\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline
{\em Electronic Journal of Differential Equations},
Conference 13, 2005, pp. 35--47.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{35}

\begin{document}

\title[\hfilneg EJDE/Conf/13 \hfil semilinear evolution equations]
{A system of semilinear evolution equations with
homogeneous boundary conditions for thin plates coupled with
membranes}

\author[J. Hern\'andez\hfil EJDE/Conf/13 \hfilneg]
{Jairo Hern\'andez}  

\address{Universidad del Norte, Km 5 via a Puerto Colombia,
 Barranquilla, Colombia}
\email{jahernan@uninorte.edu.co}


\date{}
\thanks{Published May 30, 2005.}
\subjclass[2000]{74H20, 74H25, 74K15}
\keywords{Plates, membranes, coupled structures,  transmission problems,
\hfill\break\indent 
semilinear evolution equations}

\begin{abstract}
 In this work we consider a semilinear initial  boundary-value
problem modelling an elastic thin plate (in the context of
the so-called Kirchhoff-Love theory) coupled with an elastic
membrane, regarding homogeneous boundary conditions. By means of
the theory of strongly continuous semigroups of linear operators
applied to abstract semilinear initial valued problems \cite{Pazy},
we obtain existence and uniqueness of a weak solution which is
defined in a suitable way.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theo}{Theorem}[section]
\newtheorem{Def}[theo]{Definition}
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{remark}[theo]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this  work we consider a semilinear evolution problem which
we pose  as follows: Let $\Omega$ and $\Omega_m$ be two open bounded
connected subsets of $\mathbb{R}^2$ with sufficiently smooth boundary
$\partial\Omega$ and $\partial\Omega_m$  so that $\Omega_m\subset\subset\Omega$.
Let $\Omega_p:=\Omega\setminus\overline{\Omega}_m$ and
 $\Gamma_1:=\partial\Omega_m$. We decompose  $\partial\Omega$ in two
connected  parts $\Gamma_2$ and $\Gamma_3$ with  $\Gamma_2\cap\Gamma_3=\emptyset$, $\sigma_1(\Gamma_2)\neq 0$
 and $\sigma_1(\Gamma_3)\neq 0$, where  $\sigma_1$ is the surface measure
on $\partial\Omega$, induced
  by the Lebesgue measure on  $\mathbb{R}$  (see figure \ref{figura-1}).
Then we consider the system of partial differential equations
\begin{gather}
\begin{aligned}
&\rho_ph\frac{\partial^2u_{p}}{\partial
  t^2}(t,x) + \frac{h^{3}}{12} \sum_{\alpha,
  \beta \gamma, \theta=1}^{2}\frac{\partial^{2}}{\partial x_{\alpha}\partial x_{\beta}}
 \big(A_{\alpha\beta\gamma\theta}(x) \frac{\partial^{2}u_{p}}{\partial x_{\gamma}\partial
x_{\theta}}(t,x)\big)\\
&= f_{p}(t,x,u_{p}(t,x))\quad\text{in } ]0,T]\times\Omega_{p}
\end{aligned}
\label{semi-Platte-3}\\
 \rho_m\frac{\partial^2u_{m}}{\partial t^2}(t,x)-C\Delta
u_{m}(t,x)=f_{m}(t,x,u_{m}(t,x))\quad
   \text{in }]0,T]\times\Omega_{m}\,, \label{semi-Membrane-3}\\
\begin{aligned}
&\frac{h^3}{12}\sum_{\alpha, \beta, \gamma,
\theta=1}^2\nu_{\alpha}\frac{\partial}{\partial x_{\beta}} 
\big(A_{\alpha\beta\gamma\theta}\frac{\partial^2u_{p}}{\partial
x_{\gamma}\partial x_{\theta}}\big)
+\frac{h^3}{12}\frac{\partial}{\partial\vec{\tau}}
\Big( \sum_{\alpha, \beta, \gamma, \theta=1}^2 \nu_{\alpha}
\tau_{\beta}A_{\alpha\beta\gamma\theta}
 \frac{\partial^2u_{p}}{\partial x_{\gamma}\partial x_{\theta}}
\Big)\\
&=0\quad \text{on  }]0,T]\times\Gamma_{2}\,,
\end{aligned}\label{Intro-Rand2-3}\\
\begin{aligned}
&\frac{h^3}{12}\sum_{\alpha, \beta, \gamma,
\theta=1}^2\nu_{\alpha}\frac{\partial}{\partial x_{\beta}} \big(
  A_{\alpha\beta\gamma\theta}\frac{\partial^2u_{p}}{\partial x_{\gamma}
\partial x_{\theta}}\big)
+\frac{h^3}{12}\frac{\partial}{\partial\vec{\tau}}
\Big( \sum_{\alpha, \beta, \gamma, \theta=1}^2 
\nu_{\alpha}\tau_{\beta}A_{\alpha\beta\gamma\theta}
 \frac{\partial^2u_{p}}{\partial x_{\gamma}\partial x_{\theta}}\Big)\\
&+C\frac{\partial u_{m}}{\partial\vec{\nu}}
=0 \quad\text{on  }]0,T]\times\Gamma_{1}\,,
\end{aligned}\label{Intro-Rand1-3}\\
\sum_{\alpha, \beta, \gamma,
\theta=1}^2\nu_{\alpha}\nu_{\beta}A_{\alpha\beta\gamma\theta}\frac{\partial^2u_{p}}{\partial
x_{\gamma}\partial x_{\theta}}=0\quad
\text{on  }]0,T]\times(\partial\Omega_{p}\setminus\Gamma_{3}),
\label{Intro-Randp-Rand3-3}\\
u_{p}=\frac{\partial u_{p}}{\partial\vec{\nu}}=0\quad
  \text{on  }]0,T]\times\Gamma_{3},\label{Intro-Rand3-3}\\
  u_{p}=u_{m}\quad\text{on  }]0,T]\times\Gamma_{1}\,,\label{Intro-Stetigrand1-3}
\end{gather}
with the initial conditions
\begin{gather}
 u_{p}(0,\cdot)=g_{p}^{0}\quad\text{in }
 \Omega_{p},\label{Intro-up0-3}\\
  u_{m}(0,\cdot)=g_{m}^{0}\quad\text{in }
   \Omega_{m}, \label{Intro-um0-3}\\
  \frac{\partial u_{p}}{\partial t}(0,\cdot)=g_{p}^{1}\quad\text{in }
   \Omega_{p}, \label{Intro-derivada-up0-3}\\
   \frac{\partial u_{m}}{\partial t}(0,\cdot)=g_{m}^{1}\quad\text{in }
   \Omega_{m}. \label{Intro-derivada-um0-3}
\end{gather}
Equations (\ref{semi-Platte-3})-(\ref{Intro-derivada-um0-3}) describe the
vibrations of a structure which consists of a thin elastic
anisotropic plate (in the context of the so called Kirchhoff-Love
theory) with its middle surface occupying the domain $\Omega_p$,
coupled with a membrane occupying the domain $\Omega_m$ (see
figure \ref{figura-1}).

 It is supposed  that $\rho_p$ and $\rho_m$ are positive constants, where
$\rho_p$ (resp.  $\rho_m$ ) is the density of the middle surface of
the plate (resp. the membrane) and $h$ is the thickness of the
plate. The coefficients $A_{\alpha\beta\gamma\theta}$ depend on
the elastic modulus of the plate and are assumed as $C^{\infty}$
functions on $\overline{\Omega}_p$; they satisfy the symmetry
assumption
\begin{equation}\label{simetria-Aab-gamma-theta}
A_{\alpha\beta\gamma\theta}= A_{\beta\alpha\gamma\theta},\quad
A_{\alpha\beta\gamma\theta}=A_{\alpha\beta\theta\gamma},\quad
A_{\alpha\beta\gamma\theta}=A_{\gamma\theta\alpha\beta}
\end{equation}
and the coercivity hypothesis
\begin{equation}\label{condicion-coercitividad}
\sum_{\alpha,\beta,\gamma,\theta=1}^2
A_{\alpha\beta\gamma\theta}(x)\xi_{\gamma\theta}\xi_{\alpha\beta}\geq\rho\sum_{\alpha,
\beta =1}^2\xi_{\alpha\beta}^2
\end{equation}
for all $x\in\Omega_p$ and for all real matrices
$(  \xi_{\alpha\beta})_{2\times 2}$ with $\xi_{\alpha\beta}
=\xi_{\beta\alpha}$ for $\alpha,\beta\in\{1,2\}$, where
$\rho >0$ is a constant.
Moreover it is supposed that the plate is clamped on
$\Gamma_3$ (equation (\ref{Intro-Rand3-3})) and is free on
$\Gamma_2$ (see figure \ref{figura-1}).


