# Waves in flexural beams
with nonlinear adhesive interaction

G. M. Coclite  ,  G. Devillanova  and  F. Maddalena
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via
E. Orabona 4,I–70125 Bari, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The paper studies the initial boundary value problem related to the dynamic
evolution of an elastic beam interacting with a substrate through an elastic-
breakable forcing term. This discontinuous interaction is aimed to model the
phenomenon of attachement-detachement of the beam occurring in adhesion
phenomena. We prove existence of solutions in energy space and exhibit various
counterexamples to uniqueness. Furthermore we characterize some relavant
features of the solutions, ruling the main effectes of the nonlinearity due to
the elasic-breakable term on the dynamical evolution, by proving the
linearization property according to [5] and an asymtotic result pertaining the
long time behavior.

###### Key words and phrases:

Beam equation, discontinuous interaction, flexural waves

###### 2010 Mathematics Subject Classification:

35L05, 74B20, 35J25

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM). The second author is also supported by MIUR - FFABR - 2017
research grant. The authors have been partially supported by the Research
Project of National Relevance “Multiscale Innovative Materials and Structures”
granted by the Italian Ministry of Education, University and Research (MIUR
Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education,
University and Research under the Programme Department of Excellence Legge
232/2016 (Grant No. CUP - D94I18000260001)

##  1\. Introduction

In a broader sense the term adhesion refers to a physical situation in which
two material bodies, during their mechanical evolution, experience a contact
interaction which fails in (possibly bounded) regions of space-time and
restores after a while. The phenomenon strongly depends on the constitutive
properties of the involved materials and a crucial problem relies in
understanding the nature of the interaction. The manifestation of such such
phenomenon occurs at every scale, ranging from DNA molecules to the structural
engineering works. Mathematics has looked at these problems, at least in the
stationary case, since the seminal paper [1] and in some recent works (see,
e.g., [7, 8, 9, 10, 11]) one of the authors contributed to the study of the
static problem of adhesion of elastic structures by exploiting different
constitutive assumptions to the aim of characterizing, in a variational
framework, the interplay of the of debonding with other constitutive
properties.

However the dynamical problem is a different story since at the heart of the
question stays the understanding of the attachement-detachement occurrence and
how this affects the whole evolution problem. This is a subtle problem since
the analytical tools at disposal, such as spatio-temporal estimates in some
norms, seem too rough to catch exhaustive quantitative informations, even in
short time.

In [3, 4] the simplest mechanical model consisting in elastic string was
considered, whereas a discontinuous forcing term was assumed to model the
adhesive interaction of the string with a rigid substrate. The resulting
mathematical problem is then ruled by an initial boundary value problem for a
semilinear second order wave equation and the results in [3, 4] show some
tricky peculiarities of the problem. As it is well known in continuum
mechanics, the other basic model for one dimensional elastic structures is
represented by the so called Bernoulli-Navier beam governing the flexural
deformations of a slender material body. In the linear elastic framework the
equation expressing the balance of momentum is ruled by a fourth order spatial
differential operator. Analogously to [3], we assume a discontinuous forcing
term to model the adhesive interaction of the beam with the external
environment. One can visualize as a physical situation an elastic beam
connected to a rigid substrate through a foundation made of continuous
distribution of elastic-breakable springs.
We study the well posedness of the mathematical problem proving existence of
global in time solutions in the natural energy space and exploit some features
of the solutions to obtain information about the role played by the
attachment-detachment occurrence on the dynamical evolution. Indeed, in [2] it
was proved that the main effects induced by the nonlinearity at the transition
from attached to detached states consist in a loss of regularity of the
solution and in a migration of the total energy through the scales. Here we
deepen the analysis by using the linearization condition introduced by Patrick
Gérard in [5] according to which one can conclude that if the semilinear
evolution problem satisfies such condition, then the nonlinear forcing term
does not induce any new oscillation or energy concentrations ([5] ).
Furthermore we prove an asymptotic result for the long time behavior in the
case of bounded solutions, asserting the occurrence of three mutually
exclusive states: the trivial one, the totally detached state, the totally
attached state.
The paper is organized as follows. In Section 2 we formulate the initial
boundary value problem. In Section 3 we state the main results of the paper
consisting in Theorem 3.1 (Existence of solutions), Theorem 3.2
(Characerization of adhesive states), Theorem 3.3 (Long time behavior),
Theorem 3.4 (Linearization property). The proofs of these theorems are given
respectively in Sections 4, 5, 6, 7. In Section 8 we provide some examples
showing non-uniqueness and lack ofsmoothness of the solutions.

##  2\. Statement of the problem

Let us consider an elastic beam under Bernoulli-Navier constitutive
assumption, occupying in the reference configuration the interval
$[0,L]\subset\mathbb{R}$, the balance of linear momentum delivers the
semilinear initial boundary value problem

(2.1)
$\begin{cases}\rho\partial_{tt}^{2}u=-\mu\partial_{xxxx}^{4}u-\Phi^{\prime}\left(u\right)&\quad
t>0,0<x<L,\\\ \partial_{xx}^{2}u(t,0)=\partial_{xx}^{2}u(t,L)=0&\quad t>0,\\\
\partial_{xxx}^{3}u(t,0)=\partial_{xxx}^{3}u(t,L)=0&\quad t>0,\\\
u(0,x)=u_{0}(x)&\quad 0<x<L,\\\ \partial_{t}u(0,x)=u_{1}(x)&\quad
0<x<L.\end{cases}$

We shall assume that

  1. (H.1)

$\Phi\in C(\mathbb{R})\cap C^{2}(\mathbb{R}\setminus\\{1,-1\\})$, $\Phi$ is
positive, constant in $(-\infty,-1]$ and in $[1,\infty)$, convex in $[-1,1]$,
decreasing in $[-1,0]$, increasing in $[0,1]$, and $\Phi(u)\geq\kappa u^{2}$
in $[-1,1]$ for some constant $\kappa>0$;

  2. (H.2)

$u_{0}\in H^{2}(0,L)$, $u_{1}\in L^{2}(0,L)$;

  3. (H.3)

$\rho>0$ is the constant material density and $\mu>0$ is the elastic
stiffness.

