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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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\begin{document}
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\thispagestyle{empty}
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\title[The impulsive Barenblatt--Zheltov--Kochina equation]
{The one-dimensional impulsive\\ Barenblatt--Zheltov--Kochina equation\\
with a transition layer}
\author{Ivan Kuznetsov and Sergey Sazhenkov}
\address{Ivan V. Kuznetsov
\newline\hphantom{iii} Laboratory for Mathematical and Computer Modeling
\newline\hphantom{iii} in Natural and Industrial Systems,
\newline\hphantom{iii} Altai State University, Prospekt Lenina 61,
\newline\hphantom{iii} 656049 Barnaul, Russian Federation,
\newline\hphantom{iii} and
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} Siberian Division of the Russian Academy of Sciences,
\newline\hphantom{iii} pr. Acad. Lavrentyeva 15,
\newline\hphantom{iii} 630090 Novosibirsk, Russian Federation}%
\email{kuznetsov\vbox{\hrule width 0.7mm}i@hydro.nsc.ru}%
\address{Sergey A. Sazhenkov
\newline\hphantom{iii} Laboratory for Mathematical and Computer Modeling
\newline\hphantom{iii} in Natural and Industrial Systems,
\newline\hphantom{iii} Altai State University, Prospekt Lenina 61,
\newline\hphantom{iii} 656049 Barnaul, Russian Federation,
\newline\hphantom{iii} and
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} Siberian Division of the Russian Academy of
Sciences,
\newline\hphantom{iii} pr. Acad. Lavrentyeva 15,
\newline\hphantom{iii} 630090 Novosibirsk, Russian Federation}%
\email{sazhenkovs@yandex.ru}%


\thanks{\sc Kuznetsov I.V. and Sazhenkov S.A.
The one-dimensional impulsive Barenblatt--Zheltov--Kochina equation with a transition layer}
\thanks{\copyright \ 2022  Kuznetsov I.V and Sazhenkov S.A}
\thanks{\rm The work was supported by the State Assignment of the Russian
Ministry of Science and Higher Education entitled "Modern methods of
hydrodynamics for environmental management, industrial systems and
polar mechanics" (Govt. contract code: FZMW-2020-0008, 24 January
2020)}
%\thanks{\it Received 26.06.2018, Published .....}%
%\top

\semrtop \vspace{1cm} \maketitle

%\linenumbers
{\small
\begin{quote}
\noindent{\sc Abstract.} The initial-boundary value problem for the
one-dimen\-sional impulsive pseudoparabolic equation is studied. As a
coefficient in the second-order diffusion term, this equation
contains the smooth\-ed Dirac delta-function concentrated at some time
moment. From a physical view\-point, such term allows to describe
impulsive pressure drop phenomena in filtration problems. Existence
and uniqueness of solutions for fixed values of the small parameter
of smoothing is proved. After this, the limiting passage as the
small parameter tends to zero is fulfilled and rigorously justified.
As the result, the limit instantaneous impulsive
micro\-sco\-pic-macroscopic model is establish\-ed. This model is
well-posed and involves the additional equation on a transition
layer posed on a `very fast' timescale.

\noindent{\bf Key words}: pseudoparabolic equation, impulsive
equation, strong solu\-tion, Fourier series, transition layer
\end{quote}}

\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}

\newcommand{\bu}{{\bar u}}

\allowdisplaybreaks


\section{Introduction}
\label{sec.1}\indent The classical Barenblatt--Zheltov--Kochina
equation
\begin{equation}
\label{e.01}
\partial_t u = \chi \partial_{xxt}^3 u + \nu \partial_{xx}^2 u + f \quad (\chi,\nu>0)
\end{equation}
(here, in one-dimensional case) describes nonstationary filtration
of a viscous fluid in a cracky-porous ground \cite{Barenblatt-1960}.
In this framework, the sought function $u$ in \eqref{e.01} has a
physical sense of the distribution of pressure in the gallery of
cracks. Equation \eqref{e.01} also arises in studies of
non-equilibrium processes in the heat transfer \cite{Chen-1968},
where $u$ plays the role of one of the temperatures in a
two-temperature continuum. As well, some dynamical problems
regarding non-Newtonian second-order fluid reduce to boundary value
problems for equation \eqref{e.01}, and in this context $u$ means
some expression of velocity components \cite{Ting-1963}.

If processes under study are rather regular, we can confine
ourselves to consider\-ation of rather regular coefficients
$\chi=\chi(x,t)$ and $\nu=\nu(x,t)$. The relevant theory of equation
\eqref{e.01} and of its multi-dimensional version is vast and deep;
its systematic exhibition can be found, for example, in
\cite{Demidenko-2003}. On the other hand, irregular processes, like
very fast pressure drop in the cracky gallery in the problem of
filtration, lead to the cases when coefficients become highly
non-smooth. More certainly, in this case coefficient $\nu$ may
behave closely to  $\nu_0+\alpha \delta_{(t=\tau)}$, where
$\nu_0=\mbox{const}>0$, $\alpha=\mbox{const} \ne 0$, and
$\delta_{(t=\tau)}$ is the Dirac delta concentrated at the time
moment $t=\tau$ of the (instant) pressure drop. For example, such
quick capillary pressure drops were experimentally observed in
\cite{Schluter-2017} in a two-phase flow. We can also note that the
additional term $\alpha \delta_{(t=\tau)}$ may correspond to
pressure fluctuations linked with `impulsive' fast diffusion in
porous media. In this view, we remark that fast diffusion can be
simulated also with the help of nonstandard gradient growth
\cite{Cao-2018}.

In the present paper, we approximate $\delta_{(t=\tau)}$ on the
so-called transition $\varepsilon$-layer. To this end, instead of
$\delta_{(t=\tau)}$, we put into equation \eqref{e.01} the sequence
$\{K_\varepsilon^\tau\}_{\varepsilon\to 0+}$, which converges
weakly$^\star$ to $\delta_{(t=\tau)}$. We observe that the similar
approach was applied in \cite{AKS-2020,KS-2022}, where the Dirac
delta-function $\delta_{(t=\tau)}$ was encompassed in the minor
(source) term, not in the term containing a derivative.

