Abstract
A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.
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1 Introduction
Electrostatically actuated micro-electromechanical systems (MEMS) are ubiquitous in today’s electronic devices. Idealized MEMS often consist of a fixed ground plate and an elastic membrane (or plate) that are close. Keeping the two components at different potential induces a Coulomb force deflecting the membrane. In the past two decades MEMS devices have been a highly active mathematical research focus, in particular due to their interesting qualitative behaviors with respect to pull-in instabilities (as a result from a possible touching of membrane and ground plate) and the inherent challenges related to local and global well-posedness of the corresponding models. We refer to [7] and the references therein for more details on MEMS models and their mathematical investigation in general.
In this paper, we consider a model introduced in [5, 6] arising as a small aspect ratio limit of equations governing an electrostatically actuated MEMS, where the narrow gap separating the membrane and the ground plate is filled with a rarefied gas. More precisely, we consider
where \(w=w(t,x)\) denotes the varying width of the gap and \(u=u(t,x)\) is the local pressure of the gas. The (sufficiently) smooth, bounded open subset \(\Omega \) of \(\mathbb {R}^n\) with \(n\in \{1,2\}\) represents the shape of the membrane and the ground plate. The constants \(a,b,\theta _1,\theta _2>0\) and \(\sigma \ge 0\) in (1.1b) and (1.1c) as well as the initial data \(u_0\), \(w_0\), and \(w_0^{\prime }\) in (1.1d) are given. The degeneracy of (1.1a) and the singularity in (1.1b) occurring for a vanishing gap width \(w(t,x)=0\) capture the instabilities related to a touchdown of the membrane on the ground plate. A detailed account of the modeling aspects is given in [5, 6] to which we refer.
In [5] the short-time existence of solutions to this MEMS model is established for the one-dimensional case \(n=1\). The approach chosen therein consists of solving first the hyperbolic equation for w (via a fixed point argument for a given, fixed u) and so reducing the coupled system (1.1) to a single fixed point equation for u which is then solved using parabolic semigroup theory. Instead of decoupling the system, we proceed differently and solve the mild formulation of (1.1) simultaneously for u and w, also relying on semigroup theory for semilinear hyperbolic and quasilinear parabolic equations described in [4, 8] respectively [2, 3]. A key ingredient for this is the observation that mild solutions to the hyperbolic equation (1.1b) enjoy a priori Hölder continuity properties with respect to time (and values in spaces of sufficiently high spatial regularity) that guarantee an evolution operator for the quasilinear parabolic equation (1.1a) in the sense of [3] (see also Remark 3.2 and Proposition A.1 below). In this way we provide a considerably shorter proof for local existence including also the case \(n=2\):
Theorem 1.1
Let \(r>0\) and \(u_0\in H^{2+r}(\Omega )\) with \(u_0>0\) in \(\Omega \) and \(u_0=\theta _1\) on \(\partial \Omega \). Let \(w_0\in H^2(\Omega )\cap C^1({\bar{\Omega }})\), \(w_0^{\prime }\in H^1(\Omega )\) with \(w_0>0\) in \(\Omega \) and \(w_0-\theta _2=w_0^{\prime }=0\) on \(\partial \Omega \). Then there is a unique solution
to (1.1) on some interval [0, T].
The initial values are compatible with the Dirichlet boundary conditions and the regularity of strong solutions at \(t=0\). In fact, the solution component u has even better regularity properties than stated in Theorem 1.1, see Remark 4.1. Moreover, the solution can be extended to a maximal solution on \([0,T_{max})\) existing as long as \(u(t)>0\) and \(w(t)>0\) in \({\bar{\Omega }}\) as well as the \(C^1\)-norm of (u, w) does not blow up. It is worth pointing out that a common feature in MEMS models [7] is the possible occurrence of finite time quenching \(\inf _{x\in \Omega }w(t,x) \rightarrow 0\) as \(t\rightarrow T_{max}\) preventing global existence of solutions.
A similar result as Theorem 1.1 can also be shown for a related fourth-order equation when the Laplacian \(\Delta \) in (1.1b) is replaced by \(-\Delta ^2+\Delta \) subject to pinned or clamped boundary conditions (see Theorem 5.1 below for details). This corresponds to a MEMS device involving an elastic plate instead of a membrane.
