficiently large and \(\mathfrak {b}_i\) small for \(i=1,2\).
Proof
Firstly, we observe that Eq. (5.2) can be rewritten as
where
for new functions \(g_1,G_1\) satisfying the same properties of g, G in Lemma 5.1. By Proposition 6.1 we reduce the equation for \( \Lambda \) to a fixed point problem
where \({\hat{T}}_1\) is defined in (6.7). Let
and define the operator \(\mathcal {L}_1[h] :={\hat{T}}_1 [h-g_1]\). We use the notation
for any \(h\in X_{\sharp ,l_1,\delta _1,\frac{1}{2}+\varepsilon }\). Observe that
Given \(h_j\in X_{\sharp ,1+l_1,\frac{1}{2}+\varepsilon }\) we consider the solution to the ODE
given explicitly by
In particular, we have
We define the vector
where \(h=(h_1,h_2,h_3)\) satisfies \(\vert \!\vert h_i\vert \!\vert _{\sharp ,1+l_1,\frac{1}{2}+\varepsilon }<\infty \) for \(i=1,2,3\). We define
Let \(\mathcal {L}_2\) the linear operator defined as \(\mathcal {L}_2[h]=\Xi \) by relation (6.10) for \(i=1,2,3\). We observe that \((\dot{\Lambda },{\dot{\xi }})\) is a solution to (6.1) if \(( \uplambda ,\Xi )\) satisfies
where \(\mathcal {A}\) is the operator
and
with \(G_1\) and \(\Theta \) defined in (6.9) and (5.6). We show that there exists a unique fixed point \(\left( \uplambda ,\Xi \right) =\left( \uplambda [\phi ],\Xi [\phi ]\right) \) in
for some L fixed large. Indeed, estimates (6.5) and (5.3) give
Also, from (5.7)
We have to verify that \(\mathcal {A}\) is a contraction. For instance, we have
where \(C_2,{{\textbf {c}}}\) is the constant appearing in (6.5) and (5.4) respectively. Since \({{\textbf {c}}}\) can be as small as required provided that \(t_0\) is fixed sufficiently large, we obtain that \(\mathcal {A}_1\) is a contraction map. By means of the Lipschitz property of \({\hat{\Theta }}\) in (5.8) we can estimate \(\mathcal {A}_2[\uplambda _1,\Xi _1]-\mathcal {A}_2[\uplambda _2,\Xi ]\) similarly. Finally, using the estimates on \({\hat{G}},{\hat{\Theta }}\) with respect to \(\Xi \), we get
As a consequence of the Banach fixed point theorem, provided that L and \(t_0\) are fixed large, the map \(\mathcal {A}\) has a unique fixed point \((\uplambda ,\Xi )\) in the space \(\mathcal {B}\). Observe that
where \(T_1\) is defined in (6.6), satisfies (3.20) thanks to (6.4). Also, the components of vector \(\xi _1=\int _{t}^\infty \Xi (s)\,ds\) satisfy (3.21). This proves the existence of a solution \((\Lambda ,\xi )\) to the system (6.1) satisfying (3.20)–(3.21). With similar estimates on \(\uplambda [\phi _1]-\uplambda [\phi _2]\) and \(\Xi [\phi _1]-\Xi [\phi _2]\), using (5.4) and (5.8), relations (6.8) follow. \(\square \)
We observe from the proof that \({\hat{T}}_1\), like an half-fractional derivative, loses 1/2-Hölder exponent but we regain it through \(g,G_1\) as a consequence of estimates on \(\psi \) from Proposition 4.1. This is the main reason to put all the terms of \(S[u_3]\) involving directly \({\dot{\mu }}\) in the outer error (2.25). Indeed, to get \({\dot{\Lambda }}\in C^{\varepsilon }\) it is crucial to allow H in (3.11) (and hence \(S_{\text {in}}\) in (2.24)) to depend on \({\dot{\Lambda }}\) only indirectly through \(\psi [\dot{\Lambda }]\) or \(J_1[{\dot{\Lambda }}]\).