As a consequence of (H.1), $\Phi^{\prime}$ has a jump discontinuity in $u=\pm
1$ and

$\displaystyle u\in(-\infty,-1)\cup(1,\infty)$
$\displaystyle\Rightarrow\Phi^{\prime}(u)=0,$ $\displaystyle 0<u<1$
$\displaystyle\Rightarrow 0<\Phi^{\prime}(u)\leq\lim_{u\to
1^{-}}\Phi^{\prime}(u),$ $\displaystyle-1<u<0$ $\displaystyle\Rightarrow
0>\Phi^{\prime}(u)\geq\lim_{u\to-1^{+}}\Phi^{\prime}(u).$

To fix ideas, a function satisfying such assumption is

(2.2) $\Phi(u)=\begin{cases}u^{2}&\qquad\text{if $|u|\leq 1$},\\\
1&\qquad\text{if $|u|>1$}.\end{cases}$

In particular we have for all $u\neq\pm 1$

(2.3) $\Phi^{\prime}(u)=\begin{cases}2u&\qquad\text{if $|u|<1$},\\\
0&\qquad\text{if $|u|>1$}.\end{cases}$

The forcing term $\Phi^{\prime}(u)$ is thought to model the elastic breakable
interaction between the beam and the external environment. It confers to the
problem a localized nonlinearity which affects the evolution in a significant
way.
The natural energy associated to the Problem (2.1) (i.e. to any solution $u$
to (2.1)), is given at time $t$ by the quantity

(2.4)
$E[u](t)=\int_{0}^{L}\left(\frac{\rho(\partial_{t}u(t,x))^{2}+\mu(\partial_{xx}^{2}u(t,x))^{2}}{2}+\Phi(u(t,x))\right)dx.$

In general, the lack of Lipschitz continuity in the nonlinear term
$\Phi^{\prime}$ suggests we cannot expect the existence of conservative
solutions, i.e. solutions that preserve energy. Also the physics underlying
the problem, foresees a kind of dissipation when the material is detaching
from the substrate and this is accompanied by hysteresis cycles (see e.g. [8,
Appendix B]).

##  3\. Main results

We begin by studying the well-posedness of Problem (2.1) and the regularity of
its solutions (see Definition 3.1 below). We prove the existence of Lipshitz
continuous dissipative solutions and give examples of distinct solutions to
(2.1) which do not depend continuously on the initial conditions.

######  Definition 3.1.

We say that a function $u:[0,\infty)\times[0,L]\to\mathbb{R}$ is a dissipative
solution of (2.1) if

  * ($i$)

$u\in C([0,\infty)\times[0,L])$;

  * ($ii$)

$\partial_{t}u,\,\partial_{xx}^{2}u\in L^{\infty}(0,\infty;L^{2}(0,L))$;

  * ($iii$)

$u$ is a weak solution to (2.1), i.e. for every test function $\varphi\in
C^{\infty}(\mathbb{R}^{2})$ with compact support

(3.1) $\begin{split}\int_{0}^{\infty}\int_{0}^{L}&\left(\rho
u\partial_{tt}^{2}\varphi+\mu\partial_{xx}^{2}u\partial_{xx}^{2}\varphi+h_{u}\varphi\right)dtdx\\\
&-\int_{0}^{L}\rho u_{1}(x)\varphi(0,x)dx+\int_{0}^{L}\rho
u_{0}(x)\partial_{t}\varphi(0,x)dx=0,\end{split}$

where $h_{u}\in\partial\Phi^{\prime}\left(u\right)$, that is the
subdifferential of $\Phi^{\prime}(u)$;

  * ($iv$)

$u$ may dissipate energy, i.e. for almost every $t>0$: $E[u](t)\leq E[u](0)$,
namely (see (2.4))

(3.2)
$\begin{split}E[u](t)=\int_{0}^{L}&\left(\frac{\rho(\partial_{t}u(t,x))^{2}+\mu(\partial_{xx}^{2}u(t,x))^{2}}{2}+\Phi(u(t,x))\right)dx\\\
&\qquad\qquad\leq\int_{0}^{L}\left(\frac{\rho(u_{1}(x))^{2}+\mu(\partial_{xx}^{2}u_{0}(x))^{2}}{2}+\Phi(u_{0}(x))\right)dx=E[u](0).\end{split}$

We remind that

$h\in\partial\Phi^{\prime}\left(u\right)$

means that $h:[0,\infty)\times[0,L]\to\mathbb{R}$ sastifies

$h_{u}(t,x)\begin{cases}=\Phi^{\prime}(u(t,x)),&\text{if $|u(t,x)|<1$,}\\\
=0,&\text{if $|u(t,x)|>1$,}\\\ \in[0,\Phi^{\prime}(1^{-})],&\text{if
$u(t,x)=1$,}\\\ \in[\Phi^{\prime}(-1^{+}),0],&\text{if
$u(t,x)=-1$.}\end{cases}$

Let us state the following theorem asserting the existence of dissipative
solutions.

######  Theorem 3.1 (Existence).

If $\Phi$ satisfies (H.1) and $u_{0}$, $u_{1}$ satisfy (H.2), then problem
(2.1) admits a dissipative solution in the sense of Definition 3.1.

The following result provides a sufficient condition ruling the non-detachment
of dissipative solutions in dependence on the initial data.

######  Theorem 3.2.

Let $u$ be a dissipative solution of (2.1). If

(3.3) $\left\|u_{0}\right\|_{L^{\infty}(0,L)}<1\qquad\mbox{ and }\qquad
E[u](0)<\frac{4\kappa\mu}{3\vee 2\kappa}(\leq 2\mu),$

where $\kappa$ is defined in (H.1), then

(3.4) $\left\|u\right\|_{L^{\infty}((0,\infty)\times(0,L))}<1.$

The long time behavior is a very subtle problem for evolutionary partial
differential equations, so by restricting the focus on bounded dissipative
solutions, we are able to prove the following statement.

######  Theorem 3.3 (Long Time Behavior).

Let $u$ be a dissipative solution of (2.1) and
$\\{t_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ such that $t_{n}\to\infty$. If

(3.5) $u\in L^{\infty}((0,\infty)\times(0,L)),$

then there exist a subsequence $\\{t_{n_{k}}\\}_{k\in\mathbb{N}}$ and two
constants $a,\,b\in\mathbb{R}$ such that

$u(t_{n_{k}},\cdot)\rightharpoonup u_{\infty}\quad\text{weakly in $H^{2}(0,L)$
as $k\to\infty$,}$

where

(3.6) $u_{\infty}(x)=ax+b.$

Moreover, only one of the following statements can occur

(3.7) $u_{\infty}\equiv 0$, (3.8) $u_{\infty}(x)\geq 1$ for every $x\in[0,L]$,
(3.9) $u_{\infty}(x)\leq-1$ for every $x\in[0,L]$.

The question at the basis of the last subsequent result can be formulated as
follows: How the nonlinearity characterizing the forcing term $\Phi^{\prime}$
affects the evolution problem? We retain that Gérard’s linearization condition
provides a precise mathematical tool to answer to the previous rather vague
question. Indeed the absence of further energy concentrations or oscillations
due to the nonlinearity, suspected to arise in correspondence of the
attachment-detacment process, constitutes an interesting property in itself,
also considering the nonuniqueness of the solutions $\\{u_{n}\\}_{n}$ below.