In fact, Theorem 1.1 (and Theorem 5.1) can be regarded as a special case of a more general setting including a quasilinear parabolic equation coupled to a semilinear wave equation of the form
where \(\mathcal {A}(u,w)\) are generators of analytic semigroups on a Banach space and \(-A\) is a generator of a cosine function on a Hilbert space (see Sect. 3 below for details).
In Sect. 2 we first identify (1.1) as a special case of (1.2), see (2.11) below. The latter is treated in Sect. 3 in an abstract functional analytic framework that is not restricted to the particular setting of (1.1). The main result of this research is Theorem 3.1 on the local well-posedness of (1.2) that is established using semigroup theory and then implies Theorem 1.1 for the particular case (1.1) as shown in Sect. 4. In Sect. 5 we briefly show how to apply Theorem 3.1 for the case of the fourth-order problem (5.1) including a bi-Laplacian.
2 Functional formulation of the problem
We demonstrate how to express the system (1.1) in the abstract form of problem (1.2) and list relevant properties of the functions involved.
Setting
and dropping then again the bars for simplicity, problem (1.1) is equivalent to
as long as \(w>-\theta _2\). In the following, we shall focus on (2.1) with initial data \(u_0\in H^{2+r}(\Omega )\), \(w_0\in H^2(\Omega )\cap C^1({\bar{\Omega }})\), and \(w_0^{\prime }\in H^1(\Omega )\) satisfying \(u_0+\theta _1>0\) and \(w_0+\theta _2>0\) in \({\bar{\Omega }}\) and
For technical reasons we handle the parabolic equation for u in an \(L_q\)-setting, denoting by \(H_q^s(\Omega )\) the scale of Bessel potential spaces [2] (that coincide in the Hilbert space case \(q=2\) with the Sobolev–Slobodeckii spaces \(H^s(\Omega )=H_2^s(\Omega )\)).
Since \(r>0\) and \(n\in \{1,2\}\) we may choose \(q\in (2,4)\) and \(\alpha , \beta , \mu \in (0,1)\) such that
Then, since \(q>2\ge n\) and \(\mu +\alpha >2-2/q\ge 1+n/2-n/q\), we have
and
with \(u_0+\theta _1\ge 2\varsigma >0\) and \(w_0+\theta _2\ge 2\varsigma >0\) in \({\bar{\Omega }}\) for some \(\varsigma >0\). Hence we may choose \(\varepsilon >0\) such that
Set
for \(u, w\in C^1({\bar{\Omega }})\) and v belonging to
[i.e. \( H_{q,D}^{2}(\Omega )\) incorporates homogeneous Dirichlet boundary conditions]. Using the embeddings (2.3) along with the fact that \(H_q^1(\Omega )\) is an algebra (since \(q>n\)) we obtain that the pointwise multiplications
and, choosing \(\epsilon >0\) small with \(\mu +\alpha -1-\epsilon > n/2-n/q\),
are continuous (see [1, Theorem 4.1] for the latter). Consequently, using the first multiplication for the divergence term and the second multiplication for the first-order term of \(\mathcal {A}(u,w)v\) in (2.5), we derive that
where \(C^{1-}\) means (locally) Lipschitz continuous. Moreover, since \(u_0, w_0\in C^1({\bar{\Omega }})\) satisfy (2.4), \(\mathcal {A}(u_0,w_0)\) is a second-order normally elliptic differential operator in divergence form with \(C^1\)-coefficients. Hence, it follows from e.g. [2, Theorem 4.1, Examples 4.3] that
where \(\mathcal {H}(E_1,E_0)\) denotes the set of generators of analytic semigroups on the Banach space \(E_0\) with domain \(E_1\). Also observe the identities
for complex interpolation [2, Theorem 5.2]. Setting formally
we may now reformulate (2.1a) subject to the initial and boundary conditions as quasilinear parabolic problem
in \(L_q(\Omega )\).