Now, in addition to the modification of \eqref{e.01} described above
regarding the approx\-imation of the Dirac delta, we also set
$\nu_0=1$, $\chi=1$ and $f=0$ for simplicity and formulate the
following initial-boundary value problem for the
Barenblatt--Zheltov--Kochina equation with the homogen\-eous
boundary conditions, which will be studied further in the article:
\begin{subequations}
\label{e.02}%
\begin{align}
&\partial_tu_\varepsilon
=\partial_{xxt}^3u_\varepsilon+\left(1+\alpha
K_\varepsilon^\tau(t)\right)\partial_{xx}^2u_\varepsilon,\quad(x,t)
\in \Pi_T, \label{e.02a}
\\
&u_\varepsilon(0,t)=u_\varepsilon(1,t)=0,\quad
t\in(0,T),\label{e.02b}
\\
&u_\varepsilon(x,0)=g(x),\quad x\in(0,1). \label{e.02c}
\end{align}
\end{subequations}

 In \eqref{e.02}, $\Pi_T=(0,1)\times(0,T)$ is a space-time
rectangle, $T=\mbox{const}>0$ and $\tau \in (0,T)$ are given fixed
time moments, $\alpha\in\mathbb{R}$ is a given fixed value,
$u_\varepsilon=u_\varepsilon(x,t)$ is the sought function, $g=g(x)$
is a given function, and $K_\varepsilon^\tau$ is a given smooth
kernel. For a fixed value $\tau$, kernel $K_\varepsilon^\tau =
K_\varepsilon^\tau(t)$ is supported on $\displaystyle
\left\{\tau-\frac{\varepsilon}2\leqslant
t\leqslant\tau+\frac{\varepsilon}2\right\}$ and is defined by the
formula
\begin{equation*} %\label{(2)}
K_\varepsilon^\tau (t)= \frac{1}{\varepsilon}
K\left(\frac{t-\tau}{\varepsilon}\right),\quad t\in [0,T],
\end{equation*}
where $K=K(\vartheta)$ is a nonnegative smooth even function
supported on segment $\displaystyle \left\{-\frac12\leqslant\vartheta\leqslant
\frac12\right\}$, with the mean value equal to unity, i.e.,
$\displaystyle \int_{-\frac12}^{\frac12} K(\vartheta)\,d\vartheta=1$, and
$\varepsilon>0$ is a small parameter. Thus, $K_\varepsilon^\tau$
approximates the Dirac delta-function $\delta_{(t=\tau)}$ in
weak$^\star$ sense as $\varepsilon \to 0+$, i.e., the limiting
relation $\displaystyle \lim\limits_{\varepsilon\to 0+}
\int_{\mathbb{R}} \phi(t) K_\varepsilon^\tau(t)\,dt =\phi(\tau)$
holds for any integrable in a neighborhood of $\{t=\tau\} \subset
\mathbb{R}$ function $\phi$ having the trace at the point $t=\tau$.
Note that
\begin{equation}
\varepsilon K_\varepsilon^\tau(\tau+\varepsilon\vartheta)=K(\vartheta), \quad \forall\,
\vartheta\in\left[-\frac12,\frac12\right].
\label{e.03}
\end{equation}

By analogy with the theory of generalized ordinary differential
equat\-ions \cite{Filippov-1988,Kurzweil-1958,Schwabik-1992}, we may
call \eqref{e.02a} {\it the generali\-zed
Barenblatt--Zheltov--Kochina equat\-ion} corresp\-ond\-ing to the
pressure of a liquid in cracks.

Further, in Sec. \ref{sec.2}, for any fixed rather small
$\varepsilon$ (say, $\varepsilon \in (0,\varepsilon_0]$, $\varepsilon_0\ll 1$), we construct a strong generalized
solution to problem \eqref{e.02} in the form of Fourier series
\eqref{e.07} and establish that this solution is unique and
satisfies the first and the second energy estimates uniform in
$\varepsilon$, see Proposition \ref{proposition.1}. In Sec.
\ref{sec.3}, we formulate Theorems \ref{theorem.1} and
\ref{theorem.2}, which are the main results of the paper and which
relate to the limiting transition in problem \eqref{e.02}, as
$\varepsilon\to 0+$. Theorem \ref{theorem.1} asserts the compactness
of the family $\{u_\varepsilon\}_{\varepsilon \in (0,1]}$ as
$\varepsilon \to 0+$ and the fact that the limit function
$u=\lim\limits_{\varepsilon\to 0+}u_\varepsilon$ satisfies the
homogeneous Barenblatt--Zheltov--Kochina equation outside the
section $\{t=\tau\}$. The proof of Theorem \ref{theorem.1} is given
in Secs. \ref{sec.4} and \ref{sec.6}. In order to prove compactness of
$\{u_\varepsilon\}_{\varepsilon \in (0,1]}$, we apply {the
Aubin--Lions--Simon lemma \cite[Theorem 3]{Simon-1986}, see also
\cite{Aubin-1963,Chen-2014,Lions-1961,Lions-1969,Rossi-2003}.}
Theorem \ref{theorem.2} deals with the study of the family
$\{u_\varepsilon\}_{\varepsilon \in (0,1]}$ near the section
$\{t=\tau\}$ and is proved in Sec. \ref{sec.5}. More certainly,
in Sec. \ref{sec.5}, in order to link the one-sided traces of the
limit solution at $t=\tau$, we rescale $u_\varepsilon$ on
$\displaystyle \left(0,\tau-\frac{\varepsilon}2\right)$,
$\displaystyle
\left(\tau-\frac{\varepsilon}2,\tau+\frac{\varepsilon}2\right)$, and
$\displaystyle \left(\tau+\frac{\varepsilon}2,T\right)$ and apply
{the Aubin--Lions lemma \cite[Lemma 2.48, p.
36]{Malek-1996}, \cite[Corollary 6]{Simon-1986}} to the rescaled
solution. As the result, for the rescaled solution we derive the
so-called {\it equation on the transition layer} between $t=\tau-0$
and $t=\tau+0$, see \eqref{e.11b}, which incorporates not the
`macroscopic' (`slow') time variable $t$, but the `microscopic'
(`fast') time variable $\bar{t}$, so that $\displaystyle \bar{t}\in
\left[-\frac12,\frac12\right]$, $t=\tau-0$ corresponds
to $\displaystyle \bar{t}=-\frac12+0$, and $t=\tau+0$ corresponds
to $\displaystyle \bar{t}=\frac12-0$.