######  Theorem 3.4 (Linearization Property).

Let $\\{u_{0,n}\\}_{n}\subset H^{2}(0,L),\,\\{u_{1,n}\\}_{n}\subset
L^{2}(0,L)$, $\\{u_{n}\\}_{n}$ be a sequence of dissipative solutions of (2.1)
in correspondence of such initial data, and $v_{n}$ be the dissipative
solution of the linearized problem

(3.10)
$\begin{cases}\rho\partial_{tt}^{2}v_{n}=-\mu\partial_{xxxx}^{4}v_{n}&\quad
t>0,0<x<L,\\\ \partial_{xx}^{2}v_{n}(t,0)=\partial_{xx}^{2}v_{n}(t,L)=0&\quad
t>0,\\\ \partial_{xxx}^{3}v_{n}(t,0)=\partial_{xxx}^{3}v_{n}(t,L)=0&\quad
t>0,\\\ v_{n}(0,x)=u_{0,n}(x)&\quad 0<x<L,\\\
\partial_{t}v_{n}(0,x)=u_{1,n}(x)&\quad 0<x<L.\end{cases}$

If

(3.11) $\displaystyle\text{$u_{0,n}\rightharpoonup 0$ weakly in
$H^{2}(0,L)$},$ (3.12) $\displaystyle\text{$u_{1,n}\rightharpoonup 0$ weakly
in $L^{2}(0,L)$},$ (3.13)
$\displaystyle\limsup_{n}\int_{0}^{L}\left(\frac{\rho
u_{1,n}^{2}+\mu(\partial_{xx}^{2}u_{0,n})^{2}}{2}\right)dx<\frac{4\kappa\mu}{3\vee
2\kappa}(\leq 2\mu),$

where $k$ is defined in (H.1), then the following linearization condition
holds true

(3.14)
$\left\|\partial_{t}(u_{n}-v_{n})\right\|_{L^{\infty}(0,T;L^{2}(0,L))}+\left\|\partial_{xx}^{2}(u_{n}-v_{n})\right\|_{L^{\infty}(0,T;L^{2}(0,L))}\to
0$

for every $T\geq 0$ as $n\to\infty$.

##  4\. Existence of Dissipative Solutions

This section is dedicated to the proof of Theorem 3.1.

Our argument is based on the approximation of the Neumann problem (2.1) with a
sequence of Neumann problems (4.1) characterized by smooth source terms and
smooth initial data. More precisely, let
$\\{u_{0,n}\\}_{n\in\mathbb{N}},\,\\{u_{1,n}\\}_{n\in\mathbb{N}}\subset
C^{\infty}([0,L]),\,\\{\Phi_{n}\\}_{n\in\mathbb{N}}\subset
C^{\infty}(\mathbb{R})$, for every $n\in\mathbb{N}$ consider the approximating
problems

(4.1)
$\begin{cases}\rho\partial_{tt}^{2}u_{n}=-\mu\partial_{xxxx}^{4}u_{n}-\Phi_{n}^{\prime}(u_{n})&\quad
t>0,0<x<L,\\\ \partial_{xx}^{2}u_{n}(t,0)=\partial_{xx}^{2}u_{n}(t,L)=0&\quad
t>0,\\\ \partial_{xxx}^{3}u_{n}(t,0)=\partial_{xxx}^{3}u_{n}(t,L)=0&\quad
t>0,\\\ u_{n}(0,x)=u_{0,n}(x)&\quad 0<x<L,\\\
\partial_{t}u_{n}(0,x)=u_{1,n}(x)&\quad 0<x<L,\end{cases}$

where $\\{u_{0,n}\\}_{n\in\mathbb{N}},\,\\{u_{1,n}\\}_{n\in\mathbb{N}},$
$\\{\Phi_{n}\\}_{n\in\mathbb{N}}$ are sequences of smooth approximations of
$u_{0},\,u_{1}$, and $\Phi$ respectively, i.e. they satisfy the following
requirements

(4.2) $\begin{split}&u_{0,n}\to u_{0}\quad\text{in $H^{1}(0,L)$},\quad
u_{1,n}\to u_{1}\quad\text{in
$L^{2}(0,L)$},\quad\Phi_{n}\to\Phi\quad\text{uniformly in $\mathbb{R}$},\\\
&\Phi_{n}^{\prime}\to\Phi^{\prime}\quad\text{pointwise and uniformly in
$\mathbb{R}\setminus\left((-1-\varepsilon,-1+\varepsilon)\cup(1-\varepsilon,1+\varepsilon)\right)$
for every $\varepsilon$},\\\ &|u|\geq
1+\varepsilon\Rightarrow\Phi_{n}^{\prime}(u)=0,\qquad\varepsilon>0,\>n\in\mathbb{N},\\\
&\left\|u_{0,n}\right\|_{H^{1}(0,L)}\leq
C,\quad\left\|u_{1,n}\right\|_{L^{2}(0,L)}\leq C,\quad
0\leq\Phi_{n},\,\Phi_{n}^{\prime}\leq C,\qquad n\in\mathbb{N},\\\
&u_{0,n}^{\prime\prime}(0)=u_{0,n}^{\prime\prime}(L)=u_{1,n}(0)=u_{1,n}(L)=0,\qquad
n\in\mathbb{N},\end{split}$

where $C$ is a positive constant which does not depend on $n$.

For any $n\in\mathbb{N}$, (4.1) admits a classical solution for short time
thanks to the Cauchy-Kowaleskaya Theorem (see [14]). Furthermore, for such a
problem, solutions are indeed global in time thanks to the following results.
Let $u_{n}$ be the unique classical solution to (4.1).

######  Lemma 4.1 (Energy conservation).

Classical solution to (4.1) preserves energy.

###### Proof.

Set $E_{n}:=E[u_{n}]$ the energy corresponding to $u_{n}$. We have to prove
that the function $t\mapsto E_{n}(t)$ is constant (with constant value
$E_{n}(0)$). Indeed, we have that

$\displaystyle E_{n}^{\prime}(t)=$
$\displaystyle\frac{d}{dt}\int_{0}^{L}\left(\frac{\rho(\partial_{t}u_{n})^{2}+\mu(\partial_{xx}^{2}u_{n})^{2}}{2}+\Phi_{n}(u_{n})\right)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{L}\left(\rho\partial_{t}u_{n}\partial_{tt}^{2}u_{n}+\mu\partial_{xx}^{2}u_{n}\partial_{txx}^{3}u_{n}+\Phi_{n}^{\prime}(u_{n})\partial_{t}u_{n}\right)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{L}\partial_{t}u_{n}\underbrace{\left(\rho\partial_{tt}^{2}u_{n}+\mu\partial_{xxxx}^{4}u_{n}+\Phi_{n}^{\prime}(u_{n})\right)}_{=0}dx=0.$

∎

As a consequence of energy conservation, since the functions $\Phi_{n}$ are
positive, we have the following boundedness result.

######  Corollary 4.1.

The sequences $\\{\partial_{t}u_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\partial_{xx}^{2}u_{n}\\}_{n\in\mathbb{N}}$ are bounded in
$L^{\infty}(0,\infty;L^{2}(0,L))$.

######  Lemma 4.2 ($L^{2}$ estimate).

The sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}(0,T;L^{2}(0,L))$, for every $T>0$.

###### Proof.

Using the Hölder inequality

$\displaystyle\int_{0}^{L}u_{n}^{2}(t,x)dx=$
$\displaystyle\int_{0}^{L}\left(u_{0,n}(x)+\int_{0}^{t}\partial_{s}u_{n}(s,x)ds\right)^{2}dx$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{L}u_{0,n}^{2}(x)dx+2\int_{0}^{L}\left(\int_{0}^{t}|\partial_{s}u_{n}(s,x)|ds\right)^{2}dx$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{L}u_{0,n}^{2}(x)dx+2t\int_{0}^{t}\int_{0}^{L}(\partial_{s}u_{n}(s,x))^{2}dxds$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{L}u_{0,n}^{2}(x)dx+2t^{2}\sup_{s\geq
0}\int_{0}^{L}(\partial_{s}u_{n}(s,x))^{2}dx,$

the claim follows from Corollary 4.1. ∎

The following result follows from Corollary 4.1 by a straightforward
application of Gagliardo-Nirenberg Interpolation Inequality (see for instance
[12, Theorem at page 125]).