\begin{figure}[ht]
\begin{center}
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\put(57.5,27){\line(2,0){10}} \put(57.5,29){\line(2,0){10}}
\put(57.5,31){\line(2,0){10}} \put(57.5,33){\line(2,0){10}}
\put(57.5,35){\line(2,0){10}} \put(57.5,37){\line(2,0){10}}
\put(57.5,39){\line(2,0){10}} \put(57.5,41){\line(2,0){10}}
\put(57.5,43){\line(2,0){10}} \put(57.5,45){\line(2,0){10}}
\put(57.5,47){\line(2,0){10}} \put(57.5,49){\line(2,0){10}}
\put(57.5,51){\line(2,0){10}} \put(57.5,53){\line(2,0){10}}
\put(56.5,55){\line(2,0){11}} \put(55.5,57){\line(2,0){12}}
\put(36,59){\line(2,0){31.5}} \put(36,61){\line(2,0){31.5}}
\put(36,63){\line(2,0){31.5}} \put(-17,65){\line(2,0){85}}
\put(-17,67){\line(2,0){85}} \put(-17,25){\line(2,0){9.5}}
\put(-17,27){\line(2,0){9.5}} \put(-17,29){\line(2,0){9.5}}
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\put(-17,59){\line(2,0){43}} \put(-17,61){\line(2,0){43}}
\put(-17,63){\line(2,0){43}} \thicklines \put(25,25){\oval(25,25)}
\put(24,27){$\Omega_m$} \put(24,46){$\Omega_p$} \thicklines
\put(25,25){\oval(65,65)[t]} \put(-12.5,10){$\Gamma_2$}
\put(30,60){$\Gamma_3$} \put(7,20){$\Gamma_1$}
\put(57.5,10){\vector(2,0){7}} \put(57.5,10){\vector(0,2){7}}
\put(67,10){$\vec{\nu}$} \put(59,17){$\vec{\tau}$}
\put(35,14.1){\vector(-1,1){4.5}} \put(35,14.1){\vector(-1,-1){4.5}}
\put(27,8){$\vec{\tau}$} \put(27,18){$\vec{\nu}$} \put(26,-13){}
\end{picture}
\end{center}
\caption{ $\overline{\Omega}_m$ (resp. $\overline{\Omega}_p$)
is occupied by the  membrane (resp. the middle surface of the Plate).
The Plate is clamped on  $\Gamma_3$.}\label{figura-1}
\end{figure}

 The vector $ \vec{\nu}=(\nu_1,\nu_2)$ is the unitary
outward normal to
 $\partial\Omega_p$ and $\tau=(\tau_1,\tau_2)=(-\nu_2,\nu_1)$ is
 the positive oriented unitary tangent vector.
 $C$ is a positive constant depending on the material forming the
membrane. $f_p$ (resp. $f_m$) is the pressure supported by the
plate (resp. the membrane) and depend on the transverse
displacement $u_p$ (resp. $u_m$) of the plate (resp. the
membrane). The initial conditions $g_p^0$ and $g_p^1$ (resp.
$g_m^0$ and $g_m^1$) are real functions defined on $\Omega_p$
(resp. $\Omega_m$).
The equations
(\ref{Intro-Rand1-3}) and (\ref{Intro-Stetigrand1-3}) are the
boundary conditions expressing the coupling between the plate and
the membrane.\\ \noindent We give the definition of weak
solution for our semilinear problem
(\ref{semi-Platte-3})-(\ref{Intro-derivada-um0-3}) and with help
of the theory of $C^0$-semigroups of linear operators we obtain a
result of existence and uniqueness for this type of solution.
For other works in the area of transmission problems and networks
we refer the reader to
\cite{Ali-tesis,Ali-89,Ali,Arango,Ciarlet-et-al,Hernandez,Hernandez-prep-1,
Jairo-tesis, Mercier, Nicaise-Sandig-I, Nicaise-Sandig-II}.


\section{Notation and mathematical preliminaries}
\label{preliminares-segundo-preprint}

In this section we shall present
the concepts and abstract framework that we need for the treatment
of our problem (\ref{semi-Platte-3})-(\ref{Intro-derivada-um0-3}).
We shall consider only real valued functions. Let $n$
a positive integer. For any multi-index
$\alpha=(\alpha_1,\dots,\alpha_2)$ (i.e.
$\alpha\in\mathbb{N}_{0}^{n}$, where $\mathbb{N}_{0}$ is the set
of all nonnegative integers), we write
$$
\partial^{\alpha}:=\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_1}
\dots\partial x_{n}^{\alpha_n}},\quad
\mathrm{where}\quad |\alpha|:=\alpha_1+\cdots+\alpha_n.
$$
Sometimes we write $\partial_i$ for
$\frac{\partial}{\partial x_i}$, $i=1,\dots , n$.
 For the rest of this section, let $\Omega$ be an open bounded
connected set in $\mathbb{R}^{n}$ with sufficiently smooth boundary.

 For any nonnegative integer $k$ let $C^{k}(\Omega)$
be the vector space consisting of all functions $\phi$ which,
together with all their partial derivatives
$\partial^{\alpha}\phi$ of orders $|\alpha|\leq k$, are
continuous in $\Omega$.  $C^{\infty}(\Omega)$ is the vector
space consisting of all functions $\phi$, such that $\phi\in
C^{k}(\Omega)$ for all nonnegative integer $k$.

 We write $C^{k}(\overline{\Omega})$ for the Banach space consisting
of all functions $\phi\in C^{k}(\Omega)$ for which
$\partial^{\alpha}\phi$ is bounded and uniformly continuous on
$\Omega$ for $|\alpha|\leq k$, with norm given by
$$
\|\phi\|_{_{C^{k}(\overline{\Omega})}}:= \max_{|\alpha|\leq
k } \sup_{x\in\Omega}|\partial^{\alpha}\phi(x)|.
$$
For a nonnegative integer $k$
and $1\leq p\leq\infty$ let $W^{k,p}(\Omega)$ be the usual
Sobolev space defined as
\begin{equation}\label{Sobolev_spaces-Definicion}
W^{k,p}(\Omega):=\{ u\in L^{p}(\Omega) ; \partial^{\alpha}u\in
L^{p}(\Omega) \mathrm{for all} \alpha\in\mathbb{N}_{0}^{n},
|\alpha|\leq k \},
\end{equation}
where $\partial^{\alpha}u$ is understood in distributional
(or weak) sense, with the usual norm
\begin{gather}
\|u\|_{k,p,\Omega}:=\big\{\sum_{|\alpha|\leq
k}\int_{\Omega}|\partial^{\alpha}u(x)|^pdx\big\}^{1/p}\quad\mathrm{if }
1\leq p<\infty,\\
\|u\|_{k,\infty,\Omega}:= \max_{|\alpha|\leq k}
\mathop{\rm ess\, sup}_{x\in\Omega} | \partial^{\alpha}u(x)|.
\end{gather}
As usual we shall write $H^k(\Omega):=W^{k,2}(\Omega)$.

\begin{lemma}\label{lemma-D-cerradura-Omega-denso-Wmp}
The set $\mathcal{D}(\overline{\Omega})$ of restrictions to
$\Omega$ of functions in $C_{c}^{\infty}(\mathbb{R}^{n})$ (i.e.
the set of all infinitely differentiable functions on
$\mathbb{R}^{n}$ with compact support) is dense in
$W^{k,p}(\Omega)$ for $1\leq p<\infty$.
\end{lemma}
For the proof of the above lemma, see  Adams \cite[theorem 3.18,]{Adams}.

\begin{lemma}\label{lemma-inmersion-Wmp-Lq}
If $kp=n$, then $W^{k,p}(\Omega)\hookrightarrow
L^{q}(\Omega)$ for $p\leq q<\infty$.
\end{lemma}
For the proof of the above lemma, see  Adams \cite[lemma 5.14]{Adams}.

\begin{lemma}\label{lemma-inmersion-Wmp-C-cerradura-Omega}
If $kp>n$, then $W^{k,p}(\Omega)\hookrightarrow
C^{0}(\overline{\Omega})$.
\end{lemma}

The proof of the above lemma can be found in Evans
\cite[sec. 5.6, Theorem 6]{Evans} and in Adams \cite[lemma 5.17]{Adams}.

