Vanishing aspect ratio limit for a fourth-order MEMS model

Abstract

So far most studies on mathematical models for microelectromechanical systems are focused on the so-called small aspect ratio model which is a wave or beam equation with a singular source term. It is formally derived by setting the aspect ratio equal to zero in a more complex model involving a moving boundary. A rigorous justification of this derivation is provided here when bending is taken into account.

1 Introduction

An idealized microelectromechanical system (MEMS) is made of a rigid ground plate located at height $z=-1$ and held at potential zero above which an elastic plate held at a constant potential normalized to one is suspended, the latter being at height $z=0$ at rest. Assuming that there is no variation in one of the horizontal directions, the electromechanical response of the device may be described by the elastic deformation $u=u(t,x)\ge -1$ of the elastic membrane and the electrostatic potential $\psi = \psi (t,x,z)$ between the two plates [31, Section 7.4]. The evolution of the electrostatic potential is given by the rescaled Laplace equation

$$\begin{aligned} \varepsilon ^2 \partial _x^2 \psi (t,x,z) + \partial _z^2 \psi (t,x,z) = 0 \;\;\text { in }\;\; \Omega (u(t)), \quad t>0, \end{aligned}$$

(1.1)

in the domain

$$\begin{aligned} \Omega (u(t)):= \left\{ (x,z)\in I\times (-1,\infty ):\ -1< z < u(t,x) \right\} , \quad I:= (-1,1), \end{aligned}$$

between the plates, supplemented with the boundary conditions

$$\begin{aligned} \psi (t,x,z) = \frac{1+z}{1+u(t,x)} \;\;\text { on }\;\; \partial \Omega (u(t)), \quad t>0, \end{aligned}$$

(1.2)

while that of the deformation obeys a parabolic (if $\gamma =0$) or hyperbolic (if $\gamma >0$) equation

$$\begin{aligned}&\gamma ^2 \partial _t^2 u(t,x) + \partial _t u(t,x) + \beta \partial _x^4 u(t,x) - \left( \tau + a \Vert \partial _x u(t)\Vert _2^2 \right) \partial _x^2 u(t,x) \nonumber \\&\quad = - \lambda \left( \varepsilon ^2 |\partial _x \psi (t,x,u(t,x))|^2 + |\partial _z \psi (t,x,u(t,x))|^2 \right) , \quad t>0, \quad x\in I, \end{aligned}$$

(1.3)

supplemented with clamped boundary conditions

$$\begin{aligned} u(t,\pm 1 ) = \beta \partial _x u(t,\pm 1) = 0 , \qquad t>0, \end{aligned}$$

(1.4)

and initial conditions

$$\begin{aligned} (u,\gamma \partial _t u)(0) = (u^0, \gamma u^1), \qquad x\in I. \end{aligned}$$

(1.5)

In (1.3), the terms corresponding to mechanical forces are $\beta \partial _x^4 u$, which accounts for plate bending, and $\left( \tau + a \Vert \partial _x u\Vert _2^2 \right) \partial _x^2 u$, which accounts for external stretching ($\tau >0$) and self-stretching due to moderately large oscillations ($a>0$). The right-hand side of (1.3) describes the electrostatic forces exerted on the elastic plate which are proportional to the square of the trace of the (rescaled) gradient of the electrostatic potential on the elastic plate, the parameter $\lambda$ depending on the square of the applied voltage difference before scaling. Inertial forces are accounted for by $\gamma ^2 \partial _t^2 u$ in (1.3) with $\gamma \ge 0$, and $\partial _t u$ is a damping term which governs the dynamics in the damping dominated limit $\gamma =0$. A very important parameter in (1.1)–(1.5) is the aspect ratio $\varepsilon >0$ which is proportional to the ratio height/length of the device. In applications this ratio is usually very small and therefore often taken to be zero from the outset in the mathematical analysis which has far-reaching consequences.

Indeed, setting $\varepsilon =0$ allows one to solve explicitly (1.1)–(1.2) in terms of the deformation $u_0$ as

$$\begin{aligned} \psi _0(t,x,z) = \frac{1+z}{1+u_0(t,x)},\quad (x,z)\in \Omega (u_0 (t)),\quad t >0. \end{aligned}$$

Here, the subscript zero is used to indicate the formal limit $\varepsilon \rightarrow 0$. Furthermore, setting $\varepsilon =0$ in (1.3) and using the previous formula for $\psi _0$, we are led to the so-called small aspect ratio model

$$\begin{aligned} \gamma ^2 \partial _t^2 u_0 + \partial _t u_0 + \beta \partial _x^4 u_0 - \left( \tau + a \Vert \partial _x u_0\Vert _2^2 \right) \partial _x^2 u_0 = - \frac{\lambda }{(1+u_0)^2}, \quad t>0, \quad x\in I, \end{aligned}$$

(1.6)

supplemented with clamped boundary conditions (1.4) and initial conditions (1.5). This model is studied in several works in recent years, among which we refer to [5, 7,8,9, 11,12,13,14, 16, 17, 19, 22,23,24,25,26,27,28,29].

Take notice that the above computation is formal and a reliable use of the small aspect ratio model (1.6) thus requires to justify it rigorously. Due to the intricate coupling between u and $\psi$ when $\varepsilon >0$ this is by no means obvious. The aim of this paper is to put this approximation on firm ground by providing a proof that solutions of (1.1)–(1.5) indeed converge to that of (1.4)–(1.6) as $\varepsilon \rightarrow 0$ when bending and self-stretching of the plate are taken into account, that is, when $\beta >0$ and $a\ge 0$. When these effects are not included, that is, when $\beta =a=0$, this issue is addressed in [18] for the stationary problem and in [10] for the evolution problem in the damping dominated case $\gamma =0$. It is proved in these settings that solutions to (1.1)–(1.5) converge as $\varepsilon \rightarrow 0$ to a solution to (1.4)–(1.6) in suitable topologies. To achieve this result several difficulties are faced with some differences between the stationary and evolutionary settings. In fact, the main difficulty arising in the study of (1.1)–(1.5) is the so-called pull-in instability which manifests in the following ways. It corresponds to the nonexistence of stationary solutions for values of $\lambda$ exceeding a threshold value depending on $\varepsilon$. For the evolution problem it corresponds to the occurrence of a finite time singularity for values of $\lambda$ larger than a critical value depending not only on $\varepsilon$, but also on the initial value $(u^0,\gamma u^1)$ in (1.5). The singularity becomes manifest in that the solution to (1.1)–(1.5) only exists on a finite time interval $[0,T_{m}^\varepsilon )$ and satisfies

$$\begin{aligned} \lim _{t\rightarrow T_{m}^\varepsilon } \min _{x\in [-1,1]} u(t,x) = -1, \end{aligned}$$

see [20].

Remark 1.1

A close connection between the existence of a global classical solution to the evolution problem and the existence of a weak stationary solution is well established for the small aspect ratio model (1.6) where it is expected in general and proved in some cases that the values of $\lambda$ for which both exist coincide, see [11] and the references therein. A similar connection was already uncovered for semilinear parabolic equations in a bounded domain supplemented with homogeneous Dirichlet boundary conditions [4, 32].

To overcome the aforementioned difficulties including the intricate dependence upon $\varepsilon$ and study the limiting behavior as $\varepsilon \rightarrow 0$, one has to find a range of values of $\lambda$ for which stationary solutions exist for all sufficiently small values of $\varepsilon$, respectively, to derive a lower bound on the maximal existence time $T_\mathrm{m}^\varepsilon$ which does not depend on $\varepsilon$. This approach is used in [10, 18] when $\beta =\gamma =a=0$, but can only be adapted here to the evolution problem. Indeed, in contrast to [18], the construction of stationary solutions to (1.1)–(1.4) performed in [21] for $\beta >0$ and $a\ge 0$ does not provide an interval $(0,\Lambda )$ such that stationary solutions exist for all $\lambda \in (0,\Lambda )$ and $\varepsilon$ small enough. We shall thus develop an alternative argument in Sect. 3. The second obstacle met in the analysis of the vanishing aspect ratio limit is the derivation of estimates on the solutions to (1.1)–(1.5) which are uniform with respect to $\varepsilon$ small enough and sufficient to pass to the limit in (1.1)–(1.2) and the right-hand side of (1.3). According to the analysis performed in [10, 18], a bound on u in $W_q^2(I)$ or $L_\infty (0,T,W_q^2(I))$ for some $q>2$ and $T>0$ is sufficient. For the evolution problem this bound is derived simultaneously with the aforementioned lower bound on the maximal existence time $T_\mathrm{m}^\varepsilon$. For the stationary problem the situation is again completely different as the first part of the analysis only provides a bound in $H^2(I)$. Proceeding in the study of the limit $\varepsilon \rightarrow 0$ requires a different approach to handle the weaker regularity on u and is presented in Sect. 2. As in [21] it is based on the observation that the same regularity of the right-hand side of (1.3) as in [10, 20] can be derived for less regular u. However, a more involved argument is required here.