######  Lemma 4.3 ($H^{1}$ estimate).

The sequence $\\{\partial_{x}u_{n}\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}(0,T;L^{2}(0,L))$, for every $T>0$.

######  Lemma 4.4 ($L^{\infty}$ estimate).

The sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}((0,T)\times(0,L))$, for every $T>0$.

###### Proof.

Fix $0<t<T$. Lemmas 4.1 and 4.2 and Corollary 4.1 imply that
$\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in $L^{\infty}(0,T;H^{1}(0,L))$.
Since $H^{1}(0,L)\subset L^{\infty}(0,L)$ we have

$|u_{n}(t,x)|\leq\left\|u_{n}(t,\cdot)\right\|_{L^{\infty}(0,L)}\leq
c\left\|u_{n}(t,\cdot)\right\|_{H^{1}(0,L)}\leq
c\left\|u_{n}\right\|_{L^{\infty}(0,T;H^{1}(0,L))},\quad(t,x)\in(0,T)\times(0,L),$

for some constant $c>0$ depending only on $L$. Therefore

$\left\|u_{n}\right\|_{L^{\infty}((0,T)\times(0,L))}\leq
c\left\|u_{n}\right\|_{L^{\infty}(0,T;H^{1}(0,L))},$

that gives the claim. ∎

######  Lemma 4.5 (space Lipschitz estimate).

The sequence $\\{\partial_{x}u_{n}\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}((0,T)\times(0,L))$, for every $T>0$.

###### Proof.

Fix $0<t<T$ and $0<x<L$. Lemmas 4.1, 4.2, and 4.3 imply that
$\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in $L^{\infty}(0,T;H^{2}(0,L))$.
Since $H^{1}(0,L)\subset L^{\infty}(0,L)$ we have, for every
$(t,x)\in(0,T)\times(0,L)$,

$\displaystyle|\partial_{x}u_{n}(t,x)|$
$\displaystyle\leq\left\|\partial_{x}u_{n}(t,\cdot)\right\|_{L^{\infty}(0,L)}$
$\displaystyle\leq c\left\|\partial_{x}u_{n}(t,\cdot)\right\|_{H^{1}(0,L)}\leq
c\left\|u_{n}(t,\cdot)\right\|_{H^{2}(0,L)}$ $\displaystyle\leq
c\left\|u_{n}\right\|_{L^{\infty}(0,T;H^{2}(0,L))},$

for some constant $c>0$ depending only on $L$. Therefore

$\left\|\partial_{x}u_{n}\right\|_{L^{\infty}((0,T)\times(0,L))}\leq
c\left\|u_{n}\right\|_{L^{\infty}(0,T;H^{2}(0,L))},$

that gives the claim. ∎

###### Proof of Theorem 3.1.

Thanks to Lemmas 4.1, 4.2 and [13, Theorem 5] there exists a function $u$
satisfying items ($i$) and ($ii$) in Definition 3.1 and a function $h_{u}\in
L^{\infty}((0,T)\times(0,L)),\,h_{u}\in\partial\Phi^{\prime}(u),$ such that,
passing to a subsequence,

(4.3) $\begin{split}u_{n}\rightharpoonup u&\quad\text{in
$H^{1}((0,T)\times(0,L))$ and in $L^{2}(0,T;H^{2}(0,L))$, for each $T\geq
0$},\\\ u_{n}\to u&\quad\text{in $L^{\infty}((0,T)\times(0,L))$, for each
$T\geq 0$},\\\ \Phi_{n}^{\prime}(u_{n})\rightharpoonup h_{u}&\quad\text{in
$L^{p}((0,T)\times(0,L))$, for each $T\geq 0$ and $1\leq
p<\infty$}.\end{split}$

We have yet to verify that $u$ is a weak solution of (2.1) i.e. Definition 3.1
\- item ($iii$). Let $\varphi\in C^{\infty}(\mathbb{R}^{2})$ be a test
function with compact support, since $u_{n}$ is a solution to (4.1), we have
that for every $n$

$\displaystyle\int_{0}^{\infty}\int_{0}^{L}$ $\displaystyle\left(\rho
u_{n}\partial_{tt}^{2}\varphi+\mu\partial_{xx}^{2}u_{n}\partial_{xx}^{2}\varphi+\Phi_{n}^{\prime}(u_{n})\varphi\right)dxdt$
$\displaystyle-\int_{0}^{L}\rho u_{1,n}(x)\varphi(0,x)dx+\int_{0}^{L}\rho
u_{0,n}(x)\partial_{t}\varphi(0,x)dx=0.$

Then, by taking the limit as $n\to\infty$, (3.1) follows by using (4.2) and
(4.3).

Finally, due to (4.3) and (4.2) we have

$\displaystyle\partial_{t}u_{n}\rightharpoonup\partial_{t}u$
$\displaystyle\quad\text{in $L^{p}(0,T;L^{2}(0,L))$ for each $T\geq 0$ and
$1\leq p<\infty$},$
$\displaystyle\partial_{xx}^{2}u_{n}\rightharpoonup\partial_{xx}^{2}u$
$\displaystyle\quad\text{in $L^{p}(0,T;L^{2}(0,L))$ for each $T\geq 0$ and
$1\leq p<\infty$},$ $\displaystyle\Phi_{n}(u_{n})\to\Phi(u)$
$\displaystyle\quad\text{in $L^{\infty}((0,T)\times(0,L))$, for each $T\geq
0$}.$

Therefore, Definition 3.1 \- item ($iv$) follows by the lower semicontinuity
of the $L^{2}$ norm with respect to the weak convergence by taking into
account Lemma 4.1. ∎

##  5\. Adhesive states

This section is dedicated to the proof of Theorem 3.2.