\begin{lemma}\label{trazas}
Let $ 1\leq p<\infty$. Then there exists a  linear operator
\begin{equation}\label{Operador-traza-orden-0}
\gamma_0:W^{1,p}(\Omega)\to  L^p(\partial\Omega)
\end{equation}
such that
\begin{itemize}
\item[(i)] $\gamma_0u = u|_{_{\partial\Omega}}$ if $ u\in
  W^{1,p}(\Omega)\cap C(\overline{\Omega})$.
\item[(ii)] $\|\gamma_0u\|_{_{L^p(\partial\Omega)}}\leq
  c(p,\Omega)\|u\|_{1,p,\Omega}$ for each $u\in
  W^{1,p}(\Omega)$, where $c(p,\Omega)$ is a constant
  depending only on $p$ and $\Omega$.
\end{itemize}
\end{lemma}
For the proof of the above lemma, see Evans \cite[theorem 5.5.1]{Evans}.

\begin{remark}\label{nombre-gamma-cero} \rm
We call $\gamma_0u$ the trace of order zero of $u$ on $\partial\Omega$.
\end{remark}

\begin{Def}\label{traza-orden-1-definicion} \rm
Let $j,k\in\mathbb{N}$, $k>1$, $1\leq j\leq k-1$ and $u\in
W^{k,p}(\Omega)$. We define the trace of order $j$ of $u$ on $\partial\Omega$ by
\begin{equation}\label{Operador-traza-orden-1}
\gamma_{_{j}}u:=\sum_{|\alpha|=j}\frac{j!}{\alpha_1!\cdots\alpha_n!}\gamma_0(\partial^\alpha
u)\nu_1^{\alpha_1}\cdots\nu_n^{\alpha_n},
\end{equation}
where $ \vec{\nu}=(\nu_1,\dots,\nu_n)$ is the unit outward normal
along $\partial\Omega$.
\end{Def}

\begin{remark}\label{observacion-traza-orden-1} \rm
$\gamma_{_{j}}:W^{k,p}(\Omega)\to  L^p(\partial\Omega)$ is a
 linear operator with
\begin{itemize}
\item[(i)] $\gamma_{_{j}}u=\left.\frac{\partial^j
      u}{\partial\vec{\nu}^j}\right|_{_{\partial\Omega}}:=\sum_{|\alpha|=j}\frac{j!}{\alpha_1!\cdots\alpha_n!}\partial^\alpha
u|_{_{\partial\Omega}}\nu_1^{\alpha_1}\cdots\nu_n^{\alpha_n}$ for
$j=1,\dots,k-1$ if $
  u\in W^{k,p}(\Omega)\cap C^{k-1}(\overline{\Omega})$.
\item[(ii)] $\|\gamma_ju\|_{_{L^p(\partial\Omega)}}\leq
  c(k,p,\Omega)\|u\|_{k,p,\Omega}$ for each $u\in
  W^{k,p}(\Omega)$ and for all $ j=1,\dots,k-1$.
\end{itemize}
\end{remark}

Now for $j,k\in\mathbb{N}_{0}$, $0\leq j\leq k$, and
$1\leq p<\infty$ we define the the functional
given by
\begin{equation}\label{notacion-seminorma-2-Omega}
|u|_{_{j,p,\Omega}}:=\big\{
    \sum_{|\alpha|=j}\int_{\Omega}|\partial^\alpha
    u(x)|^pdx
  \big\}^{1/p},\quad u\in W^{k,p}(\Omega).
\end{equation}
Clearly,
$|u|_{0,p,\Omega}=\|u\|_{0,p,\Omega}=\|u\|_{_{L^{^{p}}(\Omega)}}$.
We have the following statement.

\begin{lemma}\label{Normas_equivalentes_en_Sobolev_Spaces}
The functional
$$
(\!(u)\!)_{_{k,p,\Omega}}=\left\{|u|_{_{k,p,\Omega}}^{p}
+ |u|_{_{0,p,\Omega}}^{p}\right\}^{1/p}
$$
is a norm on $W^{k,p}(\Omega)$, equivalent to the usual norm
$\|\cdot\|_{k,p,\Omega}$.
\end{lemma}
The proof of the above lemma can be found in
Adams \cite[corollary 4.16]{Adams}.

 We need some crucial results of the theory of semigroups of
linear operators in Banach spaces. We refer to Pazy \cite{Pazy} or
Dautray-Lions \cite{D-L5}, chapter XVII, with respect to this
theory.

 Let $V$ (resp. $H$) be a real  separable Hilbert space
with scalar product $(\cdot|\cdot)_{V}$ (resp.
$(\cdot|\cdot)_{H}$) and norm $\|\cdot\|_{V}$ (resp.
$\|\cdot\|_{H}$). We assume $V\hookrightarrow H$ and $V$ dense in
$H$.\\ Let $a(\cdot|\cdot):V\times V\to \mathbb{R}$ be a
continuous bilinear form, $V$-coercive with respect to $H$ i.e.,
there exists $\lambda_{0}\in\mathbb{R}$
and $c_{0}>0$ such that
\begin{equation}\label{V-coerciv}
  a(v|v)+\lambda_{0}\|v\|_{H}^{2}\geq c_{0}\|v\|_{V}^{2},\quad \forall
  v\in V.
\end{equation}
We put
\begin{equation}\label{Dominio-A-bilineal-forma}
D(\mathcal{A}):=\{u\in V; V\ni v\mapsto a(u|v)
\text{is  continuous for  the  topology  of}
H \}.
\end{equation}

\begin{theo}\label{A-Propiedades}
Let $\mathcal{A}:D(\mathcal{A})\subset H\to  H$ be the
operator given by $(\mathcal{A}u|v)_{H}=a(u|v)$  $\forall
u\in D(\mathcal{A})$  and  $\forall v\in V$. Then
$-\mathcal{A}$ is the infinitesimal generator of a  $C^{0}$-
semigroup $\{T(t)\}_{t\geq 0}$
  in $H$ which satisfies
$$
\|T(t)\|_{\mathcal{L}(H)}\leq e^{\lambda_{0}t}\quad  \forall t\geq 0\,.
$$
\end{theo}
For a proof of the above theorem, see Dautray-Lions
\cite[theorem XVII.3.3]{D-L5}.

 Now we assume furthermore that $a(\cdot|\cdot)$ is symmetrical
($a(u|v)=a(v|u)$ $\forall u, v\in V$). Let $ \mathcal{H}:=V\times H$.
$ \mathcal{H}$ equipped with the scalar product defined by
$(u|v)_{\mathcal{H}}:=a(u_{1}|v_{1})+(u_{2}|v_{2})_{H}$ for
$u=(u_{1},u_{2})^{t},
v=(v_{1},v_{2})^{t}\in\mathcal{H}$ ( we write the
ele\-ments of $ \mathcal{H}$ as columns ) is a Hilbert
space (cf. Dautray-Lions \cite{D-L5}, Section VII.3.4., p. 331).

Let $D( \mathbb{A}):=D(\mathcal{A})\times V$. We define the
operator $ \mathbb{A}$ over $D( \mathbb{A})$ by
\begin{equation}\label{generador-para-ecuacion-de- ondas-AA}
  \mathbb{A}u:=\begin{pmatrix}0&-id\\ \mathcal{A}&0\end{pmatrix}
\begin{pmatrix}u_{1}\\ u_{2}\end{pmatrix}=\begin{pmatrix}-u_{2}\\
\mathcal{A}u_{1}\end{pmatrix}, \quad\forall
  u=\begin{pmatrix}u_{1}\\ u_{2}\end{pmatrix}\in D( \mathbb{A}).
\end{equation}
It follows that $D( \mathbb{A})$ is dense in $ \mathcal{H}$ and
$\mathbb{A}$ is a closed operator.

\begin{theo}\label{AA-Generador}
$-\mathbb{A}$ is the infinitesimal generator of a $C^{0}$-semigroup
in $ \mathcal{H}$.
\end{theo}
For the proof of the above theroem, see Dautray-Lions
\cite[theorem XVII.3.4]{D-L5}.

\begin{theo}\label{II-abstracto-semilineal-teorema-para-usar}
Let $-A$ be the infinitesimal generator of a $C^{0}$-semigroup of
li\-near operators on a Banach space $X$ and $u_0\in D(A)$. If
$f:[t_0,T]\times X\to  X$ is continuously  differentiable with
bounded partial derivatives then there  exists a unique classical
solution $u\in C^1([t_0,T];X)$ of the initial value problem
\begin{equation}\label{eq-II-abstracto-semilineal-teorema-para-usar}
\begin{gathered}
 \frac{du(t)}{dt}+Au(t)=f(t,u(t))\quad\text{in } X,
\text{ on } ]t_0,T]\\
u(t_0)= u_0\,.
\end{gathered}
\end{equation}
\end{theo}

The proof of this lemma ca be found in Pazy \cite[theorem 6.1.5]{Pazy}.