From now on we assume that

$$\begin{aligned} \beta >0, \quad \tau \ge 0 , \quad a\ge 0, \quad \gamma \ge 0 . \end{aligned}$$

(1.7)

For further use, given $s>0$ and $q\in [1,\infty ]$, we let $W_{q,D}^s(I)$ be the subspace of the Sobolev space $W_q^s(I)$ consisting of functions satisfying the boundary conditions (1.4) whenever meaningful, that is,

$$\begin{aligned} W_{q,D}^s(I) := \left\{ \begin{array}{ll} \left\{ v \in W_q^s(I) :\ v(\pm 1) = \partial _x v(\pm 1)=0 \right\} , &{} \quad s>(q+1)/q, \\ \\ \left\{ v \in W_q^s(I):\ v(\pm 1) =0 \right\} , &{} \quad 1/q<s<(q+1)/q, \\ \\ W_q^s(I) , &{} \quad s<1/q, \end{array} \right. \end{aligned}$$

and $H_D^s(I) := W_{2,D}^s(I)$. Also, for $\kappa \in (0,1)$, we define

$$\begin{aligned} S_q^s(\kappa ) := \left\{ v \in W_{q,D}^s(I):\ v>-1+\kappa \;\text { in }\; I \;\;\text { and }\;\; \Vert v\Vert _{W_q^s} < \frac{1}{\kappa } \right\} . \end{aligned}$$

The first result deals with the vanishing aspect ratio limit for the stationary problem, and the starting point is the existence result obtained in [21]. Let us first recall that (1.1)–(1.5) has a variational structure and that stationary solutions are critical points of the total energy

$$\begin{aligned} \mathcal {E}_\varepsilon (v) := \mathcal {E}_m(v) - \lambda \mathcal {E}_{e,\varepsilon }(v). \end{aligned}$$

(1.8)

In (1.8), the mechanical energy is defined by

$$\begin{aligned} \mathcal {E}_m(v) := \frac{\beta }{2} \Vert \partial _x^2 v\Vert _2^2 + \frac{1}{2} \left( \tau + \frac{a}{2} \Vert \partial _x v\Vert _2^2 \right) \Vert \partial _x v\Vert _2^2 \end{aligned}$$

(1.9)

and the electrostatic energy by

$$\begin{aligned} \mathcal {E}_{e,\varepsilon }(v) := \int _{\Omega (v)} \left( \varepsilon ^2 |\partial _x \psi _v|^2 + |\partial _z \psi _v|^2 \right) \ \mathrm {d}(x,z) , \end{aligned}$$

(1.10)

where $\Omega (v) := \{ (x,z)\in I \times (-1,\infty ):\ -1< z < v(x)\}$ and $\psi _v$ denotes the solution to the rescaled Laplace equation

$$\begin{aligned} \varepsilon ^2 \partial _x^2 \psi _v + \partial _z^2 \psi _v = 0 \;\;\text { in }\;\; \Omega (v), \qquad \psi _v(x,z) = \frac{1+z}{1+v(x)}, \quad (x,z)\in \partial \Omega (v). \end{aligned}$$

(1.11)

Clearly, $\psi _v$ depends on v in a nonlocal and nonlinear way. Observe that all the above quantities are well defined for $v\in H^2(I)$ satisfying $v>-1$ in $[-1,1]$. It is easy to see that, setting $\phi (x) := - (1-x^2)^2$ for $x\in [-1,1]$, there holds $\mathcal {E}_\varepsilon (\theta \phi )\rightarrow -\infty$ as $\theta \nearrow 1$. Consequently, $\mathcal {E}_\varepsilon$ is not bounded from below. Nevertheless, stationary solutions can be constructed by a constrained minimization approach. We recall the result obtained in [21].

Proposition 1.2

Let $\varepsilon >0$ and $\varrho \in (2,\infty )$. There are

• a function $u_{\varrho ,\varepsilon }\in H_D^4(I)$ satisfying $-1< u_{\varrho ,\varepsilon }<0$ in I,

• $\psi _{\varrho ,\varepsilon }\in H^2(\Omega (u_{\varrho ,\varepsilon }))$ satisfying $\psi _{\varrho ,\varepsilon } = \psi _{u_{\varrho ,\varepsilon }}$,

• and $\lambda _{\varrho ,\varepsilon }>0$

such that $(u_{\varrho ,\varepsilon }, \psi _{\varrho ,\varepsilon })$ is a stationary solution to (1.1)–(1.4) with $\lambda =\lambda _{\varrho ,\varepsilon }$. Moreover, both $u_{\varrho ,\varepsilon }$ and $\psi _{\varrho ,\varepsilon }$ are even with respect to x. In addition,

$$\begin{aligned} \mathcal {E}_m(u_{\varrho ,\varepsilon }) = \min \{ \mathcal {E}_m(u):\ u\in \mathcal {A}_{\varrho ,\varepsilon } \}, \end{aligned}$$

(1.12)

where

$$\begin{aligned} \mathcal {A}_{\varrho ,\varepsilon } := \left\{ v \in H^2_D(I):\ -1<v\le 0 \;\text { in }\; I , \ v \;\text { is even and }\; \mathcal {E}_{e,\varepsilon }(v)=\varrho \right\} . \end{aligned}$$

(1.13)

Given $\varepsilon >0$ another approach to construct (a smooth branch of) stationary solutions to (1.1)–(1.4) for small values of $\lambda$ is based on the implicit function theorem [20]. These solutions actually coincide with the ones from Proposition 1.2 having an electrostatic energy $\mathcal {E}_{e,\varepsilon }(u_{\varrho ,\varepsilon })=\varrho$ close to 2. Note, however, that Proposition 1.2 provides additional stationary solutions with large electrostatic energies as one can show that $\lambda _{\varrho ,\varepsilon }\rightarrow 0$ as $\varrho \rightarrow \infty$, see [21].

We may now state the convergence result for stationary solutions and thus provide a rigorous justification of the stationary small aspect ratio limit.

Theorem 1.3

Let $\varrho \in (2,\infty )$. There are a sequence $(\varepsilon _k)_{k\ge 1}$ with $\varepsilon _k\rightarrow 0$, $\kappa _\varrho \in (0,1)$, $u_\varrho \in S^4_2(\kappa _\varrho )$, and $\lambda _\varrho >0$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty } \left\{ |\lambda _{\varrho ,\varepsilon _k} - \lambda _\varrho | + \Vert u_{\varrho ,\varepsilon _k} - u_\varrho \Vert _{H^1} \right\} = 0, \end{aligned}$$

where $u_\varrho$ is a solution to the (stationary) small aspect ratio model

$$\begin{aligned} \beta \partial _x^4 u_\varrho - \left( \tau + a \Vert \partial _x u_\varrho \Vert _2^2 \right) \partial _x^2 u_\varrho = - \frac{\lambda _\varrho }{(1+u_\varrho )^2} \;\;\text { in }\;\; I, \qquad u_\varrho (\pm 1) = \partial _x u_\varrho (\pm 1)= 0 , \end{aligned}$$

(1.14)

satisfying

$$\begin{aligned} \int _{-1}^1 \frac{\mathrm {d}x}{1+u_\varrho (x)} = \varrho . \end{aligned}$$

In addition, $u_\varrho$ is even, with

$$\begin{aligned} \mathcal {E}_m(u_\varrho ) = \min \{ \mathcal {E}_m(u):\ u\in \mathcal {A}_{\varrho ,0} \}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {A}_{\varrho ,0} := \left\{ v \in H^2_D(I):\ -1<v\le 0 \;\text { in }\; I , \ v \;\text { is even and }\; \int _{-1}^1 \frac{\mathrm {d}x}{1+v(x)} =\varrho \right\} , \end{aligned}$$

and

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{\Omega (u_{\varrho ,\varepsilon _k})}\left\{ \left| \psi _{\varrho ,\varepsilon _k}(x,z) - \frac{1+z}{1+u_\varrho (x)} \right| ^2 + \left| \partial _z \psi _{\varrho ,\varepsilon _k}(x,z) - \frac{1}{1+u_\varrho (x)} \right| ^2 \right\} \ \mathrm {d}(x,z) = 0. \end{aligned}$$

In fact, the convergence to zero of the sequence $(\partial _z \psi _{\varrho ,\varepsilon _k} - 1/(1+u_\varrho ))_k$ also holds true in $H^1(\Omega (u_{\varrho ,\varepsilon _k}) )$.

A by-product of Theorem 1.3 is the existence of stationary solutions to the small aspect ratio model (1.14) which are minimizers to a constrained variational problem. While the latter seems to be new, the existence of solutions to (1.14) is already known, see [5, 6, 8, 9, 11, 14, 19, 25, 28, 29] where I is replaced by the unit ball of $\mathbb {R}^N$, $N\ge 1$.

We next turn to the hyperbolic evolution problem, for which we shall establish the following convergence result.