We focus next on the hyperbolic problem for w. Let \(H_{D}^{2\theta }(\Omega ):=H_{2,D}^{2\theta }(\Omega )\) be as in (2.7). Clearly, the Laplacian subject to homogeneous Dirichlet boundary conditions
is the generator of an analytic semigroup on \(L_2(\Omega )\). In fact,
Introducing
we can rewrite (2.1) now in the form
To handle the semilinear terms we define the open subsets
of \( H_{q,D}^{2\beta }(\Omega )\) respectively \(H_{D}^{1+\alpha }(\Omega )\times H_{D}^{\alpha }(\Omega )\), where \(\varepsilon >0\) stems from (2.4) and \(c_0>0\) denotes the norm of the embedding \(H_{D}^{1+\alpha }(\Omega )\hookrightarrow H_{D}^{\mu +\alpha }(\Omega )\). Recalling \(\alpha >n/2-n/q\) we find \(2\eta \in (0,1/q)\) and \(\epsilon >0\) small with \(\alpha -\epsilon -n/2>2\eta -n/q\), hence the embedding
Therefore, noticing that
due to (2.3) and (2.4b), it follows from the continuity of the multiplication (see [1, Theorem 4.1])
and (2.8) that
while (2.3), (2.4b), (2.10), and
(since \(q>2\) and \(\alpha <1/2\)) ensure that (of course, f is independent of the \(w^{\prime }\)-component)
To guarantee later on sufficient regularity of solutions it is worth noting that
whenever \({\hat{u}}\in C^1([0,T],L_2(\Omega ))\) and \({\hat{w}}\in C([0,T],H^{1+\alpha }(\Omega ))\cap C^1([0,T],H^{\alpha }(\Omega ))\) with \({\hat{w}}(t)+\theta _2\ge \varsigma \) in \(\Omega \). Also note from (2.2) tha \(H^{2+r}(\Omega )\hookrightarrow H_q^2(\Omega )\)
by the assumptions of Theorem 1.1 and (2.3). In fact, since \(u_0,w_0\in H^{2+r}(\Omega )\) and since \(H^{1+r}(\Omega )\) is an algebra, it readily follows from (2.5) that
since we may make \(\eta >0\) smaller to guarantee \(0<2\eta <\min \{r-n/2+n/q,1/q\}\). That is, the initial value \(u_0\) belongs to the domain of the \(H_{q,D}^{2\eta }(\Omega )\)-realization of the generator \(\mathcal {A}(u_0,w_0)\in \mathcal {H}(H_{q,D}^{2}(\Omega ),L_q(\Omega ))\).
The previous considerations ensure that problem (2.11) (and thus problem (1.1)) fits into the more general framework of Theorem 3.1 of the next section. We shall then continue from here in Sect. 4 and finish off the proof of Theorem 1.1 by applying Theorem 3.1.
3 Main theorem
As just pointed out above, Theorem 1.1 is a special case of a more general setting: Consider
where \(\mathcal {A}(u,w)\in \mathcal {L}(E_1,E_0)\) for some continuously and densely injected Banach couple \(E_1\hookrightarrow E_0\) is such that \(\mathcal {A}(u_0,w_0)\in \mathcal {H}(E_1,E_0)\), i.e. \(\mathcal {A}(u_0,w_0)\) with domain \(E_1\) generates an analytic semigroup on \(E_0\)], and
on a Hilbert space H with scalar product \((\cdot \vert \cdot )\). Here, positive operator means that \((Ax\vert x)\ge 0\) for \(x\in D(A)\). Let \(\sigma \ge 0\).
We formulate (3.1) as a coupled system of two first order equations relying on results for cosine functions [4, Sections 5.5 and 5.6], see also Appendix A. To this end note that (3.2) ensures that the powers \(A^z\) for \(z\in \mathbb {C}\) are well-defined closed operators (bounded for \(\textrm{Re}\,z\le 0\)). Consequently, the matrix operator
generates a strongly continuous semigroup \((e^{t\mathbb {A}})_{t\ge 0}\) on the Hilbert space
[in fact, it generates a group \((e^{t\mathbb {A}})_{t\in \mathbb {R}}\)]. Using the notion \(\textbf{w}=(w,w^{\prime })\) and setting
we can write (3.1b) as a semilinear hyperbolic Cauchy problem
in \(\mathbb {H}\). In fact, for greater flexibility (and to cope with the particular case (2.1)) we shift this problem to the interpolation space (for some \(\alpha \in [0,1)\))
where we recall (due to the Fourier series representation of \(A^{\alpha }\) or [9, Theorem 1.15.3]) that
Then, the \(\mathbb {H}_\alpha \)-realization \(\mathbb {A}_\alpha \) of \(\mathbb {A}\), given by
generates a strongly continuous semigroup \((e^{t\mathbb {A}_\alpha })_{t\ge 0}\) on \(\mathbb {H}_\alpha \) according to [3, Chapter V]. We shall then consider (3.1) in the equivalent form
In the following, let \((\cdot ,\cdot )_\theta \) be arbitrary admissible interpolation functors [3, I.Section 2.11] and set
Let \(O_\beta \subset E_\beta \) and \(\mathbf{\mathbb {O}}_\alpha \subset \mathbb {H}_\alpha \) be open sets for some \(\alpha ,\beta \in [0,1)\).