\section{Function spaces and bilinear forms for the semilinear
problem plate-membrane}\label{espacios-de-funciones-segundo-preprint}

We define the vector space (with the usual vectorial
sum and multiplication by scalars)
\begin{equation}\label{Espacio-V}
V:=\left\{ (u_p,u_m)\in
  H^2(\Omega_p)\times
  H^1(\Omega_m); u_p|_{_{\Gamma_3}}=\gamma_{_{1}}u_p|_{_{\Gamma_3}}=0, u_p|_{_{\Gamma_1}}=
  \gamma_0u_m|_{_{\Gamma_1}}\right\}
\end{equation}
(In this work we only consider real vector spaces).
The vector space $V$, endowed   with the inner product
  \begin{equation}\label{producto-escalar-V}
((u_{p},u_{m})|(v_{p},v_{m}))_{_{V}}:=(u_{p}|v_{p})_{_{H^{2}(\Omega_{p})}}
+(u_{m}|v_{m})_{_{H^{1}(\Omega_{m})}},
\end{equation}
is a separable Hilbert space. The norm in $V$ is given by
\begin{equation}\label{Norma-V}
\|(u_{p},u_{m})\|_{_{V}}:=
  \big(\|u_p\|_{2,2,\Omega_p}^2+\|u_m\|_{1,2,\Omega_m}^2\big)^{1/2}.
\end{equation}
We consider also
\begin{equation}\label{Espacio-H}
  H:=L^2(\Omega_p)\times L^2(\Omega_m)
\end{equation}
with inner product and norm given by
\begin{equation}\label{producto-escalar-H}
((u_{p},u_{m})|(v_{p},v_{m}))_{_{H}}:=(u_{p}|v_{p})_{_{L^{2}(\Omega_{p})}}
+(u_{m}|v_{m})_{_{L^{2}(\Omega_{m})}}
\end{equation}
and
\begin{equation}\label{Norma-H}
\|(u_{p},u_{m})\|_{_{H}}:={
  \left(\|u_p\|_{0,2,\Omega_p}^2+\|u_m\|_{0,2,\Omega_m}^2\right  )}^{1/2}\,.
\end{equation}
Also we consider
\begin{equation}\label{V-tilde-semigrupo-1}
  \tilde{V}:=\big\{ (\tilde{u}_p,\tilde{u}_m)\in H^{2}(\Omega_p)
\times H^{1}(\Omega_m);
  \big(\frac{1}{\sqrt{\rho_ph}}\tilde{u}_{p},\frac{1}{\sqrt{\rho_m}}
\tilde{u}_m\big)\in  V\big\},
\end{equation}
endowed with the norm
\begin{equation}\label{Norma-V-tilde-1}
 \|(\tilde{u}_p,\tilde{u}_m)\|_{\tilde{V}}:=
\big(\frac{1}{\rho_ph}\|\tilde{u}_p\|_{2,2,\Omega_p}^{2}
  +  \frac{1}{\rho_m}\|\tilde{u}_m\|_{1,2,\Omega_m}^{2}\big)^{1/2}.
\end{equation}
We have the imbedding $\tilde{V}\hookrightarrow H$ with
$\tilde{V}$ dense in $H$. Identifying $H$ with its dual $H'$ we
obtain
$\tilde{V}\mathop{\hookrightarrow} \limits^{i}H
=H'\mathop{\hookrightarrow}\limits^{i'}\tilde{V}'$,
where $i:\tilde{V}\to  H$ is the identity operator and
$i':H\to \tilde{V}'$ is the dual operator of
$i:V\to  H$. Since $i:\tilde{V}\to  H$ is
injective and its range is dense in $H$, the same holds for
$i':H\to \tilde{V}'$. Furthermore we identify $i'f$
with $f$ for $f\in H$. Therefore we regard $H$ as  subspace
of $\tilde{V}'$.

 We  consider the  symmetric  bilinear form
\begin{equation}\label{V-bilineal-forma-def}
\begin{aligned}
&a((u_{p},u_{m})|(v_{p},v_{m}))\\
&:=\frac{h^3}{12}\sum_{\alpha,\beta,\gamma,\theta=1}^2
\int_{\Omega_p}
A_{\alpha\beta\gamma\theta}\frac{\partial^2u_p}{\partial
  x_\gamma\partial x_\theta}\frac{\partial^2v_p}{\partial
  x_\alpha\partial x_\beta}dx
+ C\int_{\Omega_m}\nabla u_m\cdot\nabla v_mdx
\end{aligned}
\end{equation}
for $(u_p,u_m), (v_p,v_m)\in V$
(The symmetry is a consequence of the
assumption (\ref{simetria-Aab-gamma-theta})).
For technical reasons it is convenient to consider also
\begin{equation}\label{V-tilde-bilineal-forma-def}
\tilde{a}\left((\tilde{u}_{p},\tilde{u}_{m})|(\tilde{v}_{p},
\tilde{v}_m)\right)
:=  a\Big(\big(\frac{1}{\sqrt{\rho_ph}}\tilde{u}_p,\frac{1}{\sqrt{\rho_{m}}}
\tilde{u}_{m}\big)
  \big|\big(\frac{1}{\sqrt{\rho_ph}}\tilde{v}_p,\frac{1}{\sqrt{\rho_{m}}}
\tilde{v}_{m}\big)\Big)
\end{equation}
for
$(\tilde{u}_p,\tilde{u}_m),(\tilde{v}_p,\tilde{v}_m)\in\tilde{V}$.

\begin{lemma}\label{a-stetig-V-koerziv}
Under the assumptions introduced for the coefficients
$A_{\alpha\beta\gamma\theta}$, the bilinear form
(\ref{V-bilineal-forma-def}) (resp. (\ref{V-tilde-bilineal-forma-def}))
is continuous and $V$-coercive (resp. $\tilde{V}-coercive$) with respect
to $H$.
\end{lemma}

\begin{proof}
 From the Schwarz inequality we have the continuity of the bilinear forms
(\ref{V-bilineal-forma-def}) and (\ref{V-tilde-bilineal-forma-def}).
Now let $u=(u_p,u_m)\in V$. From Lemma
\ref{Normas_equivalentes_en_Sobolev_Spaces} we have that there exists
$c_p>0$  such that
\[
(\!(u_p)\!)_{2,2,\Omega_p}\geq c_p\|u_p\|_{2,2,\Omega_p}.
\]
Then
\begin{align*}
a(u|u)&= \frac{h^3}{12}\sum_{\alpha,\beta,\gamma,\theta=1}^2\int_{\Omega_p}A_{\alpha,\beta,\gamma,\theta}\frac{\partial^2u_p}{\partial
  x_\gamma\partial x_\theta}\frac{\partial^2u_p}{\partial
  x_\alpha\partial x_\beta}dx
+ C\int_{\Omega_m}|\nabla u_m|^2dx
\\
&\geq \frac{h^3}{12}\rho\sum_{\alpha,\beta=1}^2\int_{\Omega_p}\Bigl|\frac{\partial^2u_p}{\partial
  x_\alpha\partial x_\beta}\Bigr|^2dx + C|u_m|_{1,2,\Omega_m}^2
\\
&=\frac{h^3}{12}\rho|u_p|_{2,2,\Omega_p}^2 +  C|u_m|_{1,2,\Omega_m}^2
\\
&\geq \frac{h^3}{12}\rho c_p\|u_p\|_{2,2,\Omega_p}^2
- \frac{h^3}{12}\rho|u_p|_{0,2,\Omega_p}^2
+  C\|u_m\|_{1,2,\Omega_m}^2 -  C|u_m|_{0,2,\Omega_m}^2.
\end{align*}
With  $\lambda_0:=\max\bigl\{\frac{h^3}{12}\rho,C\bigr\}$ and
$c_0:=\min\bigl\{\frac{h^3}{12}\rho c_p,C\bigr\}$ we obtain the $V$-coerciveness
of $a(\cdot|\cdot)$ with respect to $H$. From this follows immediately the
$\tilde{V}$-coerciveness of $\tilde{a}(\cdot|\cdot)$ with respect to $H$.
\end{proof}

 Let$D(\tilde{\mathcal{A}}):= \tilde{A}^{-1}(H)$
  and
  $\tilde{\mathcal{A}}:=\tilde{A}|_{_{D(\tilde{\mathcal{A}})}}$,
  where $\tilde{A}:\tilde{V}\to \tilde{V}'$ is
  given by
  $\langle\tilde{A}\tilde{u}|\tilde{v}\rangle =
  \tilde{a}(\tilde{u}|\tilde{v})$, for all
  $\tilde{u},\tilde{v}\in\tilde{V}$. We have that
  $-\tilde{\mathcal{A}}$ is the infinitesimal generator of a
  $C^{0}$-semigroup  in $H$ (see \cite[p. 54]{Hernandez-prep-1}.