Theorem 1.4

Let $\gamma >0$ and $\lambda >0$. Given $2\alpha \in (0,1/2)$ and $\kappa \in (0,1)$ let $\mathbf{u}^0 =(u^0,u^1)\in H_D^{4+2\alpha }(I)\times H_D^{2+2\alpha }(I)$ with $u^0 \in S_2^{2+2\alpha }(\kappa )$. For $\varepsilon \in (0,1)$ let $(u_\varepsilon ,\psi _\varepsilon )$ be the unique solution to (1.1)–(1.5) on the maximal interval of existence $[0,T_\mathrm{m}^\varepsilon )$ with $\psi _\varepsilon :=\psi _{u_\varepsilon }$. There are $T>0$ and $\kappa _0\in (0,1)$ such that $T_\mathrm{m}^\varepsilon \ge T$ and $u_\varepsilon (t)\in S_2^{2+2\alpha }(\kappa _0)$ for all $(t,\varepsilon )\in [0,T]\times (0,1)$. Moreover, for any $\alpha '\in [0,\alpha )$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left\{ \sup _{t\in [0,T]} \Vert u_{\varepsilon }(t) - u_0(t)\Vert _{H^{2+2\alpha '}} + \sup _{t\in [0,T]} \Vert \partial _t u_\varepsilon (t) - \partial _t u_0(t) \Vert _{H^{2\alpha '}} \right\} = 0, \end{aligned}$$

where

$$\begin{aligned} u_0 \in W_1^2\big (0,T, H^{2\alpha '}(I)\big ) \cap C\big ([0,T],H_D^{2+2\alpha '}(I)\big ) \cap L_1\big (0,T, H_D^{4+2\alpha '}(I)\big ) \end{aligned}$$

is a strong solution to

$$\begin{aligned} \gamma ^2 \partial _t^2 u_0 + \partial _t u_0 + \beta \partial _x^4 u_0 - \left( \tau + a \Vert \partial _x u_0\Vert _2^2 \right) \, \partial _x^2 u_0 = - \frac{\lambda }{(1+u_0)^2} \;\;\text { in } (0,T)\times I, \end{aligned}$$

(1.15)

supplemented with clamped boundary conditions

$$\begin{aligned} u_0(t,\pm 1 ) = \partial _x u_0(t,\pm 1) = 0 , \qquad t \in (0,T)\, \end{aligned}$$

(1.16)

and initial conditions

$$\begin{aligned} (u_0,\partial _t u_0)(0) = (u^0,u^1), \qquad x\in I. \end{aligned}$$

(1.17)

In addition,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{\Omega (u_{\varepsilon }(t))}\left\{ \left| \psi _{\varepsilon }(t,x,z) - \frac{1+z}{1+u_0(t,x)} \right| ^2 + \left| \partial _z \psi _{\varepsilon }(t,x,z) - \frac{1}{1+u_0(t,x)} \right| ^2 \right\} \ \mathrm {d}(x,z) = 0 \end{aligned}$$

(1.18)

for all $t\in [0,T]$.

It follows from Theorem 1.4 that the initial-value problem (1.15)–(1.17) has a solution $u_0$ which exists at least on the time interval [0, T], where $T>0$ depends only on the parameters from (1.7) and on $\alpha$, $\Vert u^0\Vert _{H_D^{4+2\alpha }}$, $\Vert u^1\Vert _{H_D^{2+2\alpha }}$, and $\kappa$. Owing to the possibility of the occurrence of a finite time singularity for $u_0$ as already mentioned, the convergence stated in Theorem 1.4 is unlikely to extend to arbitrary time intervals in general.

Remark 1.5

If $\lambda$, a, $\Vert u^0\Vert _{H_D^{4+2\alpha }}$, and $\Vert u^1\Vert _{H_D^{2+2\alpha }}$ are sufficiently small, then T can be chosen arbitrarily large in the convergence result stated in Theorem 1.4.

Let us also point out that, for the evolution problem, it is the whole family $(u_\varepsilon )_{\varepsilon \in (0,1)}$ which converges toward the solution to (1.15)–(1.17), in contrast to the stationary problem, where the convergence only holds for a subsequence. This is due to the uniqueness of the solution $u_0$ to the limit problem (1.15)–(1.17).

The local well-posedness of (1.1)–(1.5) is established in [20] for $a=0$, but the proof extends to $a>0$ as $v \mapsto \Vert \partial _x v\Vert _2^2 \partial _x^2 v$ is a locally Lipschitz continuous map from $H^{s+2}(I)$ to $H^{s}(I)$ for all $s>0$. Concerning the small aspect ratio Eqs. (1.15)–(1.17), most available results are actually devoted to the situation where both bending and self-stretching are neglected, that is, $\beta =a=0$, see [12, 16, 17, 22,23,24]. As far as we know, the well-posedness of (1.15)–(1.17) including bending ($\beta >0$) and clamped boundary conditions (1.16) is only investigated in [7], when $a\ge 0$ and the length of I is sufficiently small, and in [19] when $a=0$, but the proof of local well-posedness performed in [19] extends to the case $a>0$. The corresponding solution $u_0$ has better regularity properties than stated in Theorem 1.4 as shown in the proof below. The dynamics of (1.15) was also studied with other boundary conditions, namely, pinned boundary conditions $u(t,\pm 1)=\partial _x^2 u(t,\pm 1) =0$ in [7, 13] and hinged or Steklov boundary conditions $u(t,\pm 1)=\partial _x^2 u(t,\pm 1) - d \partial _x u(t,\pm 1) = 0$, $d>0$, in [7].

The last result deals with the parabolic version of (1.1)–(1.5) corresponding to the damping dominated limit $\gamma =0$ when inertial effects are neglected.

Theorem 1.6

Let $\gamma =0$, $a=0$, and $\lambda >0$. Given $2\alpha \in (0,2)$ and $\kappa \in (0,1)$ let $u^0\in S_2^{2+2\alpha }(\kappa )$. For $\varepsilon \in (0,1)$ let $(u_\varepsilon ,\psi _\varepsilon )$ be the unique solution to (1.1)–(1.5) on the maximal interval of existence $[0,T_\mathrm{m}^\varepsilon )$ with $\psi _\varepsilon :=\psi _{u_\varepsilon }$. There are $T>0$ and $\kappa _0\in (0,1)$ such that $u_{\varepsilon }(t) \in S^{2+2\alpha }_{2}(\kappa _0) \ { \mathrm and } \ T^\varepsilon _m\ge T$ for all $(t,\varepsilon )\in [0,T]\times (0,1)$. Moreover, for any $\alpha '\in [0,\alpha )$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sup _{t\in [0,T]} \Vert u_{\varepsilon }(t) - u_0(t)\Vert _{H^{2+2\alpha '}} = 0, \end{aligned}$$

where

$$\begin{aligned} u_0 \in C^1\big ([0,T],L_2(I)\big ) \cap C\big ([0,T],H_D^{2+2\alpha '}(I)\big ) \end{aligned}$$

is a strong solution to

$$\begin{aligned} \partial _t u_0 + \beta \partial _x^4 u_0 - \tau \partial _x^2 u_0 = - \frac{\lambda }{(1+u_0)^2} \;\;\text { in } (0,T)\times I, \end{aligned}$$

(1.19)

supplemented with clamped boundary conditions

$$\begin{aligned} u_0(t,\pm 1 ) = \partial _x u_0(t,\pm 1) = 0 , \qquad t\in (0,T), \end{aligned}$$

(1.20)

and initial condition

$$\begin{aligned} u_0(0) = u^0, \qquad x\in I. \end{aligned}$$

(1.21)

In addition, the convergence properties (1.18) are still valid for all $t\in [0,T]$.

As before, the minimal existence time $T>0$ depends only on the parameters from (1.7) and on $\alpha$, $\Vert u^0\Vert _{H_D^{2+2\alpha }}$, and $\kappa$. It can be taken arbitrarily large provided $\lambda$ and $\Vert u^0\Vert _{H_D^{2+2\alpha }}$ are sufficiently small. The proof of Theorem 1.6 is similar to that of Theorem 1.4 and will thus be omitted. We just mention that the local well-posedness of (1.1)–(1.5) for $\gamma =0$ and (1.19)–(1.21) is shown in [20] and [19], respectively. Qualitative results on the behavior of solutions to (1.19)–(1.21) may be found in [26, 27].

2 The rescaled Laplace equation

This section is devoted to the study of the stability of solutions to the elliptic problem (1.1)–(1.2) as $\varepsilon \rightarrow 0$ when the function u belongs to a suitable class. In fact, given a function $v\in S_q^s(\kappa )$ for some suitably chosen parameters $q>1$, $s\ge 1$, and $\kappa \in (0,1)$, we carefully study the behavior of the solution $\psi _v$ to (1.11) as $\varepsilon \rightarrow 0$, paying special attention to the dependence upon v with the aim of deriving eventually bounds that only depend on q, s, and $\kappa$. As already mentioned such an analysis has already been performed in [10, 18] for $q>2$, $s=2$, and $\kappa \in (0,1)$, but it is not applicable for the proof of Theorem 1.3 for which we only have an $H^2$-estimate as a starting point. However, as we shall see below, bounds on $\psi _v$ can be derived when $q=2$ and $s\in (3/2,2)$.

Proposition 2.1

Consider $s\in (3/2,2)$, $\nu \in (2-s,1/2)$, $\sigma \in [0,1/2)$, and $\kappa \in (0,1)$. There is a positive constant $C_{sg} = C_{sg}(s,\nu ,\sigma ,\kappa )$ such that, given $\varepsilon \in (0,1)$ and $v\in S_2^s(\kappa )$, the corresponding solution $\psi _v$ to (1.11) satisfies

$$\begin{aligned}&\displaystyle \Vert \psi _v - b_v\Vert _{L_2(\Omega (v))} + \left\| \partial _z\psi _v - \frac{1}{1+v} \right\| _{L_2(\Omega (v))} \le C_{sg}\ \varepsilon , \end{aligned}$$

(2.1)

$$\begin{aligned}&\displaystyle \left\| \partial _x \partial _z\psi _v + \frac{\partial _x v}{(1+v)^2} \right\| _{L_2(\Omega (v))} + \left\| g_\varepsilon (v) - \frac{1}{(1+v)^2} \right\| _{H^\sigma } \le C_{sg}\ \varepsilon ^{(1-2\nu )/(3-2\nu )}, \end{aligned}$$

(2.2)

$$\begin{aligned}&\displaystyle \left\| \partial _z^2\psi _v \right\| _{L_2(\Omega (v))} \le C_{sg}\ \varepsilon ^{4(1-\nu )/(3-2\nu )}, \end{aligned}$$

(2.3)

where

$$\begin{aligned} g_\varepsilon (v)(x) := \varepsilon ^2 |\partial _x\psi _v(x,v(x))|^2 + |\partial _z \psi _v(x,v(x))|^2 \;\;\;\text { and }\;\;\; b_v(x,z) := \frac{1+z}{1+v(x)} \end{aligned}$$

for $(x,z)\in \Omega (v)$.