Remark 7.1
By remark 4.1 the outer solution \(\psi =\Psi [\psi _0]\) is smooth as a function of the initial datum \(\psi _0\), provided that \(\vert \!\vert \psi _0\vert \!\vert _\infty +\vert \!\vert \nabla \psi _0\vert \!\vert _\infty \) is sufficiently small. Thus, also the parameters \(\Lambda [\psi _0],\xi [\psi _0]\) found in Proposition 6.2 depend smoothly on \(\psi _0\), and from the proof we also obtain
7 Final argument: solving the inner problem
This section provides the final step in the proof of Theorem 1.1. At this point, given \(\phi \) satisfying (4.15), we have a solution \(\psi =\Psi [\Lambda [\phi ],\xi [\phi ],\phi ]\) to the outer problem (3.4) and parameters \(\Lambda [\phi ],\xi [\phi ]\) such that the orthogonality conditions (5.1) are satisfied. Thus, to get a solution
we need to prove the existence of \(\phi \) such that \(\vert \!\vert \phi \vert \!\vert _*<\infty \).
Proof of Theorem 1.1
We make a fixed point argument using the linear estimate (3.17). Proposition 3.1 defines a linear operator \(\mathcal {T}:h \mapsto (\phi [h],e[h])\) which is continuous between the \(L^\infty \)-weighted space described in (3.17). Thus, the solution \(\phi \) to the nonlinear inner problem satisfies
We claim that \(\mathcal {A}_\text {in}\) has a unique fixed point in the space
for some fixed constant \({{\textbf {b}}}\) large. Firstly, we prove
We recall that
Using the estimate on \(\psi \) given in Proposition 4.1, we have
and from (3.6) we get
Recalling the estimates on \(\phi \) at \(y\sim 0\) and \(y\sim R\) given by the norm (3.18), using that \(R=\mu ^{- \delta }\) with \(\delta \) satisfying (2.28) we deduce
By Lemma 2.5 we have the main error
Thanks to the previous section, H satisfies the orthogonality conditions required by Proposition 3.1. Thus, provided that \(t_0\) is large enough, we have
for \({{\textbf {b}}}\) chosen large, where C is the constant in (3.19). This proves \(\mathcal {A}_{\text {in}}(\phi )\in \mathcal {B}\). Now, we need to prove that for \(\phi ^{(1)},\phi ^{(2)} \in \mathcal {B}\) we have
for some \({{\textbf {c}}}<1\). This is a consequence of Proposition 4.2 and Proposition 6.2. Indeed, for instance we get
and similarly we get the same control on the other terms of \(H[\phi ^{(1)}]-H[\phi ^{(2)}]\). Finally, since the operator \(\mathcal {T}:X_{\nu ,4}\rightarrow X_{*}\) is continuous, where \(X_{\nu ,4}\) is defined in (3.14) for \(a=2\), by composition with \(H:X_* \rightarrow X_{\nu ,4}\) we obtain
provided that \(t_0\) is fixed sufficiently large. Thus, \(\mathcal {A}_\text {in}:\mathcal {B}\rightarrow \mathcal {B}\) is a contraction map and by Banach fixed point theorem we obtain the existence and uniqueness of \(\phi \in X_*\) such that (7.1) holds. Finally, we recall that the constant \(e_0=e_0[H]\) in the initial condition \(\phi (y,t_0)=e_0Z_0(y)\) is a linear operator of H. The existence of \(\phi \) immediately defines \(e_0\). This completes the proof of the existence of \(u=u_3 + {{\tilde{\phi }}}\) in Theorem 1.1, with the bubbling profile centered in \(x=0\in \Omega \) and parameters satisfying (1.8). \(\square \)
Remark 8.1
(Continuity of \((\phi ,e_0)\) with respect to \(\psi _0\)) We found the inner perturbation \(\phi \) and its initial datum \(\phi (y,t_0) = e_0Z_0(y)\) based on the existence of the outer solution \(\Psi [\phi ]\) given by Proposition 4.1, which in fact can be found for any initial condition \(\psi _0\in C^1({\bar{\Omega }})\). Furthermore, as a consequence of the continuity of \(\Psi [\psi _0]\) and \(\Lambda [\psi _0],\xi [\psi _0]\) found in Remarks 4.1 and 6.1 we obtain
Since we know that \(\Lambda ,{\dot{\Lambda }},\xi ,{\dot{\xi }},\psi \) depends smoothly on \(\psi _0\), by the implicit function theorem, we deduced that map \(\psi _0\mapsto (\phi [\psi _0],e_0[\psi _0])\) is \(C^1\) with respect to \(\psi _0\in C^1({\bar{\Omega }})\). This allows to prove the 1-codimensional stability of Corollary 1.1, under small perturbation. With these ingredients, we can proceed as in [11, Proof of Corollary 1.1].