 \section{Weak solution}\label{soluciones-debil-y-fuerte}

 For the function
\begin{equation}
(t,x,u)\mapsto f_p(t,x,u):
[0,T]\times\Omega_p\times\mathbb{R}\to \mathbb{R}
\label{II-semilineal-fp-annahme}
\end{equation}
we assume the following:
\begin{itemize}
\item[(i)] For all $t\in[0,T]$, $x\mapsto  f_p(t,x,u(x)):
\Omega_p\to \mathbb{R}$ is measurable, if
$u:\Omega_p\to \mathbb{R}$ is measurable.

\item[(ii)] $|f_p(t,x,u)|\leq q_{_{p}}(t,x) +  k_{_{p}}|u|$
for all $(t,x,u)\in[0,T]\times\Omega_p\times\mathbb{R}$,
where $q_{_{p}}(t,\cdot)\in  L^{2}(\Omega_p)$ for all
$t\in[0,T]$ and $k_{_{p}}>0$ is a constant.

\item[(iii)] $\frac{\partial f_p}{\partial t}(t,x,u)$
exists for all $(t,x,u)\in[0,T]\times\Omega_p\times\mathbb{R}$.
It is bounded and Lipschitz continuous on
$[0,T]\times\Omega_p\times\mathbb{R}$.

\item[(iv)] $\frac{\partial f_p}{\partial u}(t,x,u)$ exists for
all $(t,x,u)\in[0,T]\times\Omega_p\times\mathbb{R}$.
It is bounded and Lipschitz continuous on
$[0,T]\times\Omega_p\times\mathbb{R}$.
\end{itemize}
For the function
\begin{equation}
(t,x,u)\mapsto f_m(t,x,u):
[0,T]\times\Omega_m\times\mathbb{R}\to \mathbb{R}
\label{II-semilineal-fm-annahme}
\end{equation}
we assume the following:
\begin{itemize}
\item[(i)] For all $t\in[0,T]$,
$x\mapsto f_m(t,x,u(x)):\Omega_m\to \mathbb{R}$ is measurable, if
$u:\Omega_m\to \mathbb{R}$ is measurable.

\item[(ii)] $|f_m(t,x,u)|\leq q_{_{m}}(t,x) +
  k_{_{m}}|u|$, for all $(t,x,u)\in[0,T]\times\Omega_m\times\mathbb{R}$,
 where $q_{_{m}}(t,\cdot)\in  L^{2}(\Omega_m)$ for all $t\in[0,T]$
 and $k_{_{m}}>0$ a constant.

\item[(iii)] $\frac{\partial f_m}{\partial t}(t,x,u)$ exists for
all $(t,x,u)\in[0,T]\times\Omega_m\times\mathbb{R}$.
It is bounded and Lipschitz continuous on
$[0,T]\times\Omega_m\times\mathbb{R}$.

\item[(iv)] $\frac{\partial f_m}{\partial u}(t,x,u)$ exists for
all $(t,x,u)\in[0,T]\times\Omega_m\times\mathbb{R}$.
It is bounded and Lipschitz continuous on
$[0,T]\times\Omega_m\times\mathbb{R}$.
\end{itemize}
\noindent
Let $\mathbf{f}_{p}:[0,T]\times L^{2}(\Omega_p)\to
L^{2}(\Omega_p)$ and $\mathbf{f}_{m}:[0,T]\times
L^{2}(\Omega_m)\to  L^{2}(\Omega_m)$ be defined by
\begin{gather}\label{semilineal-negrita-fp-def}
[\mathbf{f}_p(t,u_p)](x):= f_p(t,x,u_p(x)) \quad \text{for }
(t,x)\in[0,T]\times\Omega_p \;  u_p\in L^{2}(\Omega_p)\,,\\
\label{semilineal-negrita-fm-def}
[\mathbf{f}_m(t,u_m)](x):= f_m(t,x,u_m(x)) \quad\text{for }
(t,x)\in[0,T]\times\Omega_m \; u_m\in L^{2}(\Omega_m).
\end{gather}
 From assumptions on (\ref{II-semilineal-fp-annahme}) and
(\ref{II-semilineal-fm-annahme}), we see that
$\mathbf{f}_p(t,u_p)\in L^{2}(\Omega_p)$ and
$\mathbf{f}_m(t,u_m)\in L^{2}(\Omega_m)$, for $u_p\in
L^{2}(\Omega_p)$ and $u_m\in L^{2}(\Omega_m)$.

 For technical reasons we introduce the following functions:
\begin{gather}\label{semilineal-negrita-fp-tilde-def}
\mathbf{\tilde{f}}_p(t,u_p):=\frac{1}{\sqrt{\rho_ph}}\mathbf{f}_p
\big(t,\frac{1}{\sqrt{\rho_ph}}u_p\big)\quad
\text{for } t\in[0,T]\; u_p\in L^{2}(\Omega_p)\,,
\\
\label{semilineal-negrita-fm-tilde-def}
\mathbf{\tilde{f}}_m(t,u_m):=\frac{1}{\sqrt{\rho_m}}\mathbf{f}_m
\big(t,\frac{1}{\sqrt{\rho_m}}u_m\big)\quad
\text{for } t\in[0,T] \; u_m\in L^{2}(\Omega_m)\,.
\end{gather}
Let us suppose that
$u_p:[0,T]\times\overline{\Omega}_p\to \mathbb{R}$ and
$u_m:[0,T]\times\overline{\Omega}_m\to \mathbb{R}$ are smooth
enough in such a way that the system (\ref{semi-Platte-3}) -
(\ref{Intro-derivada-um0-3}) for $(u_p,u_m)$ holds; i.e., we
suppose that $(u_p,u_m)$ is a classical solution of the semilinear
problem (\ref{semi-Platte-3})-(\ref{Intro-derivada-um0-3}).
Furthermore we assume that $(\tilde{u}_p(t,.),\tilde{u}_m(t,.))\in
D(\tilde{\mathcal{A}})$ for $t\in]0,T]$, where
$(\tilde{u}_p,\tilde{u}_m):=\left(\sqrt{\rho_ph}u_p,\sqrt{\rho_m}u_m\right)$.
If we multiply (\ref{semi-Platte-3}) (resp. (\ref{semi-Membrane-3}))
with $ \frac{1}{\sqrt{\rho_ph}}\tilde{v}_p$ (resp.
$\frac{1}{\sqrt{\rho_m}}\tilde{v}_m$), where
$(\tilde{v}_p,\tilde{v}_m)\in\tilde{V}$, by use of integration by
parts, (\ref{Intro-Rand2-3})-(\ref{Intro-Stetigrand1-3}) and the fact
that $\tilde{V}$ is dense in $H$ we obtain
\begin{equation}\label{semilineal-eq-justificacion-variacional-1}
\big(\frac{\partial^{2}\tilde{u}_{p}}{\partial
t^{2}}(t,\cdot),\frac{\partial^{2}\tilde{u}_m}{\partial t^{2}
}(t,\cdot)\big) +
\tilde{\mathcal{A}}(\tilde{u}_{p}(t,\cdot),\tilde{u}_m(t,\cdot))
=
\big(\mathbf{\tilde{f}}_p(t,\tilde{u}_p(t,\cdot)),
\mathbf{\tilde{f}}_m(t,\tilde{u}_m(t,\cdot))\big)
\end{equation}
in $H$, for $t\in]0,T]$. On the other hand we have
\begin{equation}\label{condiciones-iniciales-semilineal-justificacion}
 \tilde{u}_p(0,\cdot)=\tilde{g}_{p}^{0},\quad
 \tilde{u}_m(0,\cdot)=\tilde{g}_{m}^{0}, \quad
 \frac{\partial\tilde{u}_p}{\partial t}(0,\cdot)=\tilde{g}_{p}^{1},\quad
 \frac{\partial\tilde{u}_p}{\partial t}(0,\cdot)=\tilde{g}_{m}^{1},
\end{equation}
where  $\tilde{g}_{p}^{0}:=\sqrt{\rho_ph}g_{p}^{0}$,
$\tilde{g}_{m}^{0}:=\sqrt{\rho_m}g_{m}^{0} $,
$\tilde{g}_{p}^{1}:=\sqrt{\rho_ph}g_{p}^{1}$  and
$\tilde{g}_{m}^{1}:=\sqrt{\rho_m}g_{m}^{1}$.