In aggregation, the results of Theorems \ref{theorem.1} and \ref{theorem.2} give us the well-posed limit two-scale microscopic-macroscopic model \eqref{e.09b}--\eqref{e.09d}, \eqref{e.11b}--\eqref{e.11f} describing the process in which an instantaneous impulsive phenomenon manifests itself.

The main difference between the aforementioned results from
\cite{AKS-2020,KS-2022} and the results from the present paper is
that, in the present paper, the dissipative term persists on a
transition layer
$\displaystyle \left\{-\frac12\leqslant\bar{t}\leqslant\frac12\right\}$
after rescaling, since $\varepsilon
\partial_{xx}^2\bar{u}_\varepsilon$ does not vanish, as $\varepsilon\to
0+$ in equation \eqref{e.10}. This is one of the novelties in the
present paper.


We should emphasize the difference between impulsive differential equat\-ions with and without a {\it transition layer}. We address the
reader to the mono\-graphs \cite{Bainov.Simeonov-1993,Bainov.Covachev-1995,Bainov.Simeonov-1995,Laksh-1989,Samoilenko-1995,Stamov-2012}
devoted to fairly extensive studies of impulsive
differential equations without a \emph{transition layer}. In turn, impulsive differential equations with a
\emph{transition layer} are studied poorer, although there is a number of notable works as well, see
\cite{Coutinho-2009,Feketa-2021,Filippov-1988,Griffiths-1999,Klinshov-2021,Kurzweil-1958,Miller-2003,Nedeljkov-2012,
Schwabik-1992}. Moreover, the scaling on a transition
layer is linked with the theory of boundary transition layer
\cite{Kuehn-2015,Nayfeh-2000,Verhulst}. Also, it is important to mention a rather new type of equations, namely, non-instantaneous
differential equations \cite{Agarwal-2017,Wang-2018}, where a
transition layer is given a priori, without a limiting passage.

Remark that the straightforward substitution of $\displaystyle
\delta_{(t=\tau+0)}=w^\star\mbox{-}\lim\limits_{\varepsilon\to 0}
K_\varepsilon^\tau$ for $K_\varepsilon^\tau$ into \eqref{e.02a}
yields the {\it instantaneous impulsive} pseudoparabolic equation
\begin{equation*}
\partial_tu=\partial_{xxt}^3u+
\left(1+\alpha\delta_{(t=\tau+0)}(t)\right)\partial_{xx}^2u,\quad(x,t)
\in \Pi_T,
\end{equation*}
which is equivalent in the sense of distributions to the system
consisting of the homogeneous pseudoparabolic equation
\begin{equation} \label{impulse01}
\partial_tu=\partial_{xxt}^3u+ \partial_{xx}^2u\quad \mbox{in } \Pi_T\setminus \{t=\tau\}
\end{equation}
 and the
 {\it impulsive condition} on the section $\{t=\tau\}$:
\begin{equation} \label{impulse02}
u(x,\tau+0)- (1+\alpha) \partial_{xx}^2u(x,\tau+0)=
u(x,\tau-0)-\partial_{xx}^2u(x,\tau-0)\quad \mbox{for }x\in(0,1).
\end{equation}
One can notice that the system \eqref{impulse01}--\eqref{impulse02} does not coincide with the system \eqref{e.09b}, \eqref{e.11b}, \eqref{e.11d}, \eqref{e.11f}.

Let us now give a simple example \cite{Coutinho-2009} demonstrating the same feature, which manifests the
difference between impuls\-ive differential equations with a
\emph{transition layer} and instantaneous ones. Con\-sider the
first-order ordinary differential equation
\begin{equation}
\label{(10)}%
\frac{df_\varepsilon(x)}{dx}=\alpha
f_\varepsilon(x)\delta_\varepsilon(x),\quad x\in{\mathbb R},
\end{equation}
with
\begin{equation*}
\alpha=\mbox{const}, \quad
\delta_\varepsilon(x)=\left\{\begin{array}{lll} \displaystyle
\frac1{\varepsilon}&\text{ if }&\displaystyle x\in \Bigl(-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\Bigr),
\\[2ex]
0 & \text{ if }& \displaystyle |x|\geqslant \frac{\varepsilon}{2}\,,
\end{array}\right. \quad \varepsilon >0.
\end{equation*}
It is easy to deduce that the function $\delta_\varepsilon$
weakly$^\star$ approximates the Dirac delta function
$\delta_{(x=0)}$ concentrated at the origin ($x=0$) and to calculate
explicitly the solution of \eqref{(10)} for every fixed
$\varepsilon>0$:
\[
f_\varepsilon(x)=C\exp\left(\alpha
\int\limits_0^x\delta_\varepsilon(y)\,dy\right)= \left\{
\begin{array}{lll}
\displaystyle C\exp\Bigl(-\frac\alpha2\Bigr)&\text{ if }& \displaystyle  x\leqslant
-\frac{\varepsilon}{2},
\\[2ex]
\displaystyle C\exp\Bigl(\frac{\alpha x}{\varepsilon}\Bigr)&\text{
if }& \displaystyle  -\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2},
\\[2ex]
\displaystyle C\exp\Bigl(\frac{\alpha}2\Bigr)&\text{ if
}& \displaystyle  x\geqslant \frac{\varepsilon}{2}.
\end{array}
\right.
\]
If $C\ne 0$, one establishes that the limit function
$f=\lim\limits_{\varepsilon\to 0+}f_\varepsilon$ meets the jump
condition
\begin{equation} \label{(11)}
f(0+)-f(0-)=\tanh\left(\frac\alpha2\right)\bigl[f(0+)+f(0-)\bigr].
\end{equation}
At the same time, the solution of equation $\displaystyle
\frac{df(x)}{dx}=\alpha f(x)\delta_{(x=0)}$ meets the standard
impulsive condition
\begin{equation} \label{(12)}
f(0+)-f(0-)=\frac\alpha2\bigl[f(0+)+f(0-)\bigr],
\end{equation}
and we immediately notice the discrepancy between \eqref{(11)} and
\eqref{(12)}, since these two conditions coincide only for
$\alpha=0$, i.e., in the trivial case.