8 Invertibility theory for the nonlocal linear problem
In this section we prove Proposition 6.1. We deduce the result by Laplace transform method combined with asymptotic estimates of the heat kernel \(p_t^{\Omega }\) associated to \(\Omega \). It turns out that the operator \(\mathcal {J}[{\dot{\Lambda }}]\) is similar to a half-fractional integral of \({\dot{\Lambda }}\). Thus, roughly speaking, we expect the inverse operator to behave as a fractional derivative of order 1/2. In fact, Proposition 6.1 can be seen as a precise statement of this idea.
For later purpose we recall some facts about the Dirichlet heat kernel. For the definition and properties we follow [22, 32]. A function \(p_t^{\Omega }(x,y)\) continuous on \(\bar{\Omega }\times {\bar{\Omega }} \times {{\mathbb {R}}}^+\), \(C^2\) in x and \(C^1\) in t is called Dirichlet heat kernel for the problem
if, for any \(y \in \Omega \), satisfies
and
uniformly for every function \(u_0 \in C_0({\bar{\Omega }})\). The existence of the Dirichlet heat kernel is a classical result by Levi [43]. It has the following basic properties:
-
\(p_t^{\Omega }(x,y)\ge 0\), \(p_t^{\Omega }(x,y)=p_t^{\Omega }(y,x)\) and \(p_t^{\Omega }(x,y)=0\) if \(x\in {\partial } \Omega \);
-
for any \(y\in \Omega \) the function \(p_t^{\Omega }(x,y)\in C^{\infty }({{\mathbb {R}}}^+\times \Omega )\);
-
it satisfies \( {\partial } _t p_t^{\Omega }(x,y)=\Delta _x p_t^{\Omega }(x,y)\) for \((x,y,t)\in \Omega \times \Omega \times {{\mathbb {R}}}^+\).
Also, from [32, Theorem 10.13] and its proof, the heat kernel \(p_t^{\Omega }(x,y)\) admits the expansion
where \(\lambda _k\) is the k-th Dirichlet eigenvalue of \(-\Delta \) on \(\Omega \) and \(\phi _k\) the corresponding eigenfunction and also for \(n\ge 1\) (see [32, Remark 10.15])
The series (8.1) converges absolutely and uniformly in \([\varepsilon ,\infty ]\times \Omega \times \Omega \) for any \(\varepsilon >0\), as well as in the topology of \(C^\infty ({{\mathbb {R}}}^+\times \Omega \times \Omega )\).
Before starting the proof of Proposition 6.1, we recall an estimate on the short time behavior of the heat kernel \(p_\tau ^\Omega (x,y)\) due to Varadhan [55, Theorem 4.9]. We will use it in the following form as in Hsu [37, Corollary 1.6].
Lemma 8.1
(Short time estimate of \(p_\tau ^{\Omega }\)) Let \(\varepsilon >0\) fixed such that \(B_\varepsilon (0)\subset \Omega \). Then, there exists \(\tau _0>0\) such that, for \(y\in B_{\varepsilon }(0)\) and \(\tau \in (0,\tau _0)\) we have
where \(\delta <\delta _0\) is independent of y and
Proof
Recall the identities in [55, p. 675]
where
From (8.4) for \(\tau \in (0,\tau _0)\) we have
for all \(x,y \in \Omega \), where \(0\le c(\tau _0)=o(1)\) as \(\tau _0\rightarrow 0\). In particular, fix \(x=0\) and consider \(y \in B_{\varepsilon }(0)\subset \Omega \) for a small \(\varepsilon >0\). Then, choosing \(\tau _0\) smaller if needed, we have
Thus for \(y \in B_\varepsilon (0)\)
and (8.5) says
Thus, we have for \(\tau <\tau _0\) small and \(y \in B_\varepsilon (0)\)
for any \(\delta <\delta _0\) independent of y. \(\square \)
We mention that the uniform bound (8.6) holds for y ranging in any convex subset of the domain, see [37, p.374–375]. Also, for any \(\tau >0\) and \(x,y \in \Omega \) we have the upper bound
as a consequence of the maximum principle. Thus, Varadhan’s estimate (8.3) is a precise statement about the idea that for small times the heat kernel “does not feel the boundary”. We refer to Kac [38] and Dodziuk [22] for statements with the same flavor. In the proof of Proposition 6.1 we need the following lemma.