\noindent We suppose
\begin{equation}\label{gp-gm-supuestos-semigrupos-homogen}
(i)\;  (g_{p}^{0},g_{m}^{0})\in A^{-1}(H), \quad
 (ii)\;  (g_{p}^{1},g_{m}^{1})\in V
\end{equation}
where $A:V\to  V'$ is given by $\langle Au | v \rangle = a(u|v)$,
for all $u,v\in V$.

Equations
(\ref{semilineal-eq-justificacion-variacional-1}) and
 (\ref{condiciones-iniciales-semilineal-justificacion}) motivate the
 following definition:
Consider the Hilbert space $\mathcal{H}:=\tilde{V}\times H$ endowed
with the inner product
\begin{equation}\label{producto-escalar-pm}
\Big(\begin{pmatrix}(\tilde{u}_{p}^{1} ,\tilde{u}_{m}^{1})\\[3pt]
(\tilde{u}_{p}^{2},\tilde{u}_{m}^{2})\end{pmatrix} \Big|
  \begin{pmatrix}(\tilde{v}_{p}^{1},\tilde{v}_{m}^{1})\\[3pt]
(\tilde{v}_{p}^{2},\tilde{v}_{m}^{2})\end{pmatrix}\Big)_{\mathcal{H}}
:=
  a((\tilde{u}_{p}^{1} ,\tilde{u}_{m}^{1})|(\tilde{v}_{p}^{1} ,
\tilde{v}_{m}^{1}))
+((\tilde{u}_{p}^{2} ,\tilde{u}_{m}^{2})|(\tilde{v}_{p}^{2}
  ,\tilde{v}_{m}^{2}))_{H}.
\end{equation}
Moreover let $D(\tilde{\mathbb{A}}):=D(\tilde{\mathcal{A}})\times\tilde{V}$
and $ \tilde{\mathbb{A}}:=\begin{pmatrix}0&-id\\
\tilde{\mathcal{A}}&0\end{pmatrix}$.
It follows from theorem \ref{AA-Generador}
that $-\tilde{\mathbb{A}}$ is the infinitesimal generator of a
$C^{0}$-semigroup of contractions in $\mathcal{H}$.
We put
\begin{equation}\label{F-def-eq-semilineal-pm-0}
\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}):=\begin{pmatrix}0\\[3pt]
\big(\mathbf{\tilde{f}}_{p}(t,\mathbf{\tilde{u}}_{p}^{1}),
  \mathbf{\tilde{f}}_{m}(t,\mathbf{\tilde{u}}_{m}^{1})\big)\end{pmatrix}
\quad\text{for }
 \tilde{\mathbb{U}}:=\begin{pmatrix}(\mathbf{\tilde{u}}_{p}^{1},
\mathbf{\tilde{u}}_{m}^{1})\\[3pt]
 (\mathbf{\tilde{u}}_{p}^{2},\mathbf{\tilde{u}}_{m}^{2})
\end{pmatrix}\in\mathcal{H},
\end{equation}
\begin{equation}\label{desvelo5}
  \tilde{\mathbb{G}}:=\begin{pmatrix}(\tilde{g}_{p}^{0},\tilde{g}_{m}^{0})\\[3pt]
(\tilde{g}_{p}^{1},\tilde{g}_{m}^{1})\end{pmatrix}.
\end{equation}
Next we define weak solution for our
semilinear problem.

\begin{Def}\label{II-definicion-debilloesung-semilineal-semigrupos} \rm
Assume that \eqref{simetria-Aab-gamma-theta},
\eqref{condicion-coercitividad}, \eqref{II-semilineal-fp-annahme},
 \eqref{II-semilineal-fm-annahme} and
\eqref{gp-gm-supuestos-semigrupos-homogen} are satisfied.
 We say that a function $(\mathbf{u}_p,\mathbf{u}_m)\in
C^1([0,T];V)\cap C^{2}([0,T];H)$ is a weak solution of the
semilinear problem \eqref{semi-Platte-3}-\eqref{Intro-derivada-um0-3}
if the function
$$
(\mathbf{\tilde{u}}_p,\mathbf{\tilde{u}}_m)
:=\big(\sqrt{\rho_ph}\mathbf{u}_p,\sqrt{\rho_m}\mathbf{u}_m\big)\in
C^1([0,T];\tilde{V})\cap C^{2}([0,T];H)
$$
has the following properties:
\begin{equation}\label{II-semilineal-debilsolucion-pm-def}
\begin{aligned}
(i) & \big(\frac{d^{2}\mathbf{\tilde{u}}_{p}(t)}{dt^{2}},\frac{d^{2}
\mathbf{\tilde{u}}_{m}(t)}{dt^{2}}\big)
   +\tilde{\mathcal{A}}(\mathbf{\tilde{u}}_{p}(t),\mathbf{\tilde{u}}_{m}(t))=
   \big(\mathbf{\tilde{f}}_{p}(t,\mathbf{\tilde{u}}_{p}(t)),
\mathbf{\tilde{f}}_{m}(t,\mathbf{\tilde{u}}_{m}(t))\big)\\
      &\text{in } H, \text{ on } ]0,T]
  \\
  (ii) & (\mathbf{\tilde{u}}_{p}(0),\mathbf{\tilde{u}}_{m}(0))
=(\tilde{g}_{p}^{0},\tilde{g}_{m}^{0}).
  \\
  (iii)& \big(\frac{d\mathbf{\tilde{u}_p}}{dt}(0),
\frac{d\mathbf{\tilde{u}_m}}{dt}(0)\big)=
  (\tilde{g}_{p}^{1},\tilde{g}_{m}^{1}).
\end{aligned}
\end{equation}
\end{Def}

\begin{lemma}\label{semilineal-lema-F-diferenciable-con-continuidad}
Assume \eqref{simetria-Aab-gamma-theta},
\eqref{condicion-coercitividad}, \eqref{II-semilineal-fp-annahme}
and \eqref{II-semilineal-fm-annahme}. Then the function
$(t,\mathbb{U})\mapsto\mathbb{F}(t,\mathbb{U}):[0,T]\times\mathcal{H}\to
\mathcal{H}$
which is defined by \eqref{F-def-eq-semilineal-pm-0}, is continuously
differentiable with bounded partial derivatives.
\end{lemma}

\begin{proof}
\textbf{1.}\; The assumptions (\ref{II-semilineal-fp-annahme})(i),(ii)
and (\ref{II-semilineal-fm-annahme})(i),(ii) lead to
$$
\mathbf{\tilde{f}}_{p}(t,\mathbf{\tilde{u}}_{p}^{1})\in
L^2(\Omega_p)\quad\mathrm{and}\quad\mathbf{\tilde{f}}_{m}(t,\mathbf{\tilde{u}}_{m}^{1})\in
L^{2}(\Omega_m)
$$
for $\mathbf{\tilde{u}}_{p}^{1}\in
L^{2}(\Omega_p)$ and $\mathbf{\tilde{u}}_{m}^{1}\in
L^{2}(\Omega_m)$ and for all $t\in[0,T]$
(cf. \cite[theorem 2.1]{Appell}).
  Then we have
$\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\in\mathcal{H}$
for $(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$.

\noindent\textbf{2.}\; It follows from (\ref{II-semilineal-fp-annahme})(iii) that
$$\frac{\partial f_{p}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(\cdot)\big)
\in L^{2}(\Omega_p)\quad
\forall t\in[0,T]\; \forall\mathbf{\tilde{u}}_{p}^{1}\in
L^{2}(\Omega_p).
$$
Let $t\in[0,T]$. For
$\tau\in\mathbb{R}$ with $-t\leq\tau\leq T-t$ we have
\begin{equation}\label{semilineal-eq-abschaetzung-I-diffbar-F}
\begin{aligned}
&\big\|\frac{\mathbf{\tilde{f}}_{p}(t+\tau,\mathbf{\tilde{u}}_{p}^{1})-
\mathbf{\tilde{f}}_{p}(t,\mathbf{\tilde{u}}_{p}^{1})}{\tau} -
\frac{1}{\sqrt{\rho_ph}}\frac{\partial f_{p}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(\cdot)\big)
\big\|_{_{L^{2}(\Omega_p)}}^{2}
\\
&=\int_{\Omega_p}\frac{1}{\rho_ph}\Big|\int_{0}^{1}\big[\frac{\partial
f_p}{\partial
t}\big(t+\xi\tau,x,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(x)\big)
-\frac{\partial f_p}{\partial t}
\big(t,x,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(x)\big)\big]d\xi
\Big|^{2}dx
\\
& \leq \int_{\Omega_p}\frac{1}{\rho_ph}
\Big[\int_{0}^{1}\big|\frac{\partial f_p}{\partial
t}\big(t+\xi\tau,x,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(x)\big)
-\frac{\partial f_p}{\partial t}
\big(t,x,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(x)\big)\big|d\xi
\Big]^{2}dx
\\
& \leq \frac{1}{\rho_ph}\text{const.}
\mu_{_{p}}(\Omega_p)\tau^{2}\xrightarrow[\tau\to 0]{}0
\end{aligned}
\end{equation}
The above inequality because the Lipschitz continuity of
$\frac{\partial f_p}{\partial t}$.