Concluding this introduction, let us remark that pseudoparabolic
equat\-ions are of Sobolev type \cite{Alshin-2011,Demidenko-2003}
and are also applied for regularization of forward-backward parabolic
equations
 \cite{Barenblatt-1993,NovickCohen-1991,Plotnikov-1999,Smarrazzo-2022}.
The presence of $\partial_{xxt}^3u_\varepsilon$ is essential in our
study. The purely parabolic equation of the form \eqref{e.02a},
i.e., the equation where the third order derivative
$\partial_{xxt}^3 u_\varepsilon$ is discarded, requires an
additional research and this question lays beyond the present
article. As well, an interesting direction of further research may
be devoted to inclusion of a non-local term of the Fredholm type
into \eqref{e.02a}. Specifically, inserting the term
\begin{equation*}
K_\varepsilon^\tau(t)
\int_0^TK_\varepsilon^\tau(s)\partial_{xx}^2u_\varepsilon(x,s)\,ds
\end{equation*}
on the place of $K_\varepsilon^\tau(t)\partial_{xx}^2u_\varepsilon(x,t)$,
we encounter the generalized Fredholm type integro-differential
pseudo\-parabolic equa\-tion, which may have a significant place in the theory of
non-local in time partial differential equations
\cite{Starovoitov-2018,Starovoitov-2021,Yuldashev-2018} and in the theory of
impulsive integro-differential equations \cite{Lakshmikantham}.

\section{Well-posedness of problem \eqref{e.02}.\\ The first and the second energy estimates}
\label{sec.2}

We start our study of problem \eqref{e.02} by establishing the following:
\begin{proposition}
\label{proposition.1} Whenever $g\in W^{2,2}(0,1) \cap W^{1,2}_0(0,1)$, there is a
unique strong generalized solution $u_\varepsilon=u_\varepsilon(x,t)$ to problem
\eqref{e.02}. Moreover, the solution satisfies the uniform in
$\varepsilon$ first and second energy estimates
\begin{align}
\label{e.04} & \|u_\varepsilon\|_{L^\infty(0,T;L^2(0,1))}^2 +\|
\partial_xu_\varepsilon\|_{L^\infty(0,T;L^2(0,1))}^2
\\
&\hspace{37mm} \leqslant C\bigl(\|g\|_{L^2(0,1)}^2 +\|\partial_x
g\|_{L^2(0,1)}^2\bigr) \stackrel{\mathrm{def}}{=} M_1, \;\;
\forall\, \varepsilon\in (0,\varepsilon_0],\nonumber
\end{align}
\begin{equation} \label{e.05}
\|\partial_x u_\varepsilon\|_{L^\infty(0,T;L^2(0,1))}^2
+\|\partial_{xx}^2 u_\varepsilon\|_{L^\infty(0,T;L^2(0,1))}^2
\leqslant M_2,
\end{equation}
and the bound
\begin{equation}
\label{e.06} \|\partial_x u_\varepsilon\|_{L^\infty(\Pi_T)}
\leqslant M_3,
\end{equation}
where $C,M_1,M_2,M_3=\emph{const}>0$ do not depend on $\varepsilon$.
\end{proposition}

The notion of strong generalized solution is quite standard:
\begin{definition}
\label{definition1} A function $u_\varepsilon$: $\Pi_T\mapsto
\mathbb{R}$ is called a \textbf{strong generalized solution} of problem
\eqref{e.02} if
\begin{enumerate}
\item[{\bf (i)}] $u_\varepsilon,\,\partial_t u_\varepsilon \in L^\infty(0,T;W^{2,2}(0,1) \cap W^{1,2}_0(0,1))$,
\item[{\bf (ii)}] $u_\varepsilon$ satisfies \eqref{e.02a} a.e. in $\Pi_T$ and \eqref{e.02c} in the strong sense, namely, 
$$\|u_\varepsilon(\cdot,t)-g(\cdot)\|_{W^{2,2}(0,1)} \underset{t\to 0+}{\longrightarrow} 0.$$
\end{enumerate}
\end{definition}

The proof of Proposition \ref{proposition.1} can be divided into four stages and is given in Sections \ref{sec.4a}--\ref{sec.3.3a} further.

\subsection{Existence}
\label{sec.4a}
Since we deal with the linear case, we apply the Fourier series
method. Taking into account homogeneous conditions, a solution is
sought as a Fourier series:
\begin{equation}
\label{e.07} u_\varepsilon(x,t)=\sum\limits_{n=1}^\infty
c_{\varepsilon,n}(t)\varphi_n(x),
\end{equation}
where $\varphi_n(x)=\sin(\lambda_nx)$, $\lambda_n=n\pi$. Therefore,
the coefficients satisfy the set of Cauchy problems ($n\in\mathbb{N}$)
\begin{subequations}
\label{e.08}%
\begin{align}
&c_{\varepsilon,n}'(t)=\frac{-\lambda_n^2}{1+\lambda_n^2}(1+\alpha
K_\varepsilon^\tau(t))c_{\varepsilon,n}(t),\quad t\in(0,T),\\
&c_{\varepsilon,n}(0)=g_n,
\end{align}
\end{subequations}
where, in turn, $g_n$ are the Fourier coefficients of initial data
$g\in W^{2,2}(0,1) \cap W^{1,2}_0(0,1)$.

Now, let us apply the results from
\cite{Showalter-1970,Showalter-1977} to \eqref{e.08}. Namely, for
every $t\in[0,+\infty)$ and $\varepsilon>0$ we introduce the linear
operator $E_\varepsilon(t)$: $L^2(0,1) \mapsto L^2(0,1)$ by the
rule:
\begin{equation*}
E_\varepsilon(t)v=\sum\limits_{n=1}^\infty
\exp\left(-\frac{\lambda_n^2}{1+\lambda_n^2}
\left(t+\alpha\int_0^tK_\varepsilon^\tau(s)\,ds\right)\right)v_n
\varphi_n(x),\quad t>0,
\end{equation*}
for an arbitrary function $v$ with
$\|v\|^2_{L^2(0,1)}=\sum\limits_{n=1}^\infty v_n^2<\infty$.