###### Proof of Theorem 3.2.

Since $u$ is continuous by (3.3) there exists $\tau>0$ such that in the short
time interval $[0,\tau]$ we have

$|u(t,x)|<1,\qquad(t,x)\in[0,\tau]\times[0,L].$

So we can define $\tau^{*}$ as follows

$\tau^{*}=\sup\left\\{\tau>0\;|\;|u(t,x)|<1\mbox{ for all
}(t,x)\in[0,\tau]\times[0,L]\right\\}.$

We claim that

(5.1) $\tau^{*}=\infty.$

Observe that, due to (H.1),

$\Phi(u(t,x))\geq\kappa u^{2}(t,x),\qquad(t,x)\in[0,\tau^{*})\times[0,L].$

Therefore

$E[u](t)\geq\int_{0}^{L}\left(\frac{\rho(\partial_{t}u(t,x))^{2}+\mu(\partial_{xx}^{2}u(t,x))^{2}}{2}+\kappa(u(t,x))^{2}\right)dx,\qquad
t\in[0,\tau^{*}),$

and, in particular,

$E[u](t)\geq\frac{\mu\left\|\partial_{xx}^{2}u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}}{2}+\kappa\left\|u(t,\cdot)\right\|_{L^{2}(0,L)}^{2},\qquad
t\in[0,\tau^{*}).$

Since $u$ is dissipative, using the Sobolev embedding $H^{1}(0,L)\subset
L^{\infty}(0,L)$ (see [6, Theorem 8.5]) and Lemma 4.3, we have from every
$t\in[0,\tau^{*})$

$\displaystyle\left\|u(t,\cdot)\right\|_{L^{\infty}(0,L)}^{2}\leq$
$\displaystyle\frac{\left\|u(t,\cdot)\right\|_{H^{1}(0,L)}^{2}}{2}=\frac{\left\|u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}+\left\|\partial_{x}u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}}{2}$
$\displaystyle\leq$
$\displaystyle\frac{3}{4}\left\|u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}+\frac{\left\|\partial_{xx}^{2}u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}}{4}$
$\displaystyle\leq$
$\displaystyle\left(\frac{3}{4\kappa}\vee\frac{1}{2}\right)\left(\frac{\left\|\partial_{xx}^{2}u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}}{2}+\kappa\left\|u(t,\cdot)\right\|_{L^{2}(0,L)}^{2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\left(\frac{3}{4\kappa}\vee\frac{1}{2}\right)E(t)\leq\frac{1}{\mu}\left(\frac{3}{4\kappa}\vee\frac{1}{2}\right)E(0)<1,$

(where the last inequality holds thanks to (3.3)) that proves (5.1). ∎

##  6\. Long Time Behavior

This section is dedicated to the proof of Theorem 3.3.

Let $u$ be a dissipative solution of (2.1) satisfying (3.5).

STEP 1. We begin by deducing the effective asymptotic problem.

Consider the functions

$u_{\tau}(t,x)=u(\tau t,x),\qquad\tau>0,\>t\geq 0,\>x\in[0,L].$

$u_{\tau}$ is a dissipative solution of the initial boundary value problem

(6.1)
$\begin{cases}\displaystyle\frac{\rho\partial_{tt}^{2}u_{\tau}}{\tau^{2}}=-\mu\partial_{xxxx}^{4}u_{\tau}-\Phi^{\prime}\left(u_{\tau}\right)&\quad
t>0,0<x<L,\\\
\partial_{xx}^{2}u_{\tau}(t,0)=\partial_{xx}^{2}u_{\tau}(t,L)=0&\quad t>0,\\\
\partial_{xxx}^{3}u_{\tau}(t,0)=\partial_{xxx}^{3}u_{\tau}(t,L)=0&\quad
t>0,\\\ u_{\tau}(0,x)=u_{0}(x)&\quad 0<x<L,\\\ \partial_{t}u_{\tau}(0,x)=\tau
u_{1}(x)&\quad 0<x<L,\end{cases}$

in the sense of Definition 3.1, namely

  * ($iii$)

for every test function $\varphi\in C^{\infty}(\mathbb{R}^{2})$ with compact
support

(6.2)
$\begin{split}\int_{0}^{\infty}\int_{0}^{L}&\left(-\frac{\rho\partial_{t}u_{\tau}}{\tau^{2}}\partial_{t}\varphi+\mu\partial_{xx}^{2}u_{\tau}\partial_{xx}^{2}\varphi+h_{\tau}\varphi\right)dtdx-\int_{0}^{L}\frac{\rho
u_{1}(x)}{\tau}\varphi(0,x)dx=0,\end{split}$

where $h_{\tau}\in\partial\Phi^{\prime}\left(u_{\tau}\right)$, that is the
subdifferential of $\Phi^{\prime}(u_{\tau})$;

  * ($iv$)

$u_{\tau}$ may dissipate energy, i.e. for almost every $t>0$:

(6.3)
$\begin{split}\int_{0}^{L}&\left(\frac{\rho(\partial_{t}u_{\tau}(t,x))^{2}}{2\tau^{2}}+\frac{\mu(\partial_{xx}^{2}u_{\tau}(t,x))^{2}}{2}+\Phi(u_{\tau}(t,x))\right)dx\\\
&\qquad\leq\int_{0}^{L}\left(\frac{\rho(u_{1}(x))^{2}}{2}+\frac{\mu(\partial_{xx}^{2}u_{0}(x))^{2}}{2}+\Phi(u_{0}(x))\right)dx.\end{split}$

Thanks to (H.1), (3.5), and (6.3),

$\displaystyle\text{$\\{u_{\tau}\\}_{\tau>0}$ is bounded in
$L^{\infty}(0,\infty;H^{2}(0,L))$},$
$\displaystyle\text{$\\{h_{\tau}\\}_{\tau>0}$ is bounded in
$L^{\infty}((0,\infty)\times(0,L))$},$

there exists two functions $U\in L^{\infty}(0,\infty;H^{2}(0,L)),\,H\in
L^{\infty}((0,\infty)\times(0,L))$ such that, passing to a subsequence,

$\displaystyle u_{\tau}\overset{\star}{\rightharpoonup}U\qquad$
$\displaystyle\text{weakly-$\star$ in
$L^{\infty}_{loc}((0,\infty)\times(0,L))$ as $\tau\to\infty$},$ $\displaystyle
h_{\tau}\overset{\star}{\rightharpoonup}H\qquad$
$\displaystyle\text{weakly-$\star$ in
$L^{\infty}_{loc}((0,\infty)\times(0,L))$ as $\tau\to\infty$}.$

Using (6.3)

$\displaystyle\text{$\\{\partial_{t}u_{\tau}/\tau\\}_{n\in\mathbb{N}}$ is
bounded in $L^{\infty}(0,\infty;L^{2}(0,L))$},$

therefore sending $\tau\to\infty$ in (6.2) we get

(6.4)
$\int_{0}^{\infty}\int_{0}^{L}\left(\mu\partial_{xx}^{2}U\partial_{xx}^{2}\varphi+H\varphi\right)dtdx=0,$

namely $U=U(x)$, $H=H(x)$ and the effective asymptotic problem is

(6.5) $\begin{cases}-\mu\partial_{xxxx}^{4}U=H,&\quad 0<x<L,\\\
\partial_{xx}^{2}U(0)=\partial_{xx}^{2}U(L)=0,&{}\textstyle\\\
\partial_{xxx}^{3}U(0)=\partial_{xxx}^{3}U(L)=0.&{}\textstyle\end{cases}$

STEP 2. We exploit more subtle characterizations of the limit functions $U$
and $H$. To this aim we fix a sequence
$\\{t_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ such that $t_{n}\to\infty$ and
study the convergence of the sequence

$\\{u(t_{n},\cdot)\\}_{n\in\mathbb{N}}.$

Since we have the dissipation inequality (3.2) and the assumption (3.5), we
gain

$\displaystyle\text{$\\{u(t_{n},\cdot)\\}_{n\in\mathbb{N}}$ is bounded in
$H^{2}(0,L)$},$
$\displaystyle\text{$\\{h_{u}(t_{n},\cdot)\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}(0,L)$}.$