\noindent\textbf{3.}\;
It follows from (\ref{II-semilineal-fm-annahme})(iii) that
$$
\frac{\partial f_{m}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_m}}\mathbf{\tilde{u}}_{m}^{1}(\cdot)\big)\in
L^{2}(\Omega_m)\quad \forall t\in[0,T]\; \forall\mathbf{\tilde{u}}_{m}^{1}\in
L^{2}(\Omega_m).
$$
Let $t\in[0,T]$. For
$\tau\in\mathbb{R}$ with $-t\leq\tau\leq T-t$ we have as above
\begin{equation}\label{paso3-semilineal-eq-abschaetzung-I-diffbar-F}
\big\|\frac{\mathbf{\tilde{f}}_{m}(t+\tau,\mathbf{\tilde{u}}_{m}^{1})-
\mathbf{\tilde{f}}_{m}(t,\mathbf{\tilde{u}}_{m}^{1})}{\tau} -
\frac{1}{\sqrt{\rho_m}}\frac{\partial f_{m}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_m}}\mathbf{\tilde{u}}_{m}^{1}(\cdot)\big)
\big\|_{_{L^{2}(\Omega_m)}}^{2}
\end{equation}
approaches zero as $\tau\to  0$.

\noindent\textbf{4.}\;
 Let $(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$
with $
\tilde{\mathbb{U}}:=\begin{pmatrix}(\mathbf{\tilde{u}}_{p}^{1},
\mathbf{\tilde{u}}_{m}^{1})\\
  (\mathbf{\tilde{u}}_{p}^{2},\mathbf{\tilde{u}}_{m}^{2})\end{pmatrix}$.
We consider the operator
$D_{1}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\in\mathcal{L}(\mathbb{R};\mathcal{H})$
which is defined by
\begin{equation}\label{tildeF-derivada-t-semilineal}
D_{1}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\tau
:=\begin{pmatrix}0 \\
\big(\frac{1}{\sqrt{\rho_ph}}\frac{\partial f_{p}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(\cdot)\big)\tau
, \frac{1}{\sqrt{\rho_m}}\frac{\partial f_{m}}{\partial
t}\big(t,\cdot,\frac{1}{\sqrt{\rho_m}}\mathbf{\tilde{u}}_{m}^{1}(\cdot)\big)
\tau \big)\end{pmatrix}
\end{equation}
For $(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$ and
from (\ref{semilineal-eq-abschaetzung-I-diffbar-F}) and
(\ref{paso3-semilineal-eq-abschaetzung-I-diffbar-F}) we have that
\begin{equation}\label{prueba-existe-parcial-t-tilde-F-semilineal}
\frac{\|\tilde{\mathbb{F}}(t+\tau,\tilde{\mathbb{U}}) -
\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}) -
D_{1}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\tau
\|_{\mathcal{H}}}{|\tau|}
\xrightarrow[-t\leq\tau\leq T-t,\,
\tau\neq 0,\, \tau\to 0] 0.
\end{equation}
Then there exists the partial derivative of
$\tilde{\mathbb{F}}$ with respect to $t$ for all
$(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$ and it is equal
to $D_{1}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})$.
By the Lipschitz continuity of $\frac{\partial f_p }{\partial
t}$ and $\frac{\partial f_m }{\partial t}$ it can be showed that
\begin{equation}\label{otra-Lipschitz-parcial-t-tilde-F-semilineal}
\| D_{1}\tilde{\mathbb{F}}(t_1,\tilde{\mathbb{U}}_1) -
D_{1}\tilde{\mathbb{F}}(t_2,\tilde{\mathbb{U}}_2)\|_{_{\mathcal{L}(\mathbb{R};\mathcal{H})}}
\leq \mathrm{const.}\big( |t_1-t_2| +
\|\tilde{\mathbb{U}}_1-\tilde{\mathbb{U}}_2\|_{_{\mathcal{H}}}\big).
\end{equation}
Then the maping
$$
(t,\tilde{\mathbb{U}})\mapsto D_1\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})
:[0,T]\times\mathcal{H}\to \mathcal{L}(\mathbb{R};\mathcal{H})
$$
is continuous.
The boundedness of $\frac{\partial f_p}{\partial t}$ and
$\frac{\partial f_m}{\partial t}$ implied by
the boundedness of $D_1\tilde{\mathbb{F}}$.

\noindent\textbf{5.}\;
 From (\ref{II-semilineal-fp-annahme})(iv) and
(\ref{II-semilineal-fm-annahme})(iv) we have
$$
\frac{\partial f_{p}}{\partial u}
\big(t,\cdot,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1}(\cdot)\big)
\mathbf{\tilde{v}}_{p}^{1} \in L^{2}(\Omega_p)
$$
and
$$
\frac{\partial f_{m}}{\partial u}\big(t,\cdot,\frac{1}{\sqrt{\rho_m}}
\mathbf{\tilde{u}}_{m}^{1}(\cdot)\big)\mathbf{\tilde{v}}_{m}^{1}
\in L^{2}(\Omega_m)
$$
for all $t\in[0,T]$ and all
$(\mathbf{\tilde{u}}_{p}^{1},\mathbf{\tilde{u}}_{m}^{1}),
(\mathbf{\tilde{v}}_{p}^{1},\mathbf{\tilde{v}}_{m}^{1})\in H$.
 For $t\in[0,T]$,
$\tilde{\mathbb{U}}:=\begin{pmatrix}(\mathbf{\tilde{u}}_{p}^{1},
\mathbf{\tilde{u}}_{m}^{1})\\[3pt]
  (\mathbf{\tilde{u}}_{p}^{2},\mathbf{\tilde{u}}_{m}^{2})\end{pmatrix}
\in\mathcal{H}$
  and $\tilde{\mathbb{V}}:=\begin{pmatrix}(\mathbf{\tilde{v}}_{p}^{1},
\mathbf{\tilde{v}}_{m}^{1})\\[3pt]
  (\mathbf{\tilde{v}}_{p}^{2},\mathbf{\tilde{v}}_{m}^{2})\end{pmatrix}
\in\mathcal{H}$
  we put
\begin{equation}\label{semilin-def-D2tildeF}
D_{2}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\tilde{\mathbb{V}}
:=\begin{pmatrix}0 \\ \big( \frac{1}{\rho_ph}\frac{\partial
f_{p}}{\partial u}\big(t,\cdot,\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}
_{p}^{1}(\cdot)\big)\mathbf{\tilde{v}}_{p}^{1}
, \frac{1}{\rho_m}\frac{\partial f_{m}}{\partial
u}\big(t,\cdot,\frac{1}{\sqrt{\rho_m}}\mathbf{\tilde{u}}_{m}^{1}(\cdot)\big)
\mathbf{\tilde{v}}_{m}^{1} \big)\end{pmatrix}
\end{equation}
 Since $ \frac{\partial f_p}{\partial u}$ (resp. $
\frac{\partial f_m}{\partial u}$) is bounded on
$[0,T]\times\Omega_p\times\mathbb{R}$
(resp. $[0,T]\times\Omega_m\times\mathbb{R}$), we see that
$D_{2}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\in\mathcal{L}(\mathcal{H})
$ for all $(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$.