The remainder of this section is devoted to the proof of Proposition 2.1.

Let $s\in (3/2,2]$, $\kappa \in (0,1)$, and let $v\in S_2^s(\kappa )$ be fixed. We denote the corresponding solution to (1.11) by $\psi =\psi _v$ and recall that $\psi \in H^s(\Omega (v))$ by [21, Corollary 4.2]. As in previous works, to handle the dependence of the domain on v, we map the domain $\Omega (v)$ onto the rectangle $\Omega := (-1,1)\times (0,1)$ with the help of the transformation

$$\begin{aligned} T_v(x,z) := \left( x , \frac{1+z}{1+v(x)} \right) , \qquad (x,v)\in \Omega (v). \end{aligned}$$

We then define

$$\begin{aligned} \Phi (x,\eta ) = \Phi _v(x,\eta ) := \psi \circ T_v^{-1}(x,\eta ) - \eta = \psi (x,-1+\eta (1+v(x))) - \eta , \end{aligned}$$

(2.4)

for $(x,\eta )\in \Omega$, and recall that $\Phi \in H^{\sigma }(\Omega )$ for each $\sigma <s$ and $\partial _\eta \Phi \in H^1(\Omega )$ according to [21, Proposition 4.1]. As a consequence of (1.11) we realize that $\Phi$ solves the Dirichlet problem

$$\begin{aligned} \mathcal {L}_v \Phi = f_v \;\;\text { in }\;\;\Omega , \qquad \Phi =0 \;\;\text { on }\;\; \partial \Omega , \end{aligned}$$

(2.5)

the operator $\mathcal {L}_v$ and the function $f_v$ being given by

$$\begin{aligned} \begin{aligned} \mathcal {L}_v w&:= \varepsilon ^2 \partial _x^2 w - 2\varepsilon ^2\eta V(x)\ \partial _x \partial _\eta w + \left[ \frac{1}{(1+v(x))^2} +\varepsilon ^2\eta ^2 V(x)^2 \right] \partial _\eta ^2 w \\&\qquad +\, \varepsilon ^2 \eta \ \left( V(x)^2 - \partial _x V(x) \right) \partial _\eta w \end{aligned} \end{aligned}$$

(2.6)

and

$$\begin{aligned} f_v(x,\eta ) := \varepsilon ^2 \eta \ \left( \partial _x V(x) - V(x)^2 \right) , \end{aligned}$$

(2.7)

with

$$\begin{aligned} V := \partial _x \ln {(1+v)} = \partial _x v/(1+v). \end{aligned}$$

(2.8)

We now prove some useful identities involving $\Phi$.

Lemma 2.2

We set $\Gamma := \partial _\eta \Phi (.,1) \in H^{1/2}(I)$. Then $V \in H^{s-1}(I)$ and $\Gamma ^2\in H^{2-s}(I)$ with

$$\begin{aligned} \varepsilon ^2 \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta \Phi }{1+v} \right\| _{L_2(\Omega )}^2&= \varepsilon ^2 \int _\Omega V (\Phi +\eta ) \left( \partial _x \Phi - \eta V \partial _\eta \Phi \right) \ \mathrm {d}(x,\eta ) \nonumber \\&\qquad - \,\varepsilon ^2 \int _\Omega \eta V^2 \Phi \ \mathrm {d}(x,\eta ), \end{aligned}$$

(2.9)

and

$$\begin{aligned} Q^2 := \varepsilon ^2 \left\| \partial _x\partial _\eta \Phi - \eta V \partial _\eta ^2 \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta ^2 \Phi }{1+v} \right\| _{L_2(\Omega )}^2 = \varepsilon ^2 \left( \mathcal {R}_1^* +\mathcal {R}_2^* \right) , \end{aligned}$$

(2.10)

where

$$\begin{aligned} \mathcal {R}_1^*&:= \frac{1}{2}\, \langle \partial _x V , \Gamma ^2 \rangle _{H^{s-2},H^{2-s}} + \int _\Omega V \left( \partial _x\partial _\eta \Phi - \eta V \partial _\eta ^2\Phi \right) \partial _\eta \Phi \ \mathrm {d}(x,\eta ), \end{aligned}$$

(2.11)

$$\begin{aligned} \mathcal {R}_2^*&:= \langle \partial _x V , \Gamma \rangle _{H^{s-2},H^{2-s}} - \int _\Omega \eta V^2 \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ). \end{aligned}$$

(2.12)

Proof

It first follows from the continuity of pointwise multiplication (see [1, Theorem 4.1 & Remark 4.2(d)])

$$\begin{aligned} H^{1/2}(I) \cdot H^{1/2}(I) \longrightarrow H^{\sigma }(I), \qquad 0 \le \sigma < 1/2, \end{aligned}$$

that $\Gamma ^2$ belongs to $H^\sigma (I)$ for all $\sigma \in [0,1/2)$ and in particular to $H^{2-s}(I)$.

Step 1 To prove (2.9) and (2.10) we first consider the case where v is more regular, namely we assume that $v\in H^3(I)$ in addition to being in $S_2^s(\kappa )$, so that $\Phi \in H^2(\Omega )$ by [10, Proposition 5]. We then multiply (2.5) by $\Phi$ and integrate over $\Omega$. Using Green’s formula and the boundary condition $\Phi =0$ on $\partial \Omega$, we obtain

$$\begin{aligned} \varepsilon ^2 \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta \Phi }{1+v} \right\| _{L_2(\Omega )}^2 = \varepsilon ^2 \int _\Omega \eta (V^2 - \partial _x V) \Phi \left( 1 - \partial _\eta \Phi \right) \ \mathrm {d}(x,\eta ). \end{aligned}$$

Moreover,

$$\begin{aligned} - \int _\Omega \eta \Phi \partial _x V\ \mathrm {d}(x,\eta ) = \int _\Omega \eta V \partial _x \Phi \ \mathrm {d}(x,\eta ) \end{aligned}$$

and, employing Green’s formula twice,

$$\begin{aligned} \int _\Omega \eta \Phi \partial _\eta \Phi \ \partial _x V\ \mathrm {d}(x,\eta ) = -\frac{1}{2} \int _\Omega \Phi ^2 \partial _x V\ \mathrm {d}(x,\eta ) = \int _\Omega V \Phi \partial _x \Phi \ \mathrm {d}(x,\eta ). \end{aligned}$$

Combining the above identities leads us to

$$\begin{aligned} \varepsilon ^2 \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta \Phi }{1+v} \right\| _{L_2(\Omega )}^2&= \varepsilon ^2 \int _\Omega V \Phi \left( \partial _x \Phi - \eta V \partial _\eta \Phi \right) \ \mathrm {d}(x,\eta ) \\&\qquad +\, \varepsilon ^2 \int _\Omega \eta V \left( \partial _x \Phi + V \Phi \right) \ \mathrm {d}(x,\eta ) . \end{aligned}$$

We finally use once more the homogeneous Dirichlet boundary conditions of $\Phi$ and Green’s formula to prove that

$$\begin{aligned} \int _\Omega \eta V^2 \Phi \ \mathrm {d}(x,\eta )&= 2 \int _\Omega \eta V^2 \Phi \ \mathrm {d}(x,\eta ) - \int _\Omega \eta V^2 \Phi \ \mathrm {d}(x,\eta ) \\&= - \int _\Omega \eta ^2 V^2 \partial _\eta \Phi \ \mathrm {d}(x,\eta ) -\int _\Omega \eta V^2 \Phi \ \mathrm {d}(x,\eta ) \end{aligned}$$

and complete the proof of (2.9).