Therefore there exist two functions $u_{\infty}\in H^{2}(0,L),\,h_{\infty}\in
L^{\infty}(0,L)$ such that passing to a subsequence

(6.6) $\begin{split}&u(t_{n},\cdot)\rightharpoonup
u_{\infty}\qquad\text{weakly in $H^{2}(0,L)$ as $n\to\infty$},\\\
&u(t_{n},\cdot)\to u_{\infty}\qquad\text{a.e. in $(0,L)$ as $n\to\infty$},\\\
&h_{u}(t_{n},\cdot)\overset{\star}{\rightharpoonup}h_{\infty}\qquad\text{weakly-$\star$
in $L^{\infty}(0,L)$ as $n\to\infty$}.\end{split}$

Due to the result in STEP 1, we know that the functions $u_{\infty}$ and
$h_{\infty}$ must satisfy the effective problem

(6.7) $\begin{cases}-\mu\partial_{xxxx}^{4}u_{\infty}=h_{\infty},&\quad
0<x<L,\\\
\partial_{xx}^{2}u_{\infty}(0)=\partial_{xx}^{2}u_{\infty}(L)=0,&{}\textstyle\\\
\partial_{xxx}^{3}u_{\infty}(0)=\partial_{xxx}^{3}u_{\infty}(L)=0.&{}\textstyle\end{cases}$

Moreover, by (6.6) we have also that

(6.8) $h_{\infty}\in\partial\Phi^{\prime}(u_{\infty}).$

By multiplying (6.7) by $u_{\infty}$, integrating over $(0,L)$, and recalling
(6.8), we get

$\int_{0}^{L}\mu(\partial_{xx}^{2}u_{\infty})^{2}dx=-\int_{0}^{L}h_{\infty}u_{\infty}dx\leq
0,$

it follows that

$\partial_{xx}^{2}u_{\infty}\equiv 0.$

Therefore, we can conclude that (3.6) holds.

Using (3.6) in (6.7) we have also that

$h_{\infty}\equiv 0,$

hence, due to (6.8) only one within (3.7),(3.8), (3.9) can occur.

##  7\. Linearization Property

###### Proof of Theorem 3.4.

Thanks to (3.11)

$u_{0,n}\to 0\quad\text{uniformly in $[0,L]$}.$

Therefore using also (3.13), the assumptions of Theorem 3.2 are fulfilled,
hence

(7.1) $\left\|u_{n}\right\|_{L^{\infty}((0,\infty)\times(0,L))}<1.$

In addition, the function

$w_{n}=u_{n}-v_{n}$

is a dissipative solution of

(7.2)
$\begin{cases}\rho\partial_{tt}^{2}w_{n}=-\mu\partial_{xxxx}^{4}w_{n}-\Phi^{\prime}(u_{n})&\quad
t>0,0<x<L,\\\ \partial_{xx}^{2}w_{n}(t,0)=\partial_{xx}^{2}w_{n}(t,L)=0&\quad
t>0,\\\ \partial_{xxx}^{3}w_{n}(t,0)=\partial_{xxx}^{3}w_{n}(t,L)=0&\quad
t>0,\\\ w_{n}(0,x)=\partial_{t}w_{n}(0,x)=0&\quad 0<x<L.\end{cases}$

Multiplying (7.2) by $\partial_{t}w_{n}$ we gain

$\displaystyle\frac{d}{dt}\int_{0}^{L}\frac{\rho(\partial_{t}w_{n})^{2}+\mu(\partial_{xx}^{2}w_{n})^{2}}{2}dx=$
$\displaystyle\int_{0}^{L}\Phi^{\prime}(u_{n})\partial_{t}w_{n}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\rho}\int_{0}^{L}\Phi^{\prime}(u_{n})^{2}dx+\int_{0}^{L}\frac{\rho(\partial_{t}w_{n})^{2}}{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\rho}\int_{0}^{L}\Phi^{\prime}(u_{n})^{2}dx+\int_{0}^{L}\frac{\rho(\partial_{t}w_{n})^{2}+\mu(\partial_{xx}^{2}w_{n})^{2}}{2}dx$

and applying the Gronwall Lemma

(7.3)
$\int_{0}^{L}\frac{\rho(\partial_{t}w_{n}(t,x))^{2}+\mu(\partial_{xx}^{2}w_{n}(t,x))^{2}}{2}dx\leq\frac{1}{2\rho}\int_{0}^{t}\int_{0}^{L}e^{t-s}\Phi^{\prime}(u_{n}(s,x))^{2}dsdx.$

Thanks to Theorem 3.1 and [13, Theorem 5], (7.1) there exists a function $u$
satisfying items ($i$) and ($ii$) in Definition 3.1 such that, passing to a
subsequence,

(7.4) $\begin{split}&\left\|u\right\|_{L^{\infty}((0,\infty)\times(0,L))}\leq
1,\\\ &u_{n}\rightharpoonup u\quad\text{in $H^{1}((0,T)\times(0,L))$ and in
$L^{2}(0,T;H^{2}(0,L))$, for each $T\geq 0$},\\\ &u_{n}\to u\quad\text{in
$L^{\infty}((0,T)\times(0,L))$, for each $T\geq 0$},\\\
&\Phi^{\prime}(u_{n})\rightharpoonup\Phi^{\prime}(u)\quad\text{in
$L^{p}((0,T)\times(0,L))$, for each $T\geq 0$ and $1\leq
p<\infty$}.\end{split}$

Therefore $u$ is a distributional solution of

(7.5)
$\begin{cases}\rho\partial_{tt}^{2}u=-\mu\partial_{xxxx}^{4}u-\Phi^{\prime}(u)&\quad
t>0,0<x<L,\\\ \partial_{xx}^{2}u(t,0)=\partial_{xx}^{2}u(t,L)=0&\quad t>0,\\\
\partial_{xxx}^{3}u(t,0)=\partial_{xxx}^{3}u(t,L)=0&\quad t>0,\\\
u(0,x)=\partial_{t}u(0,x)=0&\quad 0<x<L.\end{cases}$

Since $u$ takes values in $[-1,1]$ and $\Phi$ is $C^{2}$ therein we can
differentiate (7.5) and get

$\rho\partial_{ttt}^{3}u=-\mu\partial_{txxxx}^{4}u-\Phi^{\prime\prime}(u)\partial_{t}u.$

Multiplying by $\partial_{tt}^{2}u$, using (H.1) and ($ii$) in Definition 3.1
through a regularization argument we get

$\displaystyle\frac{d}{dt}\int_{0}^{L}\frac{\rho(\partial_{tt}^{2}u)^{2}+\mu(\partial_{txx}^{3}u)^{2}}{2}dx=$
$\displaystyle\int_{0}^{L}\Phi^{\prime\prime}(u)\partial_{t}u\partial_{tt}^{2}udx$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\rho}\int_{0}^{L}\Phi^{\prime\prime}(u)^{2}(\partial_{t}u)^{2}dx+\int_{0}^{L}\frac{\rho(\partial_{tt}^{2}u)^{2}}{2}dx$
$\displaystyle\leq$ $\displaystyle
c+\int_{0}^{L}\frac{\rho(\partial_{tt}^{2}u)^{2}+\mu(\partial_{txx}^{3}u)^{2}}{2}dx.$