Lemma 8.2
Define the function
where \(p_{\tau }^{\Omega }(x,y)\) denotes the Dirichlet heat kernel associated to \(\Omega \) and \(G_\gamma (x,y)\) the Green function of the operator \(-\Delta -\gamma \) on \(\Omega \). Then \(I(\tau )\) has the following asymptotic behavior:
for some constant \(c_{i,*}\) and \(i=1,2,3\).
Proof
Step 1 (Asymptotic for \(\tau \rightarrow \infty \)). We recall that the heat kernel \(p_\tau ^{\Omega }(x,y)\) admits the series expansion (8.1) which converges absolutely and uniformly in the domain \([\varepsilon ,\infty )\times \Omega \times \Omega \) for any \(\varepsilon >0\), as well as in the topology \(C^\infty ({{\mathbb {R}}}^+\times \Omega \times \Omega )\). By the uniform convergence with respect to \(y\in \Omega \) we obtain for \(\tau >0\)
Multiplying equation (2.7) by \(\phi _k\) and integrating by parts we get
that gives
We plug (8.10) into (8.9). Finally, from (8.2) we obtain the asymptotic behavior (8.8) for \(\tau \rightarrow \infty \).
Step 2 (Asymptotic for \(\tau \rightarrow 0^+\)). Firstly, we split
We analyze \(I_1(\tau )\). For the region \(B_{\varepsilon }(0)\) we invoke Varadhan’s estimate (8.6) and we obtain
for some \(c>0\), and by (8.7) we have
From these bounds we conclude
In the region \(\Omega \setminus B_\varepsilon (0)\) by (8.7) we get
We conclude that
Now, we estimate the term \(I_2(\tau )\). We treat it similarly to \(I_1(\tau )\) but we get a lower order term in the expansion since \(H_\gamma (y,0)\) is not singular. We use decomposition (2.10) for \(H_\gamma (y,0)\) and we consider the integral over \(B_{\varepsilon }(0)\). Using the cosine expansion we get
Thus, we compute the integral associated to the first term with Varadhan’s estimate (8.3) and the upper bound (8.7):
for an explicit constant \(c_{2,*}\). The same computation on the remainder \(O\left( \vert y\vert ^3\right) \) gives a term of order \(O\left( \tau ^{3/2}\right) \). Another Taylor expansion at \(y=0\) gives
where \(D_{yy}h_\gamma (0,0)\) denotes the Hessian of \(h_\gamma (\cdot ,0)\) evaluated in \(y=0\). Integrating the first term on \(B_\varepsilon (0)\) against \(p_t^\Omega (0,y)\) and using (8.3)–(8.7) we see by symmetry of the integrand \(p_t^{{{\mathbb {R}}}^3}(0,y)\nabla _y h_\gamma (0,0)\cdot y\) that the integral gives an exponentially decaying term. The second term in (8.12) can be treated similarly to (8.11) and gives a term of order \(c_{3,*}\tau (1+o(1))\) for some explicit constant \(c_{3,*}\). The integral of \(p_{\tau }^\Omega (y,0) H_\gamma (y,0)\) on the complement can be treated as before and gives an exponentially decay term for \(\tau \rightarrow 0\). Thus, we obtain that
We conclude that \(I(\tau )=I_1(\tau )+I_2(\tau )\) has the asymptotic (8.8) for \(\tau \rightarrow 0^+\). \(\square \)
We start here the main proof of Proposition 6.1.