Theorem 3.1
Let \(\alpha ,\beta ,\mu \in [0,1)\) and \(\tau \in (\beta ,1]\). Consider initial values \(u_0\in O_\beta \cap E_\tau \) and \((w_0,w_0^{\prime })\in \mathbb {O}_\alpha \), let
and suppose (3.2). Moreover, assume that
(a) There is a unique mild solution
to the Cauchy problem (3.4) on some interval [0, T].
(b) If
or if
then \(u\in C^1((0,T],E_0)\cap C((0,T],E_1)\) is a strong solution to (3.1a).
(c) Let \(u_0\in E_1\). If (3.8) is satisfied and
or if (3.9) is satisfied, then
is a strict solution to (3.1a). In this case, if
or if
then
is a strong solution to (3.1b) provided that \((w_0,w_0^{\prime })\in \mathbb {O}_\alpha \cap D(A)\times D(A^{1/2})\).
We emphasize that one may rely on the regularity properties (3.7) and (3.11) when checking (3.12) or (3.13).
Proof
(i) It follows from (3.5) and [3, I.Theorem 1.3.1] that there are \(\varepsilon >0\), \(\kappa \ge 1\), and \(\omega >0\) such thatFootnote 1
and
for some constant \(c=c(u_0,w_0)>0\). Let \(\rho \in (0,\min \{\tau -\beta ,1-\mu \})\) and let
where \(i:D(A^{(\alpha +1)/2})\hookrightarrow D(A^{(\alpha +\mu )/2})\) is the natural inclusion. Writing \(z=(u,\textbf{w})\) with \(\textbf{w}=(w,w^{\prime })\) in the following and noticing that \(z_0=(u_0,\textbf{w}_0)\in O_\beta \times \mathbb {O}_\alpha \) with open subsets \(O_\beta \subset E_\beta \) and \(\mathbb {O}_\alpha \subset \mathbb {H}_\alpha \), it follows from (3.6) that we may assume without loss of generality that
for some constant \(c_1=c_1(z_0)>0\) and that
Given \(T\in (0,1)\) (to be specified later) we then introduce
where \(z_0=(u_0,\textbf{w}_0)\) and the notation \(\textbf{w}=(w,w^{\prime })\) is used throughout. Then \(\mathcal {V}_T\) is a complete metric space when equipped with the metric
Then, for \(z=(u,\textbf{w})\in \mathcal {V}_T\), we have by interpolation [see (3.3)]
and it thus follows from (3.15) and \(\rho <1-\mu \) that
for some constant \(r(u_0,\textbf{w}_0)>0\) (independent of \(z\in \mathcal {V}_T\)) and from (3.14) that
Now, [3, II.Corollary 4.4.2] and (3.18) imply that for each \(z=(u,\textbf{w})\in \mathcal {V}_T\), the operator \(\mathcal {A}(u,w)\) generates a parabolic evolution operator
on \(E_0\) with regularity subspace \(E_1\) and that we may apply the results of [3, II.Section 5]. Introduce now
for \(t\in [0,T]\) and \(z=(u,\textbf{w})\in \mathcal {V}_T\) recalling \(u_0\in O_\beta \cap E_\tau \) and \(\textbf{w}_0=(w_0,w_0^{\prime })\in \mathbb {H}_\alpha \). Then, mild solutions to (3.4) correspond to fixed points of the operator \(\Gamma =(\Gamma _1,\mathbf{\Gamma }_2)\).