Thanks to the Gronwall Lemma

$\int_{0}^{L}\frac{\rho(\partial_{tt}^{2}u(t,x))^{2}+\mu(\partial_{txx}^{3}u(t,x))^{2}}{2}dx\leq
c(e^{t}-1),\qquad t\geq 0.$

As a consequence $u$ is an energy preserving solution of (7.5) and then it
must be the trivial one. Eventually, (7.3) concludes the proof. ∎

##  8\. Non-uniqueness and lack of smoothness

This section is devoted to exploit some qualitative properties of (2.1)
through explicit analytical examples evidencing the lack of uniqueness and
smoothness of solutions. A key mechanism ruling these phenomena relies in the
transition between the two configurations induced by the discontinuity
affecting the forcing term $\Phi^{\prime}$. In particular, the first two
examples show the lack of uniqueness while the last one and the numerical
experiments enlighten the occurrences of lack of smoothness.

######  Example 8.1.

Let $\varepsilon>0$ and set $\rho=\mu=1$. Consider the function

$\Phi_{\varepsilon}(u)=\begin{cases}\frac{2-\varepsilon}{2}u^{2},&\quad\text{if
$|u|\leq 1,$}\\\
\frac{2-\varepsilon}{\varepsilon}\left((1+\varepsilon)\left(u-\frac{1}{2}\right)-\frac{u^{2}}{2}\right),&\quad\text{if
$1\leq u\leq 1+\varepsilon,$}\\\
\frac{\varepsilon-2}{\varepsilon}\left((1+\varepsilon)\left(u+\frac{1}{2}\right)+\frac{u^{2}}{2}\right),&\quad\text{if
$-1-\varepsilon\leq u\leq-1,$}\\\
\frac{(2-\varepsilon)(1+\varepsilon)}{2},&\quad\text{if $|u|\geq
1+\varepsilon.$}\end{cases}$

We have

$\Phi_{\varepsilon}^{\prime}(u)=\begin{cases}(2-\varepsilon)u,&\quad\text{if
$|u|\leq 1,$}\\\
\frac{2-\varepsilon}{\varepsilon}(1+\varepsilon-u),&\quad\text{if $1\leq u\leq
1+\varepsilon,$}\\\
\frac{\varepsilon-2}{\varepsilon}(1+\varepsilon+u),&\quad\text{if
$-1-\varepsilon\leq u\leq-1,$}\\\ 0,&\quad\text{if $|u|\geq
1+\varepsilon.$}\end{cases}$

The functions

$u_{\varepsilon}(t,x)=(1-\varepsilon)\cos\left(\sqrt{2-\varepsilon}\,t\right),\qquad
v_{\varepsilon}(t,x)=1+\varepsilon$

solve

(8.1)
$\displaystyle\begin{cases}\partial_{tt}^{2}u_{\varepsilon}=-\partial_{xxxx}^{4}u_{\varepsilon}-\Phi_{\varepsilon}^{\prime}(u_{\varepsilon}),&\quad
t>0,0<x<L,\\\
\partial_{xx}^{2}u_{\varepsilon}(t,0)=\partial_{xx}^{2}u_{\varepsilon}(t,L)=0,&\quad
t>0,\\\
\partial_{xxx}^{3}u_{\varepsilon}(t,0)=\partial_{xxx}^{3}u_{\varepsilon}(t,L)=0,&\quad
t>0,\\\ u_{\varepsilon}(0,x)=1-\varepsilon,&\quad 0<x<L,\\\
\partial_{t}u_{\varepsilon}(0,x)=0,&\quad 0<x<L,\end{cases}$ (8.2)
$\displaystyle\begin{cases}\partial_{tt}^{2}v_{\varepsilon}=-\partial_{xxxx}^{4}v_{\varepsilon}-\Phi_{\varepsilon}^{\prime}(v_{\varepsilon}),&\quad
t>0,0<x<L,\\\
\partial_{xx}^{2}v_{\varepsilon}(t,0)=\partial_{xx}^{2}v_{\varepsilon}(t,L)=0,&\quad
t>0,\\\
\partial_{xxx}^{3}v_{\varepsilon}(t,0)=\partial_{xxx}^{3}v_{\varepsilon}(t,L)=0,&\quad
t>0,\\\ v_{\varepsilon}(0,x)=1+\varepsilon,&\quad 0<x<L,\\\
\partial_{t}v_{\varepsilon}(0,x)=0,&\quad 0<x<L.\end{cases}$

As $\varepsilon\to 0$ we have

$u_{\varepsilon}(t,x)\to u(t,x)=\cos\left(\sqrt{2}\,t\right),\qquad
v_{\varepsilon}(t,x)\to v(t,x)=1,$

and $u$ and $v$ provides two different solutions of (2.1) in correspondence of
the initial data

$u_{0}(x)=1,\qquad u_{1}(x)=0.$

The energies associated to (8.1) and (8.2) are

$\displaystyle E_{\varepsilon}[u_{\varepsilon}](t)$
$\displaystyle=\int_{0}^{L}\left(\frac{(\partial_{t}u_{\varepsilon}(t,x))^{2}+(\partial_{xx}^{2}u_{\varepsilon}(t,x))^{2}}{2}+\Phi_{\varepsilon}(u_{\varepsilon}(t,x))\right)dx=\frac{(2-\varepsilon)(1-\varepsilon)^{2}}{2}L,$
$\displaystyle E_{\varepsilon}[v_{\varepsilon}](t)$
$\displaystyle=\int_{0}^{L}\left(\frac{(\partial_{t}v_{\varepsilon}(t,x))^{2}+(\partial_{xx}^{2}v_{\varepsilon}(t,x))^{2}}{2}+\Phi_{\varepsilon}(v_{\varepsilon}(t,x))\right)dx=\frac{(2-\varepsilon)(1+\varepsilon)}{2}L,$

respectively.