8.1 Proof of Proposition 6.1
Firstly, we observe that \(J(0,t_0)=h(t_0)\) is in general not compatible with a null initial condition. For this reason it is natural to solve the problem for \(\mathcal {J}\) starting from \(t=t_0-1\). We look for \(\Lambda (t)\) for \(t \in (t_0-1,\infty )\). The function \(\mathcal {J}\) is a solution to the problem
such that
where
and
where \(\eta \) is a smooth function such that \(\eta (t_0-1)=0\), \(\eta (t_0)=1\) and
for any \(\nu \le 1\). This choice gives an extension \(h^*(t)\in C^{\varepsilon }\) with
Let \(s{:=}t-(t_0-1)\) and for \(s\in (0,\infty )\) define
The function \(\mathcal {J}_0\) is a solution to
such that
Imposing the initial condition \(\mathcal {J}(x,t_0)\equiv 0\) in \(\Omega \), that is \(\mathcal {J}_0(x,0)\equiv 0\), by Duhamel’s formula we have
where
and \(p_\tau ^{\Omega }(x,y)\) denotes the heat kernel associated to \(\Omega \). The asymptotic behavior of \(I(\tau )\) is given by Lemma 8.2. We denote the Laplace transform of a function f as
We refer to the book [23] by Doetsch for classic properties of the Laplace transform. Applying the Laplace transform to (8.16), using (8.15) and the basic property
we obtain
and hence
where
By definition we have
that is well defined and analytic in the right-half plane \({\text {Re}}\xi >-\lambda _1\) thanks to Lemma 8.2. By expansion (8.8) we have
and g is integrable in \({{\mathbb {R}}}^+\) if \({\text {Re}}{\xi }>-\lambda _1\). Thus, using (8.9), in any half plane \({\text {Re}}\xi \ge c\) where \(c>-\lambda _1\) the dominated convergence theorem applies to get
At this point we can extend \({\tilde{I}}(\xi )\) analytically from \(\{\xi \in \mathbb {C}: \xi >-\lambda _1\}\) to \(\mathbb {C}{\setminus } \{-\lambda _k\}_{k=1}^\infty \). Let \(\xi =a+ib\) and rewrite the series as
Since the coefficients of the series are positive, \({\tilde{I}} (\xi )=0\) implies \(b=0\). Plugging \(b=0\) into the first series we obtain that a root \(\xi =a\) of \({\tilde{I}}\) satisfies
Hence, we deduce that the set of zeros of \({\tilde{I}}\) is given by a sequence \(\{-a_k\}_{k=1}^{\infty }\) where \(a_k\in (\lambda _k,\lambda _{k+1})\). In particular,
By standard argument [23, Theorem 33.7] on the Laplace transform, using (8.8), we have
in the half-plane \({\text {Re}}\xi >-\lambda _1\). Thus, in the same half-plane we have
As a consequence of (8.18), \({{\tilde{\sigma }}}(\xi )\) has a unique singularity at \(\xi =-\gamma \) in the half-plane of convergence. By [23, Theorem 28.3] the function \({{\tilde{\sigma }}}(\xi )\) can be represented as a Laplace transform of a function.Footnote 1 Finally, we compute the inverse Laplace transform by means of the Residue theorem defining the rectangular contour integral \(\mathcal {C}_R\) as in Fig. 1, which is suggested by the proof of [23, Theorem 35.1].
Contour integral \(\mathcal {C}_R\)
For later purpose we observe that, looking at the contour integral \( {{\mathcal {C}}}_R\), the constant \(\mathfrak {a}\in (\gamma ,\lambda _1)\) can be taken arbitrarily close to \(\lambda _1\). An application of the Riemann-Lebesgue Lemma (as in [23, p.237]) implies
Since
we obtain
We easily compute
Now, we analyze the integral (8.20). We decompose
It is easy to see (by means of another contour to avoid the standard branch) that, up to constants, the last three integral are respectively the inverse Laplace transform of \(\xi ^{-1/2},\xi ^{-3/2},\xi ^{-2}\). The integral
is absolutely convergent thanks to the second order expansion of \({\tilde{\sigma }}(\xi )\). In fact, obtaining the absolute convergence of \(R(\tau )\) (and \(R'(\tau )\)) is the main reason to use the sharp Varadhan’s estimate on the heat kernel \(p_t^{\Omega }\). Thus, from (8.20) we obtain