Operator $E_\varepsilon(t)$ depends on $t$ as on a parameter and has
the following properties.
\begin{enumerate}
\item[(i)] $E_\varepsilon(0)=I$,\vspace{2mm}
\item[(ii)] $\Vert E_\varepsilon\Vert \leqslant \exp(|\alpha|)$.\\[1ex]
\textit{Proof of assertion (ii)}. By Parseval’s identity, we have
\begin{align*}
\Vert E_\varepsilon(t)v\Vert_{L^2(0,1)}^2 =
\sum\limits_{n=1}^\infty v_n^2
\exp\left(-\frac{2\lambda_n^2}{1+\lambda_n^2}
\left(t+\alpha\int_0^tK_\varepsilon^\tau(s)\,ds\right)\right) \\
\leqslant \exp(2|\alpha|)\sum\limits_{n=1}^\infty v_n^2 =\exp(2|\alpha|)\Vert
v\Vert_{L^2(0,1)}^2,
\end{align*}
and the estimate follows.\vspace{2mm}
\item[(iii)] $E_\varepsilon(\cdot)v\in C([0,+\infty);L^2(0,1))$ for any $v\in L^2(0,1)$.\\[1ex]
\textit{Proof of assertion (iii)}. On the strength of Parseval's identity followed by simple estimation, we have
\begin{align*}
& \int_0^1( E_\varepsilon(t_2)v -E_\varepsilon(t_1)v)^2\,dx \\
& \hspace{5mm} \leqslant \frac12\sum\limits_{n=1}^\infty
\Bigg[\exp\left(-\frac{\lambda_n^2}{1+\lambda_n^2}
\left(t_2+\alpha\int_0^{t_2}K_\varepsilon^\tau(s)\,ds\right)\right)
\\
& \hspace{10mm} -\exp\left(-\frac{\lambda_n^2}{1+\lambda_n^2}
\left(t_1+\alpha\int_0^{t_1}K_\varepsilon^\tau(s)\,ds\right)\right)
\Biggr]^2v_n^2\rightarrow 0,\quad \text{as }t_2-t_1\rightarrow 0,
\end{align*}
since the series in the right hand side of the inequality are uniformly continuous in $t:=t_1,t_2$ with respect to $n$ (but not with respect to $\varepsilon$, though).

Thus assertion (iii) follows.\vspace{2mm}
\item[(iv)] For any fixed $\varepsilon>0$, function $u_\varepsilon=E_\varepsilon(\cdot)g$ is a strong generalized solution of problem \eqref{e.02} in the sense of Definition \ref{definition1}.\\[1ex]
\textit{The proof of assertion (iv)} is analogous to the one in \cite[Secs. 4,\,5]{Showalter-1970},
where a more general linear case is treated.
\end{enumerate}

\subsection{The first energy estimate uniform in $\boldsymbol\varepsilon$. Uniqueness}
\label{sec.2.1} We multiply \eqref{e.02a} by the solution $u_\varepsilon$,
integrate over $(0,1)\times (0,t)$, where $t\in (0,T)$,  integrate
by parts in $x$, taking into account \eqref{e.02c}, and then use
\eqref{e.02b} to get the first energy identity
\begin{align}
& \frac{1}{2} \int_0^1\left(|u_\varepsilon(x,t)|^2 + |\partial_x
u_\varepsilon(x,t)|^2\right)\,dx  -\frac{1}{2} \int_0^1
\left(|g(x)|^2+|\partial_xg(x)|^2\right)dx \label{FEE-1}
\\
& \hspace{3mm} =-\int_0^t \int_0^1 \left(1+\alpha
K_\varepsilon^\tau(t')\right)
\left\vert\partial_xu_\varepsilon(x,t')\right\vert^2\,dxdt' \nonumber
\\
& \hspace{3mm} \leqslant |\alpha|\int_0^t\int_0^1K_\varepsilon^\tau(t')
\left\vert\partial_xu_\varepsilon(x,t')\right\vert^2\,dxdt', \quad
\forall\,t\in [0,T]. \nonumber
\end{align}
Applying the Gr\"{o}nwall--Bellman lemma to this inequality, we
immediately estab\-lish estimate \eqref{e.04}. Uniqueness of solution $u_\varepsilon$ follows immediately from \eqref{e.04} due to linearity of the problem.


\subsection{The second energy estimate uniform in $\boldsymbol \varepsilon$}
\label{sec.2.2} \indent Since $u_\varepsilon=E_{\varepsilon}(t) g$,
$\partial_{xx}^2g\in L^2(0,1)$ and
$\partial_{xx}^2u_\varepsilon(\cdot,t)\in L^2(0,1)$, we are eligible
to multiply \eqref{e.02a} by $-\partial_{xx}^2 u_\varepsilon$,
integrate over $(0,1)\times (0,t)$, where $t\in (0,T)$,  integrate
by parts in $x$ in the first two terms, taking into account
\eqref{e.02c}, and then use \eqref{e.02b} to get {\it the second
energy identity}
\begin{align}
& \frac{1}{2} \int_0^1 \bigl(|\partial_{x} u_\varepsilon(x,t)|^2
+|\partial_{xx}^2u_\varepsilon(x,t)|^2\bigr)\,dx -\frac{1}{2}\int_0^1 \bigl(|\partial_{x}g(x)|^2 +
|\partial_{xx}^2g(x)|^2 \bigr)\,dx \nonumber
\\
& \hspace{4mm} =-\int_0^t \int_0^1 \left(1+\alpha
K_\varepsilon^\tau(t')\right)\vert\partial_{xx}^2
u_\varepsilon(x,t')\vert^2\,dx dt'\nonumber
\\
& \hspace{4mm} \leqslant \int_0^t \int_0^1|\alpha|
K_\varepsilon^\tau(t')\vert\partial_{xx}^2
u_\varepsilon(x,t')\vert^2\,dx dt',\quad \forall\,t\in [0,T].
\nonumber
\end{align}
Applying the Gr\"{o}nwall--Bellman lemma to this inequality, we
arrive at the second energy estimate \eqref{e.05}.