We next infer from [21, Eqs. (4.14)–(4.16)] that

$$\begin{aligned} Q^2 = \varepsilon ^2 \left\| \partial _x\partial _\eta \Phi - \eta V \partial _\eta ^2 \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta ^2 \Phi }{1+v} \right\| _{L_2(\Omega )}^2 = \varepsilon ^2 \left( \mathcal {R}_1 +\mathcal {R}_2 \right) , \end{aligned}$$

with

$$\begin{aligned} \mathcal {R}_1&:= \int _\Omega \eta \left( \partial _x V - V^2 \right) \partial _\eta \Phi \ \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ), \\ \mathcal {R}_2&:= \int _\Omega \eta \left( \partial _x V - V^2 \right) \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ). \end{aligned}$$

Arguing as in the derivation of [21, Eq. (4.17)], a first use of Green’s formula gives

$$\begin{aligned} \mathcal {R}_1&= \frac{1}{2} \int _\Omega \eta \partial _x V \partial _\eta \left[ (\partial _\eta \Phi )^2 \right] \ \mathrm {d}(x,\eta ) - \int _\Omega \eta V^2 \partial _\eta \Phi \partial _\eta ^2 \Phi \ \mathrm {d}(x,\eta ) \\&= \frac{1}{2} \int _{-1}^1 \Gamma ^2 \partial _x V\ \mathrm {d}x - \frac{1}{2} \int _\Omega \partial _x V (\partial _\eta \Phi )^2\ \mathrm {d}(x,\eta ) - \int _\Omega \eta V^2 \partial _\eta \Phi \partial _\eta ^2 \Phi \ \mathrm {d}(x,\eta ) . \end{aligned}$$

Since the homogeneous Dirichlet boundary conditions on $\Phi$ entail that $\partial _\eta \Phi (\pm 1,\eta )=0$ for $\eta \in (0,1)$, we use once more Green’s formula to transform the second integral on the right-hand side of the above identity and thus obtain

$$\begin{aligned} \mathcal {R}_1&= \frac{1}{2} \int _{-1}^1 \Gamma ^2 \partial _x V\ \mathrm {d}x + \int _\Omega V \partial _\eta \Phi \partial _x\partial _\eta \Phi \ \mathrm {d}(x,\eta ) - \int _\Omega \eta V^2 \partial _\eta \Phi \partial _\eta ^2 \Phi \ \mathrm {d}(x,\eta ). \end{aligned}$$

Therefore $\mathcal {R}_1$ coincides with $\mathcal {R}_1^*$. Finally, note that

$$\begin{aligned} \mathcal {R}_2&= \int _\Omega \partial _x V\ \partial _\eta \left( \eta \partial _\eta \Phi \right) \ \mathrm {d}(x,\eta ) - \int _\Omega \partial _x V \partial _\eta \Phi \ \mathrm {d}(x,\eta ) - \int _\Omega \eta V^2 \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ) \\&= \int _{-1}^1 \partial _x V \Big [ \eta \partial _\eta \Phi \Big ]_{\eta =0}^{\eta =1}\ \mathrm {d}x - \int _{-1}^1 \partial _x V \left[ \Phi (x,1)-\Phi (x,0) \right] \ \mathrm {d}x - \int _\Omega \eta V^2 \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ) \\&= \int _{-1}^1 \Gamma \partial _x V\ \mathrm {d}x - \int _\Omega \eta V^2 \partial _\eta ^2\Phi \ \mathrm {d}(x,\eta ), \end{aligned}$$

so that $\mathcal {R}_2 = \mathcal {R}_2^*$ as claimed.

Step 2 Finally, extending the validity of (2.9) and (2.10) to functions v belonging only to $S_2^s(\kappa )$ is done by an approximation argument as in [21, Section 4] to which we refer. $\square$

Proof of Proposition 2.1

On the one hand, one easily checks that the functions $\eta \mapsto -\eta$ and $\eta \mapsto 1-\eta$ solve (2.5) in $\Omega$ with $-\eta \le \Phi (x,\eta ) = 0 \le 1-\eta$ for all $(x,\eta )\in \partial \Omega$. We then deduce from the comparison principle that

$$\begin{aligned} - \eta \le \Phi (x,\eta ) \le 1 - \eta , \qquad (x,\eta )\in \Omega . \end{aligned}$$

(2.13)

On the other hand, $H^s(I)$ is continuously embedded in $C^1([-1,1])$ and $H^{s-1}(I)$ is an algebra since $s>3/2$. Thus, as $v\in S_2^s(\kappa )$, there is $c_1(\kappa )>0$ depending only on s and $\kappa$ such that

$$\begin{aligned} \Vert v\Vert _{C^1([-1,1])} + \Vert V \Vert _{\infty } + \Vert V\Vert _{H^{s-1}} \le c_1(\kappa ). \end{aligned}$$

(2.14)

It now follows from (2.9), (2.13), (2.14), and the Cauchy–Schwarz inequality that

$$\begin{aligned}&\varepsilon ^2 \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta \Phi }{1+v} \right\| _{L_2(\Omega )}^2 \\&\qquad \le \frac{\varepsilon ^2}{2} \int _\Omega \left( \partial _x \Phi - \eta V \partial _\eta \Phi \right) ^2 \ \mathrm {d}(x,\eta ) + \frac{\varepsilon ^2}{2} \int _\Omega V^2 \left[ (\Phi +\eta )^2 - 2 \eta \Phi \right] \ \mathrm {d}(x,\eta ) \\&\qquad \le \frac{\varepsilon ^2}{2} \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \varepsilon ^2 \left\| V \right\| _2^2 \\&\qquad \le \frac{\varepsilon ^2}{2} \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + c_1(\kappa )^2 \varepsilon ^2. \end{aligned}$$

We have thus shown that

$$\begin{aligned} \varepsilon ^2 \left\| \partial _x \Phi - \eta V \partial _\eta \Phi \right\| _{L_2(\Omega )}^2 + \left\| \frac{\partial _\eta \Phi }{1+v} \right\| _{L_2(\Omega )}^2 \le c(\kappa ) \varepsilon ^2. \end{aligned}$$

(2.15)

We next estimate $\mathcal {R}_1^*$. Fix $\nu \in (2-s,1/2)$. Then continuity of pointwise multiplication (see [1, Theorem 4.1 & Remark 4.2(d)])

$$\begin{aligned} H^{1/2}(I) \cdot H^\nu (I) \longrightarrow H^{2-s}(I) \end{aligned}$$

and (2.14) imply that

$$\begin{aligned}&\left| \langle \partial _x V , \Gamma ^2 \rangle _{H^{s-2},H^{2-s}} \right| \le \Vert \partial _x V \Vert _{H^{s-2}} \Vert \Gamma ^2\Vert _{H^{2-s}}\\&\quad \le c \Vert V\Vert _{H^{s-1}}\ \Vert \Gamma \Vert _{H^{1/2}} \Vert \Gamma \Vert _{H^\nu } \le c(\kappa ) \Vert \Gamma \Vert _{H^{1/2}} \Vert \Gamma \Vert _{H^\nu }. \end{aligned}$$

We next use the continuity of the trace from in for all and the fact that the complex interpolation space coincides with (up to equivalent norms) to estimate and obtain

(2.16)

Combining the above estimate with (2.16) for gives

Since and we further infer from (2.14), (2.15), and the definition (2.10) of Q that

hence

(2.17)

It also follows from (2.14), (2.15), the definition (2.10) of Q, and the Cauchy–Schwarz inequality that

Combining (2.11), (2.17), and the above estimate, we end up with

(2.18)

We next turn to and first deduce from (2.14) and (2.16) (with that

Arguing as in the proof of (2.17) leads us to

(2.19)

In addition, it follows from (2.14), the definition (2.10) of Q, and the Cauchy–Schwarz inequality

which gives, together with (2.19),

(2.20)

We now infer from (2.10), (2.18), and (2.20) that

and further deduce from Young’s inequality (using and ) that

Recalling (2.15) we have thus established that

(2.21)

(2.22)

with . Several consequences can be derived from (2.21)–(2.22). First, it readily follows from the boundary condition for all that

which, together with (2.21), gives

(2.23)

Another straightforward consequence of (2.14), (2.21), (2.22), and the continuity (2.16) of the trace operator is

(2.24)

Now, owing to the relationship (2.4) between and , we realize that, for and ,

Combining these formulas with (2.21)–(2.23) leads to (2.1), the first part of (2.2), and (2.3). As for the second part of (2.2) we first recall that, thanks to (1.11), there holds for from which we deduce that , . Thus

Given , continuity of pointwise multiplication (see [1, Theorem 4.1 & Remark 4.2(d)])

and (2.14) entail that

Therefore, thanks to (2.24),

which completes the proof of (2.2). $\square$

Remark 2.3

As mentioned before, estimates similar to (2.21) and (2.22) have already been derived in [10, 18] when v is more regular, namely for some and . The estimates obtained therein are better in the sense that they involve higher powers of . A rough explanation for this discrepancy is that we use several times Green’s formula in the proof of Proposition 2.1 to handle less regular functions v. This procedure somewhat mixes the x-derivative and -derivative which do not decay in the same way with respect to and results in the weaker estimates (2.21) and (2.22).

3 Small aspect ratio limit: the stationary case

Fix . Starting from the outcome of Proposition 1.2 the first step of the proof of Theorem 1.3 is to establish bounds on which do not depend on small enough.

Lemma 3.1

There are , , and such that, for ,

(3.1)

(3.2)

(3.3)

Proof

Setting

we recall that by [21, Proposition 2.7] and satisfies

(3.4)

see [21, Lemma 2.8]. Introducing , , it readily follows from the continuity of and (3.4) that continuously maps (0, 1) in . Consequently, there is $\theta _{\varrho ,\varepsilon }\in (0,1)$ such that $\mathcal {E}_{e,\varepsilon }(\theta _{\varrho ,\varepsilon }\phi )=\varrho$ and thus $\theta _{\varrho ,\varepsilon }\phi$ belongs to the set $\mathcal {A}_{\varrho ,\varepsilon }$ introduced in (1.13). The variational characterization (1.12) of $u_{\varrho ,\varepsilon }$ then entails

$$\begin{aligned} \mathcal {E}_m(u_{\varrho ,\varepsilon }) \le \mathcal {E}_m(\theta _{\varrho ,\varepsilon }\phi ) \le \mathcal {E}_m(\phi ). \end{aligned}$$

(3.5)

Furthermore, it follows from [20, Equation (6.12)] that there are $\varepsilon _0>0$ and $\Lambda >0$ such that (1.1)–(1.4) has no stationary solution for $\lambda >\Lambda$ and $\varepsilon \in (0,\varepsilon _0)$. Consequently, $\lambda _{\varrho ,\varepsilon }\in (0,\Lambda ]$ for $\varepsilon \in (0,\varepsilon _0)$ which, together with (3.5), gives (3.2).