(ii) We claim that \(\Gamma : \mathcal {V}_T\rightarrow \mathcal {V}_T\) is a contraction for \(T\in (0,1)\) sufficiently small. To see this, let \(z=(u,\textbf{w})\in \mathcal {V}_T\). It then follows from (3.19a), (3.16), (3.18), and [3, II.Theorem 5.3.1] that (for some \(c>0\) depending only on the parameters in (3.18))
(we recall that \(\rho <\tau -\beta \)) and thus, in particular,
for \(0\le s\le t\le T\) with \(T\in (0,1)\) sufficiently small. Moreover, we deduce from (3.16) that \(\textbf{F}(z)\in C([0,T],\mathbb {H}_\alpha )\) so that (3.19b), the assumption \(\textbf{w}_0\in \mathbb {H}_\alpha \), and Proposition A.1 entail that
with
and
for \(t\in [0,T]\) with \(T\in (0,1)\) sufficiently small since \((e^{t\mathbb {A}_\alpha })_{t\ge 0}\) is strongly continuous on \(\mathbb {H}_\alpha \). In particular, (3.20) implies
Consequently, \(\Gamma : \mathcal {V}_T\rightarrow \mathcal {V}_T\) is well-defined.
(iii) To check the Lipschitz property consider \(z=(u,\textbf{w})\in \mathcal {V}_T\) and \({\hat{z}}=({\hat{u}},{\hat{\textbf{w}}})\in \mathcal {V}_T\). Then (3.18) and [3, II.Theorem 5.2.1] imply
for \(t\in [0,T]\) with \(T\in (0,1)\) sufficiently small, where we used (3.15) and (3.16) for the last estimate. Moreover, due to (3.16) we have
for \(t\in [0,T]\) with \(T\in (0,1)\) sufficiently small. Consequently, taking into account (3.20), we deduce that
Banach’s fixed point theorem now ensures that there is a unique \(z=(u,\textbf{w})\in \mathcal {V}_T\) with \(z=\Gamma (z)\); that is, \((u,\textbf{w})\) is a mild solution to (3.1).
This proves statement (a) of Theorem 3.1.
(iv) Setting for \(t\in [0,T]\)
we see that u is a mild solution to the linear Cauchy problem
If (3.8) holds, then \({\tilde{g}}\in C([0,T], E_\eta )\) with \(\eta >0\) and we infer from [3, II.Theorem 1.2.2] that
is a strong solution to (3.1a). If (3.9) holds, then (3.16) and (3.17) imply \({\tilde{g}}\in C^\rho ([0,T], E_0)\) and we obtain again that u is a strong solution to (3.1a) with regularity properties as above in view of [3, II.Theorem 1.2.1].
This proves statement (b) of Theorem 3.1.
(v) Let now \(u_0\in E_1\). Then (3.9) or (3.10) both imply (3.11) due to [3, II.Theorem 1.2.1] respectively [3, II.Theorem 1.2.2]; that is,
is a strict solution to (3.1a). Set now \(\tilde{\textbf{F}}(t):=\textbf{F}(z(t))\) and note that \(\textbf{w}\) is a mild solution to the linear Cauchy problem
Then (3.12) implies \(\tilde{\textbf{F}}\in C([0,T],D(\mathbb {A}))\) while (3.13) yields \(\tilde{\textbf{F}}\in C^1([0,T],\mathbb {H})\). In either case we derive from Proposition A.1 that
is a strong solution to (3.1b) provided that \((w_0,w_0^{\prime })\in D(\mathbb {A})\). This proves statement (c) of Theorem 3.1. \(\square \)
Remark 3.2
It is worth emphasizing that one of the key ingredients of the proof of Theorem 3.1 is the observation that the first component w of a mild solution \(\textbf{w}=(w,w^{\prime })\) to the hyperbolic equation (3.4b) enjoys a Hölder continuity property with respect to time and values in spaces of sufficiently high spatial regularity [see (3.17)] as stated in Proposition A.1. In fact, this ensures the Hölder continuity of the operator \(t\mapsto \mathcal {A}(u(t),w(t))\) and thus that the associated evolution operator is well-defined according to [3, II.Corollary 4.4.2].
4 Proof of Theorem 1.1
We can now complete the proof of Theorem 1.1. From Sect. 2 we know that problem (1.1) is equivalent to (2.1) (recalling that (u, w) is identified with \((u-\theta _1,w-\theta _2)\)) which, in turn, is a special case of (3.1), see (2.11).