######  Example 8.2.

For every $\varepsilon>0$, the solutions $u_{\varepsilon}$ and
$v_{\varepsilon}$ of the two following problems

(8.3)
$\displaystyle\begin{cases}\partial_{tt}^{2}u_{\varepsilon}=-\partial_{xxxx}^{4}u_{\varepsilon}-\Phi^{\prime}(u_{\varepsilon}),&\quad
t>0,0<x<L,\\\
\partial_{xx}^{2}u_{\varepsilon}(t,0)=\partial_{xx}^{2}u_{\varepsilon}(t,L)=0,&\quad
t>0,\\\
\partial_{xxx}^{3}u_{\varepsilon}(t,0)=\partial_{xxx}^{3}u_{\varepsilon}(t,L)=0,&\quad
t>0,\\\ u_{\varepsilon}(0,x)=1+\varepsilon,&\quad 0<x<L,\\\
\partial_{t}u_{\varepsilon}(0,x)=\varepsilon,&\quad 0<x<L,\end{cases}$ (8.4)
$\displaystyle\begin{cases}\partial_{tt}^{2}v_{\varepsilon}=-\partial_{xxxx}^{4}v_{\varepsilon}-\Phi^{\prime}(v_{\varepsilon}),&\quad
t>0,0<x<L,\\\
\partial_{xx}^{2}v_{\varepsilon}(t,0)=\partial_{xx}^{2}v_{\varepsilon}(t,L)=0,&\quad
t>0,\\\
\partial_{xxx}^{3}v_{\varepsilon}(t,0)=\partial_{xxx}^{3}v_{\varepsilon}(t,L)=0,&\quad
t>0,\\\ v_{\varepsilon}(0,x)=1-\varepsilon,&\quad 0<x<L,\\\
\partial_{t}v_{\varepsilon}(0,x)=0,&\quad 0<x<L,\end{cases}$

are

$u_{\varepsilon}(t,x)=\varepsilon t+1+\varepsilon,\qquad
v_{\varepsilon}(t,x)=(1-\varepsilon)\cos(\sqrt{2}t).$

We have

$\displaystyle\left\|u_{\varepsilon}(0,\cdot)-v_{\varepsilon}(0,\cdot)\right\|_{L^{2}(0,L)}+\left\|\partial_{t}u_{\varepsilon}(0,\cdot)-\partial_{t}v_{\varepsilon}(0,\cdot)\right\|_{L^{2}(0,L)}=3\varepsilon\sqrt{L},$
$\displaystyle\lim_{t\to\infty}u_{\varepsilon}(t,x)=\infty,\qquad\limsup_{t\to\infty}v_{\varepsilon}(t,x)=1-\varepsilon.$

Moreover, as $\varepsilon\to 0$,

$u_{\varepsilon}(t,x)\to 1,\qquad v_{\varepsilon}(t,x)\to\cos(\sqrt{2}t).$

The energies associated to (8.3) and (8.4) are

$\displaystyle E[u_{\varepsilon}](t)$
$\displaystyle=\int_{0}^{L}\left(\frac{(\partial_{t}u_{\varepsilon}(t,x))^{2}+(\partial_{xx}^{2}u_{\varepsilon}(t,x))^{2}}{2}+\Phi(u_{\varepsilon}(t,x))\right)dx=\frac{\varepsilon^{2}+2}{2}L,$
$\displaystyle E[v_{\varepsilon}](t)$
$\displaystyle=\int_{0}^{L}\left(\frac{(\partial_{t}v_{\varepsilon}(t,x))^{2}+(\partial_{xx}^{2}v_{\varepsilon}(t,x))^{2}}{2}+\Phi(v_{\varepsilon}(t,x))\right)dx=(1-\varepsilon)^{2}L,$

respectively.

######  Example 8.3.

Consider the function

(8.5) $u(t,x)=\begin{cases}\sqrt{2}\sin(\sqrt{2}t),&\qquad\text{if $0\leq
t\leq\frac{\pi}{4\sqrt{2}}$},\\\ \sqrt{2}t+1-\frac{\pi}{4},&\qquad\text{if
$t\geq\frac{\pi}{4\sqrt{2}}$}.\end{cases}$

Clearly, $u$ solves the problem

$\begin{cases}\partial_{tt}^{2}u=-\partial_{xxxx}^{4}u-\Phi^{\prime}(u),&\quad
t>0,x\in(0,L),\\\ \partial_{xx}^{2}u(t,0)=\partial_{xx}^{2}u(t,L)=0,&\quad
t>0,\\\ \partial_{xxx}^{3}u(t,0)=\partial_{xxx}^{3}u(t,L)=0,&\quad t>0,\\\
u(0,x)=0,&\quad x\in(0,L),\\\ \partial_{t}u(0,x)=2,&\quad
x\in(0,L),\end{cases}$

but

$u\in C^{1}([0,\infty)\times[0,L])\setminus C^{2}([0,\infty)\times[0,L]).$

Indeed

$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{-}}u\left(t,x\right)=1,\qquad$
$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{+}}u\left(t,x\right)=1,$
$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{-}}\partial_{t}u\left(t,x\right)=\sqrt{2},\qquad$
$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{+}}\partial_{t}u\left(t,x\right)=\sqrt{2},$
$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{-}}\partial_{tt}^{2}u\left(t,x\right)=-2,\qquad$
$\displaystyle\lim_{t\to\frac{\pi}{4\sqrt{2}}^{+}}\partial_{tt}^{2}u\left(t,x\right)=0.$

The energy associated to (8.5) is

$E[u](t)=\int_{0}^{L}\left(\frac{(\partial_{t}u(t,x))^{2}+(\partial_{x}u(t,x))^{2}}{2}+\Phi(u(t,x))\right)dx=2L,$

for every $t\geq 0$.

## References

  * [1] R. Burridge and J. B. Keller.  Peeling, slipping and cracking—some one-dimensional free-boundary problems in mechanics.  SIAM Rev., 20(1):31–61, 1978.
  * [2] G. M. Coclite, G. Fanizza, and F. Maddalena.  Regularity and energy transfer for a nonlinear beam equation.  Appl. Math. Lett., 115:106959, 2021.
  * [3] G. M. Coclite, G. Florio, M. Ligabò, and F. Maddalena.  Nonlinear waves in adhesive strings.  SIAM J. Appl. Math., 77(2):347–360, 2017.
  * [4] G. M. Coclite, G. Florio, M. Ligabò, and F. Maddalena.  Adhesion and debonding in a model of elastic string.  Comput. Math. Appl., 78(6):1897–1909, 2019.
  * [5] P. Gérard.  Oscillations of concentration effects in semilinear dispersive wave equations.  J. Funct. Anal., 141(1):60–98, 1996.
  * [6] E. H. Lieb and M. Loss.  Analysis, volume 14 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, second edition, 2001.
  * [7] F. Maddalena and D. Percivale.  Variational models for peeling problems.  Interfaces Free Bound., 10(4):503–516, 2008.
  * [8] F. Maddalena, D. Percivale, G. Puglisi, and L. Truskinovsky.  Mechanics of reversible unzipping.  Contin. Mech. Thermodyn., 21(4):251–268, 2009.
  * [9] F. Maddalena, D. Percivale, and F. Tomarelli.  Adhesive flexible material structures.  Discrete Contin. Dyn. Syst. Ser. B, 17(2):553–574, 2012.
  * [10] F. Maddalena, D. Percivale, and F. Tomarelli.  Local and non-local energies in adhesive interaction.  IMA J. Appl. Math., 81(6):1051–1075, 2016.
  * [11] F. Maddalena, D. Percivale, and F. Tomarelli.  Elastic-brittle reinforcement of flexural structures, 2021.
  * [12] L. Nirenberg.  On elliptic partial differential equations.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:115–162, 1959.
  * [13] J. Simon.  Compact sets in the space $L^{p}(0,T;B)$.  Ann. Mat. Pura Appl. (4), 146:65–96, 1987.
  * [14] M. E. Taylor.  Partial differential equations I. Basic theory, volume 115 of Applied Mathematical Sciences.  Springer, New York, second edition, 2011.