\subsection{The bound on ${\boldsymbol \partial}_{\boldsymbol x} {\boldsymbol u}_{\boldsymbol \varepsilon}$ uniform in $\boldsymbol\varepsilon$}
\label{sec.3.3a} \indent The bound \eqref{e.06} directly
follows from the two energy estimates and the Newton--Leibnitz
formu\-la.

Proposition \ref{proposition.1} is fully proved.

\section{The main results}
\label{sec.3}
The main results of the article are the following Theorems \ref{theorem.1} and \ref{theorem.2}. In Theorem
\ref{theorem.1} we deal with the limit
$u(x,t)=\lim\limits_{\varepsilon\to0+}u_\varepsilon$ apart from the section
$\{t=\tau\}$. In Theorem \ref{theorem.2} we formulate the initial-boundary value problem on the `transition layer' between $t=\tau-0$ and $t=\tau+0$. Solution of this problem links $u(x,\tau-0)$ with
$u(x,\tau+0)$.
%In the present section, we are going to describe the scheme of the main result.

\begin{theorem}
\label{theorem.1} The family $\{u_\varepsilon\}_{\varepsilon \in
(0,\varepsilon_0]}$ is relatively compact in
$L^2(0,T;W_0^{1,2}(0,1))$ and relatively weakly$^\star$ compact in
$L^\infty(0,T;W^{2,2}(0,1))$, as $\varepsilon \to 0+$. In other
terms, there exist a subsequence from
$\{u_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]}$, still labeled by $\varepsilon$,
and a limit function $u\in L^\infty(0,T;W^{2,2}(0,1)\cap W^{1,2}_0(0,1))$ such that
\begin{subequations}
\begin{align}
&\label{e.09a} u_\varepsilon \underset{\varepsilon\to
0+}{\longrightarrow} u \quad \mbox{strongly in }
L^2(0,T;W_0^{1,2}(0,1))
\\[-1.1ex]
&\hspace{17mm}\mbox{ and weakly$^\star$ in }
L^\infty(0,T;W^{2,2}(0,1)),\nonumber
\\
&\label{e.09b}\partial_t
u=\partial_{xxt}^3u+\partial_{xx}^2u\quad\mbox{for }(x,t)\in
\Pi_T\setminus\{t=\tau\},
\\
&\label{e.09c}u(0,t)=u(1,t)=0\quad \text{for }
x\in(0,1),\,t\in(0,\tau)\cup(\tau,T),
\\
&\label{e.09d}u(x,0)=g(x)\quad\mbox{for } x\in(0,1).
\end{align}
\end{subequations}
\end{theorem}

Equation \eqref{e.09b} is understood a.e. on $(0,1)\times (0,\tau)$ and a.e. on $(0,1)\times (\tau,T)$. Initial condition \eqref{e.09d} is understood in the sense of the strong trace, namely, 
$$\|u(\cdot,t)-g(\cdot)\|_{W^{2,2}(0,1)} \underset{t\to 0+}{\longrightarrow} 0.$$


In Sections \ref{sec.4} and \ref{sec.5}, the identities for the
one-sided limits are proved:
\[
u(x,\tau-0)=\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau-0)=
\lim\limits_{\varepsilon\to0+}
u_\varepsilon\Bigl(x,\tau-\frac\varepsilon2\Bigr),
\]
\[
u(x,\tau+0)=\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau+0)=
\lim\limits_{\varepsilon\to0+}
u_\varepsilon\Bigl(x,\tau+\frac\varepsilon2\Bigr).
\]
Therefore, following the idea presented in
\cite{Kurzweil-1958,KS-2022,V-2002}, along with the family\linebreak
$\{u_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]}$, in order to link $u(x,\tau-0)$
and $u(x,\tau+0)$, we use rescaling on the transition layer
$\displaystyle
\Bigl[\tau-\frac{\varepsilon}2,\tau+\frac{\varepsilon}2\Bigr]$ and
deal with the rescaled solutions
$$\bar{u}_\varepsilon(x,\bar{t})=u_\varepsilon(x,\tau+\varepsilon\bar{t}),$$
where $\displaystyle
\bar{u}_\varepsilon\!\!:\Pi
\mapsto \mathbb{R}$ and we denote $\displaystyle \Pi:=(0,1)\times \Bigl(-\frac12,\frac12\Bigr)$.

For every fixed $\varepsilon>0$, with the help
of a new variable $\displaystyle
\bar{t}=\frac{t-\tau}{\varepsilon}\in\Bigl[-\frac12,\frac12\Bigr]$
and multiplication by $\varepsilon$, from \eqref{e.02a} we deduce
the rescaled equation
\begin{align}
&\label{e.10}\partial_{\bar{t}}\bar{u}_\varepsilon
=\partial_{xx\bar{t}}^3\bar{u}_\varepsilon+\left(\varepsilon+\alpha
K(\bar{t})\right)
\partial_{xx}^2\bar{u}_\varepsilon,\quad (x,\bar{t})\in\Pi.
\end{align}


\begin{theorem}
\label{theorem.2} The family of rescaled solutions
$\left\{\bar{u}_\varepsilon\right\}_{\varepsilon\in (0,\varepsilon_0]}$ is relati\-vely
compact in $\displaystyle L^2(\Pi)$ and relatively weakly$^\star$
compact in $\displaystyle
L^\infty\Bigl(-\frac12,\frac12;W^{2,2}(0,1)\Bigr)$. In other
terms, there exist a subsequence from
$\{\bar{u}_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]}$, still labeled by
$\varepsilon$, and a limit function $\displaystyle \bar{u}\in
L^\infty\Bigl(-\frac12,\frac12;W^{2,2}(0,1)\cap W^{1,2}_0(0,1)\Bigr)$ such that
\begin{subequations}
\begin{align}
&\label{e.11a}\bar{u}_\varepsilon \underset{\varepsilon \to
0+}{\longrightarrow} \bar{u} \quad \mbox{strongly in }
L^2(\Pi)
\\[-1ex]
&\hspace{18mm} \mbox{and weakly$^\star$ in }
L^\infty\Bigl(-\frac12,\frac12;W^{2,2}(0,1)\Bigr).\nonumber
 \end{align}
Moreover, on the transition layer we write out the pseudo-parabolic
equation supple\-mented with the homogeneous boundary conditions and
the initial condit\-ion:
\begin{align}
&
%
\label{e.11b}\partial_{\bar{t}}\bar{u}=\partial_{xx\bar{t}}^3\bar{u}+\alpha
K(\bar{t})\partial_{xx}^2\bar{u}\mbox{ for }(x,\bar{t})\in\Pi,
\\
&
%
\label{e.11c}\bar{u}(0,\bar{t})=\bar{u}(1,\bar{t})=0\text{ for }
\bar{t}\in\left(-\frac12,\frac12\right),
\\
&
%
\label{e.11d}\bar{u}\left(x,-\frac12+0\right)=u(x,\tau-0)\mbox{ for
} x\in(0,1).
\end{align}
Finally, the matching condition for the \emph{transition layer solution} $\bar{u}$ on the section $\displaystyle \Bigl\{\bar{t}=\frac12-0\Bigr\}$ and the \emph{outer solution} $u$ on the section $\{t=\tau+0\}$ holds true as follows:
\begin{equation}
\label{e.11f} u(x,\tau+0)=\bar{u}\left(x,\frac12-0\right) \mbox{ for
} x\in(0,1).
\end{equation}
\end{subequations}
\end{theorem}