Next, since $u_{\varrho ,\varepsilon }\in \mathcal {A}_{\varrho ,\varepsilon }$, we infer from Proposition 1.2, (3.2), and [21, Lemma 3.3] that

$$\begin{aligned} 0 = \max _{[-1,1]} u_{\varrho ,\varepsilon } \ge \min _{[-1,1]} u_{\varrho ,\varepsilon } \ge - 1 + \frac{\kappa _0^2}{\varrho (2\kappa _0 + \varrho )^2}, \end{aligned}$$

whence (3.1) by making $\kappa _0$ smaller, if necessary.

Finally, using again that $u_{\varrho ,\varepsilon }\in \mathcal {A}_{\varrho ,\varepsilon }$, we deduce from (3.1), (3.2), and (3.4) that

$$\begin{aligned} 0 \le \varrho - \int _{-1}^1 \frac{\mathrm {d}x}{1+u_{\varrho ,\varepsilon }} \le \varepsilon ^2 \int _{-1}^1 \frac{|\partial _x u_{\varrho ,\varepsilon }|^2}{1+u_{\varrho ,\varepsilon }}\ \mathrm {d}x \le \frac{\varepsilon ^2}{\kappa _0} \Vert \partial _x u_{\varrho ,\varepsilon }\Vert _2^2 \le \frac{\varepsilon ^2}{\kappa _0^3}, \end{aligned}$$

which completes the proof. $\square$

Thanks to the just derived bounds and the analysis performed in Sect. 2 we are in a position to prove Theorem 1.3.

Proof of Theorem 1.3

Owing to the compact embedding of $H^2(I)$ in $H^s(I)$, $s\in [1,2)$, and in $C^1([-1,1])$, we infer from Lemma 3.1 that there are a sequence $(\varepsilon _k)_{k\ge 1}$ with $\varepsilon _k\rightarrow 0$, $\lambda _\varrho \in [0,1/\kappa _0]$, and $u_\varrho \in H_D^2(I)$ such that

$$\begin{aligned} u_{\varrho ,\varepsilon _k} \rightharpoonup u_\varrho \;\;\text { in }\;\; H^2(I) \end{aligned}$$

(3.6)

and

$$\begin{aligned} \lim _{k\rightarrow \infty } \left\{ |\lambda _{\varrho ,\varepsilon _k} - \lambda _\varrho | + \Vert u_{\varrho ,\varepsilon _k} - u_\varrho \Vert _{H^s} + \Vert u_{\varrho ,\varepsilon _k} - u_\varrho \Vert _{C^1([-1,1])} \right\} = 0 \end{aligned}$$

(3.7)

for any $s\in [1,2)$. It readily follows from (3.1), (3.3), and (3.7) that

$$\begin{aligned} -1 + \kappa _0 \le u_\varrho (x)\le 0, \quad x\in I,\;\;\text { and }\;\; \int _{-1}^1 \frac{\mathrm {d}x}{1+u_\varrho (x)} = \varrho . \end{aligned}$$

(3.8)

Now, fix $s\in (3/2,2)$ and $\nu \in (2-s,1/2)$. According to (3.1) and (3.2), $u_{\varrho ,\varepsilon _k}$ belongs to $S_2^s(\kappa _0)$ and we infer from Proposition 2.1, (3.1), and (3.8) that, for $\sigma \in [0,1/2)$,

$$\begin{aligned} \left\| g_{\varepsilon _k}(u_{\varrho ,\varepsilon _k}) - \frac{1}{(1+u_\varrho )^2} \right\| _{H^{\sigma }}&\le \left\| g_{\varepsilon _k}(u_{\varrho ,\varepsilon _k}) - \frac{1}{(1+u_{\varrho ,\varepsilon _k})^2} \right\| _{H^{\sigma }}\\&\quad +\, \left\| \frac{1}{(1+u_{\varrho ,\varepsilon _k})^2} - \frac{1}{(1+u_\varrho )^2} \right\| _{H^{\sigma }} \\&\le c(\kappa _0) \varepsilon _k^{(1-2\nu )/(3-2\nu )} + \left\| \frac{(2+ u_\varrho + u_{\varrho ,\varepsilon _k}) (u_\varrho - u_{\varrho ,\varepsilon _k})}{(1+u_{\varrho ,\varepsilon _k})^2 (1+u_\varrho )^2} \right\| _{H^{\sigma }} \\&\le c(\kappa _0) \left[ \varepsilon _k^{(1-2\nu )/(3-2\nu )} + \Vert u_\varrho - u_{\varrho ,\varepsilon _k}\Vert _{H^1} \right] . \end{aligned}$$

We then deduce from (3.7) that

$$\begin{aligned} \lim _{k\rightarrow \infty } \left\| g_{\varepsilon _k}(u_{\varrho ,\varepsilon _k}) - \frac{1}{(1+u_\varrho )^2} \right\| _{H^{\sigma }} = 0 \end{aligned}$$

(3.9)

for all $\sigma \in [0,1/2)$. Thanks to (3.7) and (3.9) it is now straightforward to pass to the limit as $\varepsilon _k\rightarrow 0$ in the equation solved by $u_{\varrho ,\varepsilon _k}$ and conclude that $u_\varrho$ is a weak solution in $H_D^2(I)$ to (1.14). However, since the right-hand side of (1.14) belongs to $H^2_D(I)$, classical elliptic regularity results entail that $u_\varrho$ belongs to $H_D^4(I)$ and is a classical solution to (1.14).

To check the minimizing property of $u_\varrho$, we consider $v\in \mathcal {A}_{\varrho ,0}$ and observe that the function $\vartheta \mapsto \mathcal {E}_{e,\varepsilon }(\vartheta v)$ continuously maps (0, 1] onto $(2,\mathcal {E}_{e,\varepsilon }(v)]$, while $\varrho \le \mathcal {E}_{e,\varepsilon }(v)$ for $\varepsilon >0$ according to (3.4). Consequently, there is $\vartheta _{\varrho ,\varepsilon }\in (0,1]$ such that $\mathcal {E}_{e,\varepsilon }(\vartheta _{\varrho ,\varepsilon } v)=\varrho$, that is, $\vartheta _{\varrho ,\varepsilon } v\in \mathcal {A}_{\varrho ,\varepsilon }$. Recalling (1.12) gives

$$\begin{aligned} \mathcal {E}_m(u_{\varrho ,\varepsilon }) \le \mathcal {E}_m(\vartheta _{\varrho ,\varepsilon } v) \le \mathcal {E}_m(v). \end{aligned}$$

We then use the lower semicontinuity of $\mathcal {E}_m$ and (3.6) to conclude that $\mathcal {E}_m(u_\varrho ) \le \mathcal {E}_m(v)$. This inequality being valid for all $v\in \mathcal {A}_{\varrho ,0}$, we have thus proved that $u_\varrho$ is a minimizer of $\mathcal {E}_m$ on $\mathcal {A}_{\varrho ,0}$.

Finally the stated convergence properties of $\psi _{\varrho ,\varepsilon _k}$ readily follow from Proposition 2.1 and (3.7). $\square$

4 Small aspect ratio limit: the evolutionary case

We next focus on the vanishing aspect ratio limit for the evolution problem (1.1)–(1.5). As pointed out in the introduction, the proof of the parabolic case $\gamma =0$ stated in Theorem 1.6 is similar to (actually, simpler than) the hyperbolic case $\gamma >0$. We thus only treat the latter and may assume without loss of generality that $\gamma =1$. As a first step we recall the well-posedness of (1.1)–(1.5) which is established in [20]. Let $2\alpha \in (0,1/2)$ and consider the Hilbert space $\mathbb {H}_\alpha := H_D^{2+2\alpha }(I)\times H^{2\alpha }_D(I)$ and the (unbounded) linear operator $\mathbb {A}_\alpha$ on $\mathbb {H}_\alpha$ with domain $D(\mathbb {A}_\alpha ) := H^{4+2\alpha }_D(I)\times H_D^{2+2\alpha }(I)$ defined by

$$\begin{aligned} \mathbb {A}_\alpha \mathbf {w} := \begin{pmatrix} 0 &{} -w^1 \\ &{} \\ \beta \partial _x^4 w^0 - \tau \partial _x^2 w^0 &{} w^1 \end{pmatrix} , \quad \mathbf {w}=(w^0,w^1)\in D(\mathbb {A}_\alpha ). \end{aligned}$$

According to [2, Chapter V] and [3] the operator $\mathbb {A}_\alpha$ generates a strongly continuous group $(\mathrm{e}^{-t\mathbb {A}_\alpha })_{t\in \mathbb {R}}$ on $\mathbb {H}_\alpha$ and [15] entails that the damping term provides an exponential decay, that is, there are $M_\alpha >0$ and $\omega >0$ such that

$$\begin{aligned} \Vert \mathrm{e}^{-t\mathbb {A}_\alpha }\Vert _{\mathcal {L}(\mathbb {H}_\alpha )}\le M_\alpha \mathrm{e}^{-\omega t},\quad t\ge 0. \end{aligned}$$

(4.1)