Choose \(q\in (2,4)\) and \(\alpha , \beta , \mu \in (0,1)\) as in (2.2) and \(\eta \in (0,1)\) as in (2.12) and (2.16). Setting
we notice from (2.7) that \(E_\theta \doteq H_{q,D}^{2\theta }(\Omega )\) (with complex interpolation functor) for \(2\theta \not =1/q\) while the operator \(A:= -\Delta \) in \(H:=L_2(\Omega )\) with domain \(H_D^{2}(\Omega )\) satisfies (3.2), see (2.9). Moreover, (2.7) and (3.3) imply
It now follows from (2.6)–(2.16) that problem (2.11) (and thus problem (1.1)) fits into the framework of Theorem 3.1 with assumptions (3.5), (3.6), (3.8), (3.10), and (3.13) satisfied (and \(\tau =1\)). Therefore, Theorem 3.1 implies that (2.1) admits a unique solution
Since \(q>2\), this proves Theorem 1.1. \(\square \)
Remark 4.1
As shown above, u belongs in fact to \(C^1([0,T],L_q(\Omega ))\cap C([0,T],H_{q}^2(\Omega ))\) for some \(q>2\). Parabolic smoothing effects ensure additional regularity properties. For instance, the regularity of (u, w) stated in Theorem 1.1 implies that u solves a (linear) equation of the form (3.21) with \(\tilde{\mathcal {A}}\in C^\rho ([0,T],\mathcal {H}(H_D^2(\Omega ),L_2(\Omega )))\) [see (3.18)] and \(\tilde{g}\in C^\rho ([0,T],L_2(\Omega ))\) for some \(\rho >0\) [see (2.8)]. The maximal regularity result of [3, I.Theorem 1.2.1] yields \(u\in C^\rho ((0,T], H^2(\Omega ))\cap C^{1+\rho }((0,T], L_2(\Omega ))\). Moreover, since \(\tilde{g}\in C([0,T],H_{q,D}^{2\eta }(\Omega ))\) by (2.12), a higher spatial regularity of u is derived from [3, I.Theorem 1.2.2] taking into account the regularity (2.16) of the initial value.
5 A fourth-order problem
As pointed out in the introduction, Theorem 3.1 also applies to certain fourth-order wave equations. Indeed, consider
with
where \(\delta \in \{0,1\}\). Equations (5.1) govern the gap width w and the gas pressure u for a MEMS device involving an elastic plate (instead of a membrane) of shape \(\Omega \), where \(\Omega \subset \mathbb {R}^n\) with \(n\in \{1,2\}\) is assumed to be a (sufficiently) smooth, bounded open set. The elastic plate is either clamped at its boundary (corresponding to \(\delta =0\)) or is hinged along its boundary so that it is free to rotate (corresponding to \(\delta =1\)). We assume that \(D_1>0\) and \(D_2\ge 0\) and that \(a,b,\theta _1,\theta _2>0\) and \(\sigma \ge 0\).
This MEMS model was introduced in [6] and the short-time existence of solutions was established for the pinned case \(\delta =1\) (for both cases \(n=1,2\)). We derive the result for pinned and clamped boundary conditions simultaneously as a consequence of Theorem 3.1 (the assumptions on the initial values are compatible with the regularity of the solution):
Theorem 5.1
Let \(r>0\) and \(u_0\in H^{2+r}(\Omega )\) with \(u_0>0\) in \(\Omega \) and \(u_0=\theta _1\) on \(\partial \Omega \). Let \((w_0,w_0^{\prime })\in H^4(\Omega )\times H^2(\Omega )\) with \(w_0>0\) in \(\Omega \) and \(w_0-\theta _2=\mathcal {B}w_0=0\) on \(\partial \Omega \) and \(w_0^{\prime }=(1-\delta )\mathcal {B}w_0^{\prime }=0\) on \(\partial \Omega \). Then there is a unique solution
to (5.1) on some interval [0, T].