for some constants \(c_\infty , C_{i,*}\) and \(i=1,2,3\), where \(R(\tau )\) is bounded. This gives the asymptotic behavior
for any \(\mathfrak {a} \in (\gamma ,\lambda _1)\). For later purposes, we observe that \(\sigma (\tau )\) is differentiable. Indeed, differentiating \(R(\tau )\), we still obtain an absolutely convergent integral thanks to the full expansion (8.19), and an application of the dominated convergence theorem gives \(\sigma \in C^1\) with
From (8.17), taking the inverse Laplace transform of both sides, we get
that is
Proof of (6.4)
We rewrite this formula as
We choose \(\beta (0)=-c_\infty \int _{0}^{\infty }h_0^*(\tau )d\tau \). It remains to estimate
We recall that the extension \(h_0^*(s)\) has been selected so that (8.13) holds. Here and in what follows, without losing in generality we assume the same value \(c=c_i\) for \(i=1,2\). When we estimate the \(L^\infty \) norm of \(\beta \) we will only use the \(L^\infty \) norm of \(h_0^*\) and hence we get the same \(L^\infty \)-weight constant \(c_1\). Instead, when we estimate the \(C^{1/2+\varepsilon }\) we need both the \(L^\infty \) and \(C^\varepsilon \) norms of \(h_0^*\), thus we will get the same \(C^\varepsilon \)-weight constant \(c_2=\min \{c_1,c_2\}\). Thus, conditionally to \(c_i<(\lambda _1-\gamma )/(2\gamma )\), the weight constant \(c_i\) with \(i=1,2\) for \(\beta \) and \(h_0^*\) are respectively the same. We proceed with the \(L^\infty \) estimate of \(\beta \). We have
Using hypothesis (6.2) and selecting \(\mathfrak {a}\) close enough to \(\lambda _1\) so that
we get
Combining the bounds on \(\beta _1\) and \(\beta _2\) we obtain
Now we estimate the \(\left( 1/2+\varepsilon \right) \)-Hölder seminorm. In the following it is enough to assume \(\eta \in (0,1)\). We have
Let
Following the classical fractional integral estimate of Hardy and Littlewood [35, Theorem 14], we decompose
For \(s-\eta \in (\eta ,1)\) we have
For \(s-\eta \ge 1\) we get
For \(s-\eta \in (0,\eta )\) we obtain
Now we estimate \(A_2\). We have
Finally, we estimate \(A_3\). Using the \(L^\infty \) norm of \(h_0^*\) for \(\tau >1\) and \(C^{\varepsilon }\) seminorm for \(\tau <1\) we obtain
Combining the bounds on \(A_1,A_2,A_3\) and we obtain
Finally, from (8.22), (8.23) and (8.24) we obtain
Going back to the original variable t using (8.14), we obtain
and recalling (8.13) the proof of (6.4) is complete. \(\square \)
We proceed to prove the second part of Proposition 6.1: in case \(h\in X_{\sharp ,c,\frac{1}{2}+\varepsilon }\), then \(\Lambda \) is differentiable and \({\dot{\Lambda }} \in X_{\sharp ,c,\varepsilon }\).
Proof of (6.5)
In the same notation of the previous lemma, we need to prove that \(\beta _1(s),\beta _2(s)\) are differentiable and estimate the derivatives. Since
we clearly have \(\beta _1(s)\in C^{1}(0,\infty )\) and \(\beta _1'(s)=c_\infty h(s)\in X_{{\sharp ,c,\frac{1}{2}+\varepsilon }}\) by hypothesis. To analyze \(\beta _2\), following [35, Theorem 19], we introduce for any \(\epsilon \ge 0\) the function
so that \(\beta _{2,0}(s)=\beta _2(s)\). Since \(\sigma (\tau )\in C^1\), we can differentiate \(\beta _{2,\epsilon }(s)\) to obtain
Observe that we can choose the extension \(h_0^*\) such that \(h_0^*(s)=o(s^{1/2})\) for \(s\rightarrow 0\). Since \(h_0^*\in X_{\sharp ,c,\frac{1}{2}+\varepsilon }\), when \(\epsilon \rightarrow 0\) the right-hand side tends uniformly to
where
By hypothesis and the choice of the extension we have \(l(s)h_0^*(s)\in X_{\sharp ,c,\frac{1}{2}+\varepsilon }\). Also, the function g(s) is continuous since \(h_0^*(s)\in C^{\frac{1}{2}+\varepsilon }\).