Equation \eqref{e.11b} is understood a.e. in $\Pi$.
Initial condition \eqref{e.11d} and matching condition \eqref{e.11f} are understood in the sense of the
strong traces, namely,
\begin{align}
& \bigl\|\bar{u}(\cdot,\bar{t})-u(\cdot,\tau-0)\bigl\|_{W^{2,2}(0,1)}
\underset{\bar{t}\to-\frac12+0}{\longrightarrow} 0, \label{matching_01} \\
& \bigl\|u(\cdot,t)-\bar{u}\Bigl(\cdot,\frac12-0\Bigr)\bigr\|_{W^{2,2}(0,1)}
\underset{t\to \tau+0}{\longrightarrow} 0. \label{matching_02}
\end{align}

\begin{remark} \label{remark1} As it has already been mentioned in Sec. \ref{sec.1}, the system \eqref{e.09b}--\eqref{e.09d}, \eqref{e.11b}--\eqref{e.11f} is a closed limit two-scale model describing the process in which an instantaneous impulsive phenomenon manifests itself. Existence of a strong general\-ized solution $(u,\bar{u})$ to this model follows directly from Theorems \ref{theorem.1} and \ref{theorem.2} on the strength of the limiting passages, as $\varepsilon \to 0+$.

Moreover, the solution $(u,\bar{u})$ is unique. Indeed, in fact, the system \eqref{e.09b}--\eqref{e.09d}, \eqref{e.11b}--\eqref{e.11f} is a set of the three initial-boundary value problems that should be solved successively. In the first step, we solve problem \eqref{e.09b}--\eqref{e.09d} on the slow time interval $\{0<t<\tau\}$. In the second step, we solve problem \eqref{e.11b}--\eqref{e.11d} on the fast time interval $\displaystyle \Bigl\{-\frac{1}{2} < \bar{t} < \frac{1}{2} \Bigr\}$, where the initial data is taken from the first step: $\displaystyle \bar{u}\Bigl(x,-\frac{1}{2}\Bigr)=u(x,\tau-0)$. Finally, we solve problem \eqref{e.09b}, \eqref{e.09c}, \eqref{e.11f} on the slow time interval $\{\tau<t<T\}$, where the initial data is taken from the second step: $\displaystyle u(x,\tau+0)=\bar{u}\Bigl(x,\frac{1}{2}-0\Bigr)$. Each of these three problems is linear and admits the first energy estimate, analogous to estimate \eqref{FEE-1}. Therefore the solution of each of the problems is unique.
\end{remark}

\section{Proof of Theorem \ref{theorem.1} (beginning)}
\label{sec.4}
\subsection{Uniform boundedness and equi-continuity of $\boldsymbol{\{u_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]}}$}
\label{sec.3.3} {
\begin{lemma} \label{lemma1} For all
$\varepsilon\in (0,\varepsilon_0]$, the family of strong solutions
of problem \eqref{e.02} satisfies the following demands:
\begin{itemize}
\item uniform boundedness in $L^2(0,T;W^{2,2}(0,1))$;
\item integral equi-continuity
\begin{equation*}
\Vert\tau_hu_\varepsilon-u_\varepsilon\Vert_{L^2(0,T-h;W_0^{1,2}(0,1))}\leqslant Ch,\quad \forall\, h>0 \; (h\ll 1),
\end{equation*}
 where $\tau_hu_\varepsilon(x,t):=u_\varepsilon(x,t+h)$ and constant $C>0$ does not depend on $\varepsilon$ and $h$.
\end{itemize}
\end{lemma}}
\begin{proof}
{The uniform boundedness follows from \eqref{e.05}. In order to
prove the integral equi-continuity, we rewrite
\begin{align*}
& \Vert \tau_h u_\varepsilon - u_\varepsilon
\Vert_{L^2(0,T-h;W_0^{1,2}(0,1))}^2 \\
& =  \int_0^{T-h} \bigl(\Vert
u_\varepsilon(\cdot,t+h)-u_\varepsilon(\cdot,t)\Vert_{L^2(0,1)}^2
+\Vert\partial_x
u_\varepsilon(\cdot,t+h)-\partial_xu_\varepsilon(\cdot,t)\Vert_{L^2(0,1)}^2
\bigr)\,dt.
\end{align*}
Therefore,}
\begin{align*} &
\Vert
u_\varepsilon(\cdot,t+h)-u_\varepsilon(\cdot,t)\Vert_{L^2(0,1)}^2+
\Vert\partial_x
u_\varepsilon(\cdot,t+h)-\partial_xu_\varepsilon(\cdot,t)\Vert_{L^2(0,1)}^2
\\
&\hspace{3mm}=\sum\limits_{n=1}^\infty
(1+\lambda_n^2)\left(c_{\varepsilon,n}(t+h)
-c_{\varepsilon,n}(t)\right)^2
\\
&\hspace{3mm}=\sum\limits_{n=1}^\infty (1+\lambda_n^2)\int_{t}^{t+h}
c_{\varepsilon,n}'(s)\,ds\left(c_{\varepsilon,n}(t+h)-
c_{\varepsilon,n}(t)\right)
\\
&\hspace{3mm}=\sum\limits_{n=1}^\infty\lambda_n^2\int_{t}^{t+h}
c_{\varepsilon,n}(s)(1+\alpha
K_{\varepsilon}^\tau(s))\,ds\left(c_{\varepsilon,n}(t+h)
-c_{\varepsilon,n}(t)\right)
\\
&\hspace{3mm} \leqslant 2\int_{t}^{t+h}
(1+|\alpha|K_{\varepsilon}^\tau(s))\,ds\sum\limits_{n=1}^\infty
\lambda_n^2\max\limits_{t\in[0,T]}|c_{\varepsilon,n}(t)|^2
\\
&\hspace{3mm} \leqslant 2C\int_{t}^{t+h} (1+{\vert\alpha\vert}
K_{\varepsilon}^\tau(s))\,ds\sum\limits_{n=1}^\infty\lambda_n^2g_n^2.
\end{align*}
This provides
\begin{align*}
&{\Vert \tau_h u_\varepsilon-u_\varepsilon
\Vert_{L^2(0,T-h;W_0^{1,2}(0,1))}^2}
\\
&\hspace{3mm} \leqslant 2C\int_0^{T-h} \left(\int_{t}^{t+h} (1+\vert\alpha\vert
K_{\varepsilon}^\tau(s))\,ds\right)dt\sum\limits_{n=1}^\infty\lambda_n^2g_n^2\leqslant
Ch,
\end{align*}
since
\begin{align*}
&\int_{0}^{T-h}\left(\int_{t}^{t+h}K_\varepsilon^\tau(s)\,ds\right)\,dt=
\int_{0}^{h}K_\varepsilon^\tau(s)s\,ds+\int_{h}^{T-h}K_\varepsilon^\tau(s)h\,ds
\\
&\hspace{6mm}+\int_{T-h}^TK_\varepsilon^\tau(s)(T-s)\,ds\leqslant
h\int_0^TK_\varepsilon^\tau(s)\,ds=h,
\end{align*}
for $0<\tau<T$ and $h,\varepsilon\ll1$.