Let $\varepsilon , \lambda >0$, $a\ge 0$, and fix $\kappa \in (0,1)$. Consider $\mathbf {u}^0 := (u^0,u^1)\in D(\mathbb {A}_\alpha )$ such that

$$\begin{aligned} u^0\in S_2^{2+2\alpha }(\kappa ) \;\;\;\text { and }\;\;\; \Vert u^1 \Vert _{H^{2\alpha }} < \frac{1}{\kappa }. \end{aligned}$$

(4.2)

By [20, Corollary 3.4] there is a unique solution $(u_\varepsilon ,\psi _\varepsilon )$ with $\psi _\varepsilon = \psi _{u_\varepsilon }$ to (1.1)–(1.5) defined on the maximal interval of existence $[0,T_\mathrm{m}^\varepsilon )$ satisfying

$$\begin{aligned} \mathbf {u}_\varepsilon := (u_\varepsilon ,\partial _t u_\varepsilon ) \in C([0,T_\mathrm{m}^\varepsilon ),\mathbb {H}_\alpha ) , \quad \min _{x\in [-1,1]} u_\varepsilon (t,x) > -1 ,\quad t\in [0,T_\mathrm{m}^\varepsilon ), \end{aligned}$$

and

$$\begin{aligned} \mathbf {u}_\varepsilon (t) = \mathrm{e}^{-t\mathbb {A}_\alpha } \mathbf {u}^0 +\int _0^t \mathrm{e}^{-(t-s)\mathbb {A}_\alpha }\, F_\varepsilon (\mathbf {u}_\varepsilon (s))\,\mathrm {d}s, \quad t\in [0,T_\mathrm{m}^\varepsilon ), \end{aligned}$$

(4.3)

where

$$\begin{aligned} F_\varepsilon (\mathbf {w}) := \begin{pmatrix} 0\\ -\lambda \, g_\varepsilon (w^0)+ a\Vert \partial _x w^0\Vert _2^2\,\partial _x^2 w^0 \end{pmatrix}, \qquad \mathbf {w} \in \mathbb {H}_\alpha . \end{aligned}$$

(4.4)

After this preparation let us begin the study of the behavior as $\varepsilon \rightarrow 0$ by noticing that, setting $\kappa _0:=\kappa /2 \in (0,1/2)$, the continuity properties of $\mathbf {u}_\varepsilon$ and (4.2) ensure that

$$\begin{aligned} T^\varepsilon := \sup {\left\{ t\in [0,T_\mathrm{m}^\varepsilon )\, :\, u_\varepsilon (s)\in S_2^{2+\alpha }(\kappa _0)\text { and } \Vert \partial _t u_\varepsilon (s)\Vert _{H^{2\alpha }} < 1/\kappa _0\,\text { for } s\in [0,t] \right\} } > 0. \end{aligned}$$

(4.5)

Thanks to the continuity of the embeddings of $H^{2+2\alpha }(I)$ in $W_q^2(I)$ for some $q>2$ and in $W_\infty ^1(I)$ (the latter with embedding constant denoted by $c_I$), the definition of $T^\varepsilon$ guarantees that there is a positive constant $K_1$ depending only on $\kappa$ and $\alpha$ such that, for all $\varepsilon \in (0,1)$,

$$\begin{aligned} -1 + \kappa _0 \le u_\varepsilon (t,x)&\le \frac{c_I}{\kappa _0}, \qquad (t,x)\in [0,T^\varepsilon )\times [-1,1], \end{aligned}$$

(4.6)

$$\begin{aligned} \Vert u_\varepsilon (t)\Vert _{W_q^2(I)} + \Vert u_\varepsilon (t)\Vert _{W_\infty ^1(I)}&\le K_1, \qquad t\in [0,T^\varepsilon ). \end{aligned}$$

(4.7)

The next step of the proof is to show that $T^\varepsilon$ (and thus also $T_\mathrm{m}^\varepsilon$) does not collapse to zero as $\varepsilon \rightarrow 0$, so that the solutions $(u_\varepsilon ,\psi _\varepsilon )_{\varepsilon \in (0,1)}$ have a common interval of existence.

Lemma 4.1

• (i) There is $T>0$ depending only on $\lambda$, $\alpha$, a, $\kappa$, and $\Vert \mathbf {u}^0\Vert _{D(\mathbb {A}_\alpha )}$ such that $T^\varepsilon \ge T$ for all $\varepsilon \in (0,1)$.

• (ii) There is $\delta >0$ depending only on $\alpha$ and $\kappa$ such that $T^\varepsilon =T_\mathrm{m}^\varepsilon =\infty$ for all $\varepsilon \in (0,1)$ provided $(\lambda , a , \Vert \mathbb {A}_\alpha \mathbf {u}^0\Vert _{\mathbb {H}_\alpha }) \in (0,\delta )\times [0,\delta )^2$.

Furthermore, there is $K_2>0$ depending only on $\kappa$ and $\alpha$ such that

$$\begin{aligned} \left\| g_\varepsilon (u_\varepsilon (t)) \right\| _{H^{2\alpha }} \le K_2, \qquad t\in [0,T^\varepsilon ). \end{aligned}$$

(4.8)

Proof

Let $\varepsilon \in (0,1)$ and $t\in [0,T^\varepsilon )$. Since $2\alpha \in (0,1/2)$ and $u_\varepsilon (t)\in S_2^{2+2\alpha }(\kappa _0) \subset S_2^{(3+4\alpha )/2}(\kappa _0)$ we infer from Proposition 2.1 (with $s=(3+4\alpha )/2$ and $\sigma =2\alpha$), (4.6), and (4.7) that

$$\begin{aligned} \left\| g_\varepsilon (u_\varepsilon (t)) \right\| _{H^{2\alpha }}&\le \left\| g_\varepsilon (u_\varepsilon (t)) - \frac{1}{(1+u_\varepsilon (t))^2} \right\| _{H^{2\alpha }} + \left\| \frac{1}{(1+u_\varepsilon (t))^2} \right\| _{H^{2\alpha }} \nonumber \\&\le C_{sg} + C \left\| \frac{1}{(1+u_\varepsilon (t))^2} \right\| _{W_\infty ^1} \le K_3 \end{aligned}$$

(4.9)

for some positive constant $K_3$ depending only $\kappa$ and $\alpha$, hence

$$\begin{aligned} \left\| F_\varepsilon (\mathbf {u}_\varepsilon (t)) \right\| _{\mathbb {H}_\alpha } \le (\lambda +a)\, K_3, \qquad t\in [0,T^\varepsilon ), \end{aligned}$$

(4.10)

with a possibly larger constant $K_3$, but still depending only on $\alpha$ and $\kappa$. Recalling that

$$\begin{aligned} \mathrm{e}^{-t\mathbb {A}_\alpha } \mathbf {u}^0 - \mathbf {u}^0 = -\int _0^t \mathrm{e}^{-s\mathbb {A}_\alpha } \mathbb {A}_\alpha \mathbf {u}^0\,\mathrm {d}s,\quad t\ge 0, \end{aligned}$$

it follows from (4.1), (4.3), and (4.10) that, for $t\in [0,T^\varepsilon )$,

$$\begin{aligned} \left\| \mathbf {u}_\varepsilon (t) - \mathbf {u}^0 \right\| _{\mathbb {H}_\alpha }&\le \left\| \mathrm{e}^{-t\mathbb {A}_\alpha } \mathbf {u}^0 - \mathbf {u}^0 \right\| _{\mathbb {H}_\alpha } + \int _0^t \left\| \mathrm{e}^{-(t-s)\mathbb {A}_\alpha } F_\varepsilon (\mathbf {u}_\varepsilon (s) ) \right\| _{\mathbb {H}_\alpha } \mathrm {d}s \nonumber \\&\le \frac{M_\alpha }{\omega }\,\left( 1-\mathrm{e}^{-\omega t}\right) \,\left( \Vert \mathbb {A}_\alpha \mathbf {u}^0\Vert _{\mathbb {H}_\alpha } + (\lambda +a) K_3 \right) . \end{aligned}$$

(4.11)

Combining this estimate with (4.2) further gives

$$\begin{aligned} u_\varepsilon (t,x)&= u^0(x) + u_\varepsilon (t,x) -u^0(x) \ge -1+\kappa - \Vert u_\varepsilon (t) - u^0 \Vert _\infty \nonumber \\&\ge -1 + \kappa - c_I\Vert \mathbf {u}_\varepsilon (t) -\mathbf {u}^0\Vert _{\mathbb {H}_\alpha } \nonumber \\&\ge -1+\kappa - \frac{M_\alpha c_I}{\omega }\, \left( 1-\mathrm{e}^{-\omega t}\right) \left( \Vert \mathbb {A}_\alpha \mathbf {u}^0\Vert _{\mathbb {H}_\alpha } +(\lambda +a) K_3 \right) , \end{aligned}$$

(4.12)

for $(t,x)\in [0,T^\varepsilon ) \times [-1,1]$.