hence
It remains to prove that \(g(s)\in X_{\sharp ,c,\varepsilon }\). Using the asymptotic of \(\sigma '(t)\) and the assumption (6.2) with \(\mathfrak {a}\) as in (8.21) we have
We write
Using again assumption (6.2) we get
Also
and
This proves
Combining it with (8.25) we obtain
Summing up the estimates for \(\beta _1'(s)\) and \(\beta _2'(s)=l(s) h_0^*(s)+g(s)\) we obtain
Finally, in the original variable t, using (8.14) and (8.13), we obtain the bound (6.5). \(\square \)
Remark 2.1
(The initial datum \(J_1(x,t_0)\)) From the proof of Proposition 6.1 we have \(\mathcal {J}(t_0,x)=\int _0^1 h^*(s) I(x,\tau -s)\,ds\) where \(h_0^*\) is an arbitrary smooth function with \(h_0^*(t)=o(t^{1/2})\) for \(t\rightarrow 0\) and \(h_0^*(1)=h(t_0)\), connecting to h(t) at \(t=t_0\) to maintain the \(C^{\varepsilon }\) regularity of h. We observe by estimate (2.2) that
Thus, our initial datum remains positive provided that \(t_0\) is fixed sufficiently large.
Notes
We cannot have an estimate directly on \({\dot{\beta }}\) at this point. Indeed, \((\tilde{I}(\xi ))^{-1}\) is not a Laplace transform of a function since diverges as \(\vert \xi \vert \rightarrow \infty \). However, it still can be represented as the Laplace transform of a distribution, see [23, Theorem 29.3].
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Acknowledgements
The authors acknowledge the support from the Royal Society Research Professorship RP-R1-180114, United Kingdom. G. Ageno acknowledges support from the ERC/UKRI 2208 Horizon Europe Grant SWAT EP/X030644/1 and M. del Pino from the ERC/UKRI Horizon Europe Grant ASYMEVOL EP/Z000394/1. The authors express gratitude to the anonymous referee for their careful reading, which contributed to the improvement of the manuscript.
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Appendix A: Properties of the Robin function \(H_\gamma (x,x)\)
Appendix A: Properties of the Robin function \(H_\gamma (x,x)\)
In this appendix we prove some properties of the Robin function that we use in our construction. We recall that the Green function associated to the operator \(-\Delta -\gamma \) satisfies
As usual, we split
and the regular part \(H_\gamma (x,y)\) satisfies
for any fixed \(y\in \Omega \). We recall (from [14] and reference therein) the following properties of \(R_\gamma (x){:=}H_\gamma (x,x)\):
-
(1)
\(R_\gamma (x)\in C^\infty (\Omega )\)
-
(2)
\(\partial _\gamma R_\gamma (x)<0\) and belongs to \(C^{\infty }(\Omega )\).
-
(3)
for each \(\gamma \in (0,\lambda _1)\) fixed, \(R_\gamma (x)\rightarrow +\infty \) as \(x\rightarrow \partial \Omega \)
Lemma A.1
(Behavior near the first eigenvalue) The function \(H_\gamma (x,y)\) satisfies
Proof
We decompose \(H_\gamma \) as
where
and \(H_0\) satisfies
Thus, for any fixed \(y\in \Omega \), \(h_{\perp ,\gamma }(x,y)\) is the solution to
By definition of \(\alpha (y)\) we have
Testing (A.1) against \(\phi _1\) we get
Also, testing (A.4) against \(\phi _1\) and using (A.5) we obtain
Thus, we have
and plugging (A.6) in (A.3) we obtain
We notice that only the first and last term in the right-hand side depends on \(\gamma \). Hence, to prove (A.2) we just need to prove that \(h_{\perp ,\gamma }(x,y)\) is bounded as \(\gamma \rightarrow \lambda _1^{-}\). This is a consequence of the Poincaré inequality applied to functions in \(H_0^1 \) which are orthogonal to \(\phi _1\). Indeed, expanding \(h_{\perp ,\gamma }\) in the \(L^2\)-basis made of Laplacian eigenvalues we get
Now, testing equation (A.4) against \(h_{\perp ,\gamma }\), using (A.5) and Cauchy–Schwarz inequality we get
We conclude that
with the right-hand side independent of \(\gamma \). By standard elliptic estimates we get
with K independent of \(\gamma \). This concludes the proof. \(\square \)
The following lemma gives the asymptotic behavior of \(\gamma ^*(x)\) as x approaches the boundary \( {\partial } \Omega \).