 For $(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$
and $\tilde{\mathbb{V}}\in\mathcal{H}$ with
$\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}\neq 0$ we have (with
``$\mathrm{const}$" denoting different constants)
\begin{equation}\label{eq-semil-prueba-diferenciabilidad-tildeF}
\begin{aligned}
&\frac{\| \tilde{\mathbb{F}}(t,\tilde{\mathbb{U}} +
\tilde{\mathbb{V}} ) - \tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}) -
D_{2}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\tilde{\mathbb{V}}
\|_{_{\mathcal{H}}}^{2}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}}
\\
& \leq \frac{\mathrm{const}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}}
\Big\{ \int_{\Omega_p}\Big[\int_{0}^{1}\big|\frac{\partial f_p}{\partial u}
\big(t,x,\frac{1}{\sqrt{\rho_ph}}(\mathbf{\tilde{u}}_{p}^{1}(x) +
\xi\mathbf{\tilde{v}}_{p}^{1}(x)) \big)\\
& \quad- \frac{\partial f_{p}}{\partial
u}\big(t,x,\frac{\mathbf{\tilde{u}}_{p}^{1}(x)}{\sqrt{\rho_ph}}\big)
\big|d\xi\Big]^{2}
\frac{|\mathbf{\tilde{v}}_{p}^{1}(x)|^{2}}{\rho_ph}dx
\\
&\quad+ \int_{\Omega_m}\Big[\int_{0}^{1}\big|\frac{\partial f_m}{\partial u}
\big(t,x,\frac{1}{\sqrt{\rho_m}}(\mathbf{\tilde{u}}_{m}^{1}(x) +
\xi\mathbf{\tilde{v}}_{m}^{1}(x)) \big)
\\
&\quad- \frac{\partial f_{m}}{\partial
u}\big(t,x,\frac{\mathbf{\tilde{u}}_{m}^{1}(x)}{\sqrt{\rho_m}}\big)
\big|d\xi\Big]^{2}\frac{|\mathbf{\tilde{v}}_{m}^{1}(x)|^{2}}{\rho_m}dx\Big\}
\\
&\leq \frac{\mathrm{const}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}}
\big\{\frac{1}{\rho_{p}^{2}h^{2}}\int_{\Omega_p}|
\mathbf{\tilde{v}}_{p}^{1}(x)|^{4}dx
+\frac{1}{\rho_{m}^{2}}\int_{\Omega_m}|\mathbf{\tilde{v}}_{m}^{1}(x)|^{4}dx
\big\}\,.
\end{aligned}
\end{equation}
The above holds because of the Lipschitz continuity of
$\frac{\partial f_p}{\partial u}$ and $\frac{\partial f_m}{\partial u}$.
 Since
$$
\mathbf{\tilde{v}}_{p}^{1}\in H^{2}(\Omega_p)\hookrightarrow
C^{0}(\overline{\Omega}_p)\hookrightarrow L^{4}(\Omega_p)\quad
\mathrm{and}\quad\mathbf{\tilde{v}}_{m}^{1}\in
H^{1}(\Omega_m)\hookrightarrow L^{4}(\Omega_m)
$$ (see lemmas
\ref{lemma-inmersion-Wmp-Lq} and
\ref{lemma-inmersion-Wmp-C-cerradura-Omega}), from
(\ref{eq-semil-prueba-diferenciabilidad-tildeF}), we have
\begin{equation}\label{continuacion-eq-semil-prueba-diferenciabilidad-tildeF}
\begin{aligned}
&\frac{\| \tilde{\mathbb{F}}(t,\tilde{\mathbb{U}} +
\tilde{\mathbb{V}} ) - \tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}) -
D_{2}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})\tilde{\mathbb{V}}
\|_{_{\mathcal{H}}}^{2}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}}
\\
&\leq \frac{\mathrm{const.}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}
}\big(\frac{1}{\rho_{p}^{2}h^{2}}\|\mathbf{\tilde{v}}_{p}^{1}\|_{_{H^{2}
(\Omega_p)}}^{4}
+\frac{1}{\rho_{m}^{2}}\|\mathbf{\tilde{v}}_{m}^{1}\|_{_{H^{1}(\Omega_m)}}^{4}
\big)\\
&\leq \frac{\mathrm{const.}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}
}\|(\mathbf{\tilde{v}}_{p}^{1},\mathbf{\tilde{v}}_{m}^{1})\|_{_{\tilde{V}}}^{4}
\\
&\leq \frac{\mathrm{const.}}{\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}
}\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{4} =
\mathrm{const.}\|\tilde{\mathbb{V}}\|_{_{\mathcal{H}}}^{2}.
\end{aligned}
\end{equation}
It follows that the partial derivative of
$\tilde{\mathbb{F}}$ with respect to the second variable
$\tilde{\mathbb{U}}$ exists and it is equal to
$D_{2}\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}})$ for all
$(t,\tilde{\mathbb{U}})\in[0,T]\times\mathcal{H}$.
 We can show
similarly that the Lipschitz continuity (resp. the boundedness) of
$ \frac{\partial f_p}{\partial u}$ and $ \frac{\partial
f_m}{\partial u}$ leads to the continuity (resp. the boundedness) of
$$
(t,\tilde{\mathbb{U}})\mapsto
D_2\tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}):[0,T]\times\mathcal{H}\to \mathcal{L}(\mathcal{H}).
$$
So the proof is complete.
\end{proof}

\begin{lemma}\label{mejor-resultado-semilineal-lemma-con-F-U-tildeA}
Let $\tilde{\mathbb{F}}:[0,T]\times\mathcal{H}\to \mathcal{H}$
(resp. $\tilde{\mathbb{G}}$) be defined by
\eqref{F-def-eq-semilineal-pm-0} (resp. (\ref{desvelo5})). Under
assumptions \eqref{simetria-Aab-gamma-theta},
\eqref{condicion-coercitividad},
\eqref{gp-gm-supuestos-semigrupos-homogen},
\eqref{II-semilineal-fp-annahme} and
\eqref{II-semilineal-fm-annahme}, there exists a unique
 function $\tilde{\mathbb{U}}:[0,T]\to \mathcal{H}$ with the
 following properties:
\begin{equation}\label{II-Semilineal-eq-problema-pm-transf}
\begin{aligned}
(i) & \tilde{\mathbb{U}}\in C^1([0,T];\mathcal{H}).\\
(ii) & \frac{d\tilde{\mathbb{U}}(t)}{dt}+\tilde{\mathbb{A}}
\tilde{\mathbb{U}}(t)=    \tilde{\mathbb{F}}(t,\tilde{\mathbb{U}}(t))
\quad\text{in }  \mathcal{H} \quad \mathrm{on } ]0,T].\\
(iii) & \tilde{\mathbb{U}}(0)=\tilde{\mathbb{G}}.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
\textbf{1.}\; It follows from theorem \ref{AA-Generador} that
$-\tilde{\mathbb{A}}$ is the infinitesimal generator of a $C^0$-semigroup
of linear operators in $\mathcal{H}$.

\noindent\textbf{2.}\;
  From lemma \ref{semilineal-lema-F-diferenciable-con-continuidad} we
have that
  $\tilde{\mathbb{F}}:[0,T]\times\mathcal{H}\to \mathcal{H}$
  is continuously differentiable with bounded partial derivatives.

\noindent\textbf{3.}\; It can be seen that
  $\tilde{\mathbb{G}}$ belongs to $D(\tilde{\mathbb{A}})$.

\noindent\textbf{4.}
 From theorem \ref{II-abstracto-semilineal-teorema-para-usar} we have
the desired result.
\end{proof}

\begin{theo}\label{II-resultado-final-semilineal-semigrupos}
Under assumptions \eqref{simetria-Aab-gamma-theta},
\eqref{condicion-coercitividad},
\eqref{gp-gm-supuestos-semigrupos-homogen},
\eqref{II-semilineal-fp-annahme} and
\eqref{II-semilineal-fm-annahme}, there exists a unique weak
solution of the semilinear problem
\eqref{semi-Platte-3}-\eqref{Intro-derivada-um0-3}.
\end{theo}

\begin{proof}
Let $$
\tilde{\mathbb{U}}:=\begin{pmatrix}(\mathbf{\tilde{u}}_{p}^{1},
\mathbf{\tilde{u}}_{m}^{1})\\[3pt]
(\mathbf{\tilde{u}}_{p}^{2},\mathbf{\tilde{u}}_{m}^{2})\end{pmatrix}
:[0,T]\to \mathcal{H}$$
be the unique function satisfying
(\ref{II-Semilineal-eq-problema-pm-transf}) (Lemma
\ref{mejor-resultado-semilineal-lemma-con-F-U-tildeA}).
It can be showed that
$(\mathbf{\tilde{u}}_{p}^{1},\mathbf{\tilde{u}}_{m}^{1})$ belongs
to $C^{1}([0,T];\tilde{V})\cap C^2([0,T];H)$ and that it
satisfies  (\ref{II-semilineal-debilsolucion-pm-def}).
Then
$(\frac{1}{\sqrt{\rho_ph}}\mathbf{\tilde{u}}_{p}^{1},\frac{1}{\sqrt{\rho_m}}
\mathbf{\tilde{u}}_{m}^{1})$
is the desired weak solution. The uniqueness follows from the
uniqueness of $\tilde{\mathbb{U}}$.
\end{proof}

\begin{remark} \rm
For sufficiently smooth solutions in the sense of definition
 \ref{II-definicion-debilloesung-semilineal-semigrupos} we can obtain as
usual a classical pointwise solution of system
(\ref{semi-Platte-3})-(\ref{Intro-derivada-um0-3}). See \cite{Jairo-tesis}.
\end{remark}

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\end{document}