On the one hand, if T is chosen such that

$$\begin{aligned} \frac{M_\alpha }{\omega }\, \left( 1-\mathrm{e}^{-\omega T}\right) \left( \Vert \mathbb {A}_\alpha \mathbf {u}^0\Vert _{\mathbb {H}_\alpha } +(\lambda +a) K_3 \right) < \min \left\{ \frac{2-\sqrt{2}}{\kappa } , \frac{\kappa }{2 c_I} \right\} , \end{aligned}$$

then we deduce from (4.11) and (4.12) that, for $t\in [0,T)$,

$$\begin{aligned} \Vert \mathbf {u}_\varepsilon (t)\Vert _{\mathbb {H}_\alpha } = \sqrt{\Vert u_\varepsilon (t)\Vert _{H^{2+2\alpha }}^2 + \Vert \partial _t u_\varepsilon (t)\Vert _{H^{2\alpha }}^2} < \frac{1}{\kappa _0}, \end{aligned}$$

while

$$\begin{aligned} u_\varepsilon (t,x) > - 1 + \kappa - \frac{\kappa }{2} = - 1+ \kappa _0, \qquad x\in [-1,1]. \end{aligned}$$

Consequently, $u_\varepsilon (t)\in S_2^{2+2\alpha }(\kappa _0)$ and $\Vert \partial _t u_\varepsilon (t)\Vert _{H^{2\alpha }} < 1/\kappa _0$ for all $t\in [0,T)$. We have thus shown that $T^\varepsilon \ge T$ and completed the proof of the first statement of Lemma 4.1.

On the other hand, let $\delta >0$ be such that

$$\begin{aligned} \frac{M_\alpha }{\omega }\, \left( 1 +2 K_3 \right) \delta < \min \left\{ \frac{2-\sqrt{2}}{\kappa } , \frac{\kappa }{2 c_I} \right\} , \end{aligned}$$

and assume that $\lambda \in (0,\delta )$, $a\in [0,\delta )$, and $\Vert \mathbb {A}_\alpha \mathbf {u}^0\Vert _{\mathbb {H}_\alpha } \in [0,\delta )$. Arguing as above we realize that, for all $t\in [0,T^\varepsilon )$, there hold

$$\begin{aligned} u_\varepsilon (t)\in S_2^{2+2\alpha }(\kappa _0) \;\;\;\text { and }\;\;\; \Vert \partial _t u_\varepsilon (t)\Vert _{H^{2\alpha }} < \frac{1}{\kappa _0}, \end{aligned}$$

which entails that $T^\varepsilon =\infty$ and also that $T_\mathrm{m}^\varepsilon =\infty$ as claimed in the second statement of Lemma 4.1. Recalling (4.9) completes the proof of Lemma 4.1. $\square$

Proof of Theorem 1.4

According to Lemma 4.1

$$\begin{aligned} T^0 := \inf _{\varepsilon \in (0,1)} T^\varepsilon > 0. \end{aligned}$$

Fix $T\in (0,T^0)$. Recalling the definition (4.5) of $T^\varepsilon$, we realize that

$$\begin{aligned} \text { the family }\; (u_\varepsilon )_{\varepsilon \in (0,1)} \;\text { is bounded in }\; C^1\left( [0,T],H^{2\alpha }(I)\right) \;\text { and in }\; C\left( [0,T],H^{2+2\alpha }(I)\right) . \end{aligned}$$

(4.13)

Also, since

$$\begin{aligned} \partial _t^2 u_\varepsilon = - \partial _t u_\varepsilon - \beta \partial _x^4 u_\varepsilon + \left( \tau + a \Vert \partial _x u_\varepsilon \Vert _2^2 \right) \partial _x^2 u_\varepsilon - \lambda g_\varepsilon (u_\varepsilon ) \;\;\text { a.e. in }\;\; (0,T)\times I \end{aligned}$$

according to (1.3) and [20, Corollary 3.4], we infer from (4.8) and (4.13) that

$$\begin{aligned} \text { the family }\; (\partial _t u_\varepsilon )_{\varepsilon \in (0,1)} \;\text { is bounded in }\; C^1\left( [0,T],H^{2\alpha -2}(I)\right) \;\text { and in }\; C\left( [0,T],H^{2\alpha }(I)\right) . \nonumber \\ \end{aligned}$$

(4.14)

Thus, given $2\alpha '\in (0,2\alpha )$, it follows from the compactness of the embeddings of $H^{2+2\alpha }(I)$ in $H^{2+2\alpha '}(I)$ and in $W_\infty ^1(I)$, that of $H^{2\alpha }(I)$ in $H^{2\alpha '}(I)$, and the Arzelà–Ascoli theorem that there are a function

$$\begin{aligned} u_0\in C\big ([0,T],H^{2+2\alpha '}(I)\big ) \cap C^1\big ([0,T],H^{2\alpha '}(I)\big ) \end{aligned}$$

and a sequence $(\varepsilon _k)_{k\ge 1}$ of positive real numbers with $\varepsilon _k \rightarrow 0$ such that

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty } \left\{ \sup _{t\in [0,T]} \Vert u_{\varepsilon _k}(t)-u_0(t)\Vert _{H^{2+2\alpha '}} + \sup _{t\in [0,T]} \Vert u _{\varepsilon _k}(t) - u_0(t) \Vert _{W_\infty ^1} \right\} = 0, \\&\lim _{k\rightarrow \infty } \sup _{t\in [0,T]} \Vert \partial _t u_{\varepsilon _k}(t)-\partial _t u_0(t)\Vert _{H^{2\alpha '}} = 0. \end{aligned} \end{aligned}$$

(4.15)

In particular,

$$\begin{aligned} \lim _{k\rightarrow \infty } \sup _{t\in [0,T]} \Vert \mathbf {u}_{\varepsilon _k}(t) - \mathbf {u}_0(t)\Vert _{\mathbb {H}_{\alpha '}} = 0 \;\;\text { with }\;\; \mathbf {u}_0:=(u_0,\partial _t u_0). \end{aligned}$$

(4.16)

A first consequence of (4.6) and (4.15) is

$$\begin{aligned} -1+\kappa _0\le u_0(t,x)\le \frac{c_I}{\kappa _0}, \qquad (t,x)\in [0,T]\times [-1,1]. \end{aligned}$$

(4.17)

It also readily follows from Proposition 2.1 (with $s=2-\alpha$, $\nu =2\alpha$, and $\sigma =2\alpha '$), (4.6), (4.15), and (4.17) that

$$\begin{aligned} \sup _{t\in [0,T]} \left\| g_{\varepsilon _k}(u_{\varepsilon _k}(t)) - \frac{1}{(1+u_0(t))^2} \right\| _{H^{2\alpha '}}&\le C_{sg}\ \varepsilon _k^{(1-4\alpha )/(3-4\alpha )} \\&\qquad +\, C \sup _{t\in [0,T]} \left\| \frac{1}{(1+u_{\varepsilon _k}(t))^2} - \frac{1}{(1+u_0(t))^2} \right\| _{W_\infty ^1} \end{aligned}$$

with right-hand side converging to zero as $k\rightarrow \infty$, hence

$$\begin{aligned} F_{\varepsilon _k}(\mathbf {u}_{\varepsilon _k}) \longrightarrow F_0(\mathbf {u}_0) := \begin{pmatrix} 0 \\ \displaystyle {- \frac{\lambda }{(1+u_0)^2} + a \Vert \partial _x u_0\Vert _2^2\, \partial _x^2 u_0 } \end{pmatrix} \quad \text { in }\quad C([0,T],\mathbb {H}_{\alpha '}). \end{aligned}$$

We are then in a position to pass to the limit as $\varepsilon _k\rightarrow 0$ in (4.3) and deduce from (4.16) and the above convergence that

$$\begin{aligned} \mathbf {u}_0(t) = \mathrm{e}^{-t\mathbb {A}_{\alpha '}} \mathbf {u}^0 +\int _0^t \mathrm{e}^{-(t-s)\mathbb {A}_{\alpha '}}\, F_0(\mathbf {u}_0(s))\,\mathrm {d}s , \quad t\in [0,T]. \end{aligned}$$

(4.18)

In other words, $\mathbf {u}_0$ is a mild solution in $\mathbb {H}_{\alpha '}$ to

$$\begin{aligned} \partial _t \mathbf {u}_0 +\mathbb {A}_{\alpha '}\mathbf {u}_0 =F_0( \mathbf {u}_0),\quad t\in (0,T],\qquad \mathbf {u}_0(0)=\mathbf {u}^0. \end{aligned}$$

(4.19)

Furthermore, $\mathbb {H}_{\alpha '}$ is reflexive and $F_0$ is a locally Lipschitz continuous map from $S_2^{2+2\alpha '}(\kappa _0)\times H^{2\alpha '}(I)$ into itself. Thus, since $\mathbf {u}^0\in D(\mathbb {A}_{\alpha '})$, we infer from [30, Theorem 6.1.6] that $\mathbf {u}_0$ is actually a strong solution to (4.19), that is, $\mathbf {u}_0\in L_1(0,T,D(\mathbb {A}_{\alpha '}))$ is differentiable a.e. with $\partial _t \mathbf {u}_0 = (\partial _t u_0, \partial _t^2 u_0)\in L_1(0,T,\mathbb {H}_{\alpha '})$. Therefore, $\mathbf {u}_0$ is a strong solution to (1.15)–(1.17). Finally, the stated convergence of the sequence $(\psi _{\varepsilon _k})_{k}$ readily follows from (2.1) and (4.16).

In fact, we have so far proved Theorem 1.3 only for a sequence $(\varepsilon _k)_{k\ge 1}$. However, the strong solution $u_0$ to (1.15)–(1.17) is unique, see [19] for a proof when $a=0$ which extends to the case $a>0$. This ensures that the whole family $(u_\varepsilon )_{\varepsilon \in (0,1)}$ converges, thereby completing the proof of Theorem 1.3.$\square$