Proof
The proof is very much the same as for Theorem 1.1 and we thus only sketch it and point out the new aspects. Arguing as in Sect. 2 by shifting u and w we may focus on
with \(\mathcal {A}\) defined in (2.5) and g and f in (2.8) respectively (2.10). The only difference now is that we consider the fourth-order operator
where its domain \(H_\mathcal {B}^{4}(\Omega )\) incorporates the homogeneous pinned (\(\delta =0\)) or clamped (\(\delta =1\)) boundary conditions. More generally, we set for \(s\in [0,4]\)
Then \(H_{\mathcal {B}}^{4\theta }(\Omega )\) coincides with the complex interpolation space
up to equivalent norms, see [9, Theorem 4.3.3]. Moreover, \(-A\in \mathcal {H}(H_\mathcal {B}^{4}(\Omega ),L_2(\Omega ))\) is the generator of an analytic semigroup on \(L_2(\Omega )\) with exponential decay, e.g., see [2, Remarks 4.2] or [8, Theorem 7.2.7]. In fact, we have again that
Moreover, (5.3) and (3.3) entail
As in Sect. 2 we choose \(q>2\) and \(\alpha , \beta , \mu \in (0,1)\) such that
and define the open subsets
of \( H_{q,D}^{2\beta }(\Omega )\) respectively \(H_{\mathcal {B}}^{2+2\alpha }(\Omega )\times H_{\mathcal {B}}^{2\alpha }(\Omega )\), where \(c_0>0\) denotes the norm of the embedding \(H_{\mathcal {B}}^{2+2\alpha }(\Omega )\hookrightarrow H_{\mathcal {B}}^{2\mu +2\alpha }(\Omega )\) and \(\varepsilon >0\) is such that, for some \(\varsigma >0\),
Exactly as in Sect. 2 we have
and
while
for some \(\eta >0\) small enough. Moreover,
whenever \({\hat{u}}\in C^1([0,T],L_2(\Omega ))\) and \({\hat{w}}\in C([0,T],H^{2+2\alpha }(\Omega ))\cap C^1([0,T],H^{2\alpha }(\Omega ))\) with \({\hat{w}}(t)+\theta _2\ge \varsigma \) in \(\Omega \). Finally, by premise of the theorem,
and, as before, since \(u_0,w_0\in H^{2+r}(\Omega )\),
making \(\eta >0\) smaller, if necessary, so that \(0<2\eta <\min \{r,1/q\}\). Setting
it then follows from (5.4)–(5.11) that problem (5.2) (and thus problem (5.1)) fits into the framework of Theorem 3.1 with assumptions (3.5), (3.6), (3.8), (3.10), and (3.13) satisfied (and \(\tau =1\)). Therefore, Theorem 3.1 implies Theorem 5.1. \(\square \)
Notes
The notation \(\mathcal {A}\in \mathcal {H}(E_1,E_0;\kappa ,\omega )\) means that \(\omega -\mathcal {A}\in \mathcal {L}is(E_1,E_0)\) and
$$\begin{aligned} \kappa ^{-1}\le \frac{\Vert (\lambda -\mathcal {A})x\Vert _{E_0}}{\vert \lambda \vert \,\Vert x\Vert _{E_0}+\Vert x\Vert _{E_1}}\le \kappa ,\quad x\in E_1\setminus \{0\},\quad \textrm{Re}\,\lambda \ge \omega , \end{aligned}$$see [3, I.Section 1.2]. Note that \(\mathcal {H}(E_1,E_0;\kappa ,\omega )\subset \mathcal {H}(E_1,E_0)\).
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Appendix A
Appendix A
Let H be a Hilbert space and \(A: D(A)\subset H\rightarrow H\) be a closed, densely defined, self-adjoint, positive operator with bounded and compact inverse, where its domain D(A) is equipped with the graph norm. Then the square root \(A^{1/2}\) (more generally: f(A) for \(f:(0,\infty )\rightarrow \mathbb {C}\)) is a well-defined closed operator on H by Fourier series representation.
Proposition A.1
Suppose (3.2) and let \(\sigma \in \mathbb {R}\). The matrix operator
generates a strongly continuous group on the Hilbert space \(\mathbb {H}:=D(A^{1/2})\times H\) (with an exponential decay if \(\sigma >0\)). Consider \(\mathbf{w_0}\in \mathbb {H}\) and
Then
satisfies \(\textbf{w}=(w,w^{\prime })\in C([0,T],\mathbb {H})\) and
If \(\mathbf{w_0}\in D(\mathbb {A})\) and \(\textbf{F}\in C^1([0,T],\mathbb {H})+C([0,T],D(\mathbb {A}))\), then
is a strong solution to
Proof
Let \(\sigma =0\). Then
and the mild formulation for \(\textbf{w}=(w,w^{\prime })\) yields explicit formulas for both w and \(w^{\prime }\) which readily imply that \(\textbf{w}=(w,w^{\prime })\in C([0,T],\mathbb {H})\) with
If \(\sigma \not =0\), then replace f by \(f-\sigma w^{\prime }\in C([0,T],H)\) to reduce the problem to the case \(\sigma =0\). The statement about strong solutions is classical. \(\square \)
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Walker, C. On a quasilinear parabolic–hyperbolic system arising in MEMS modeling. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01465-9
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DOI: https://doi.org/10.1007/s10231-024-01465-9