Lemma A.2
The unique number \(\gamma ^*(x)\in (0,\lambda _1)\) defined by the relation
satisfies
where \(d(x,\partial \Omega )=\vert x-x'\vert \).
Proof
We divide the proof in two steps. Given \(x\in \Omega \) let \(D_x \subset \partial \Omega \) the set of points \(x'\) such that
If \(D_x\) is not a singleton we choose the unique \(x'=(x_1',x_2',x_3')\) with the property \(x_i'\le y_i'\) for all components \(i=1,2,3\) and point \(y'\in D_x\). This defines \(x'{:=}x'(x)\) uniquely.
Step 1. Firstly we prove (A.8) for domains such that, for all \(x\in \Omega \), the reflection point \(x''(x){:=}2x'(x)-x\) satisfies
We decompose
where F satisfies
and
We write
and select \(\alpha (y)\) so that \(\int _{\Omega } w_\perp (x,y) \phi _1(x)\,dx=0\). By decomposition (A.9) and (2.8) we obtain
The equation for \(w_\perp \) is
Multiplying this equation by \(w_{\perp }\) and integrating by parts we get
Using the improved Poincaré inequality
and Cauchy–Schwarz we obtain
Now, we want to estimate uniformly in y the right-hand side of
Without loss of generality, suppose \(0\in \Omega \). Let \(M{:=}2\text {diam}(\Omega )\) we have
Let \(\Omega ''=\left\{ x'' \in {{\mathbb {R}}}^3 : x''=x''(x) \text { for some } x\in \Omega \right\} \). We have
where \(M_2=2{{\,\textrm{diam}\,}}(\Omega ^{''}\cup \Omega )\) hence we get
We combine this bound with (A.11) to get
with \(K_\Omega \) independent of y and by standard elliptic estimates we get
with a possibly larger constant \(K_\Omega \). We conclude that
where
with \(w_\perp (x,y)\) bounded in \(\Omega \times \Omega \). Also we notice that
and
This proves the boundedness of B(y). Now, the equation for \(\gamma ^*(x)\) reads as
Let \(c{:=}\vert \partial _\nu \phi _1(x')\vert \). We expand \(\phi _1(x)\) at \(x'\in \partial \Omega \) to get
Since B(x) and w(x, x) are bounded, we conclude that
Step 2. Now, we modify the method in Step 1 to obtain an expansion similar to (A.12) and conclude that (A.8) is true for general smooth bounded domains. Let \(y\in \Omega _{\epsilon /4}\). Now we prove (A.8) for all smooth domains \(\Omega \). Fix \(\epsilon =\epsilon (\Omega )>0\) so small that the set \(\Omega _\epsilon {:=}\{x\in \Omega : d(x,\partial \Omega )<\epsilon \}\) possesses the property (P) and let \(\eta _\epsilon \) be a smooth cut-off function with \({{\,\textrm{supp}\,}}(\eta _\epsilon )\subset \Omega _\epsilon \) and \(\eta _\epsilon (x)\equiv 1\) for \(x\in \Omega _{\epsilon /2}\). We write
where
We notice that \(\eta _{\epsilon }(x)\eta _{\epsilon }(y)F(x,y)\), where F satisfies (A.10), is well-defined in \(\Omega \) thanks to the cut-off functions. The problem for \(F_2\) is
where
and
We decompose
where \(\beta \) is chosen such that \(\int _\Omega w_2(x,y)\phi _1(x)\,dx=0\), that gives
Next we prove that \(w_{2}(x,y)\) is uniformly bounded in \(\Omega \times \Omega \). Using the improved Poincaré inequality and standard elliptic estimates as in Step 1, we reduce the problem to estimate the \(L^2\)-norm of \(g(\cdot ,y)\) uniformly in \(y\in \Omega _{\epsilon /4}\). We have
Since \(\epsilon \) depends only on \(\Omega \) we obtain
Now, we prove the boundedness of
Indeed, we have
and
Finally, the equation for \(\gamma ^*(x)\) is
and by the boundedness of \(B_\epsilon (x)\) and \(w_2(x,x)\) we obtain (A.13). \(\square \)
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Ageno, G., Pino, M.d. Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02885-x
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DOI: https://doi.org/10.1007/s00208-024-02885-x