Lemma \ref{lemma1} is proved.
\end{proof}

\subsection{Relative compactness of $\boldsymbol{\{u_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]}}$
{in $\boldsymbol{L^2(0,T;W_0^{1,2}(0,1))}$}} \label{sec.3.4}
 Lem\-ma \ref{lemma1}
enables to apply the Aubin--Lions--Simon lemma, see \cite[Theorem
3]{Simon-1986}, where $X=W^{2,2}(0,1) \cap W^{1,2}_0(0,1)$ and $B=W_0^{1,2}(0,1)$,
so that $X$ is compactly embedded in $B$. Due to this
and Lemma \ref{lemma1}, sequence $\{u_\varepsilon\}_{\varepsilon\to
0+}$ is relatively compact in $L^2(0,T;W_0^{1,2}(0,1))$. Bound
\eqref{e.05} implies that $\{u_\varepsilon\}_{\varepsilon\to 0+}$ is
relatively weakly$^\star$ compact in $L^\infty(0,T;W^{2,2}(0,1))$.
Due to these properties, there exist a subsequ\-ence from
$\{u_\varepsilon\}_{\varepsilon\to 0+}$ and a limit function $u\in
L^\infty(0,T;W^{2,2}(0,1)\cap W^{1,2}_0(0,1))$ satisfying the limiting relation
\eqref{e.09a} and equation \eqref{e.09b} in the sense of distributions in $(0,1)\times (0,\tau)$ and in $(0,1)\times (\tau,T)$.

\subsection{Equation \eqref{e.09b} on $\boldsymbol{(0,1)\times (0,\tau)}$. Initial condition \eqref{e.09d}} \label{sec.3.5}
Note that equation \eqref{e.09b} coincides with equation \eqref{e.02a} for $\displaystyle t\in \Bigl(0,\tau-\frac{\varepsilon}{2}\Bigr)$, since $K_\varepsilon^\tau$ is supported on the segment $\displaystyle
\left\{\tau-\frac{\varepsilon}{2}\leqslant t \leqslant \tau+\frac{\varepsilon}{2}\right\}$. Therefore $u$ coincides with
$u_\varepsilon$ for $\displaystyle t\in \Bigl(0,\tau-\frac{\varepsilon}{2}\Bigr)$. Hence the initial value $g$ is attained by $u$ in the strong trace sense, as in Definition \ref{definition1} and, on $(0,1)\times (0,\tau)$, the limit function $u$ has the same regularity properties as the solution $u_\varepsilon$ in Definition \ref{definition1}. This implies that equation \eqref{e.09b} holds a.e. on $(0,1)\times (0,\tau)$.

In order to complete the proof of Theorem \ref{theorem.1}, it remains to show that equation \eqref{e.09b} holds a.e. on $(0,1)\times (\tau,T)$, as well. This is done further in Sec. \ref{sec.6}.

\section{Proof of Theorem \ref{theorem.2}}
\label{sec.5}%
%Using methods from the boundary layer theory ,
In the present section, we rescale {the solution $u_\varepsilon$} in the
time variable, correspond\-ingly, in the three domains:
\begin{itemize}
\item $\displaystyle (0,1)\times \Bigl(0,\tau-\frac\varepsilon2\Bigr)$ is mapped  into
$(0,1)\times(0,\tau)$ and a rescaled solution
$\widehat{u}_\varepsilon$, defined by Fourier coefficients
\eqref{e.12}, satisfies the problem
 \eqref{e.13}, stated further;
\item $\displaystyle (0,1)\times \Bigl(\tau-\frac\varepsilon2,\tau+\frac\varepsilon2\Bigr)$
is mapped into $\displaystyle (0,1)\times
\Bigl(-\frac12,\frac12\Bigr)$ and a rescaled solution
$\bar{u}_\varepsilon$, defined by the Fourier coefficients
\eqref{e.14}, satisfies problem
 \eqref{e.10}, \eqref{e.15a}, \eqref{e.15b};
\item $\displaystyle (0,1)\times \Bigl(\tau+{\frac\varepsilon2},T\Bigr)$ is mapped into $(0,1)\times(\tau,T)$
and a rescaled solution $\widetilde{u}_\varepsilon$, defined by the
Fourier coefficients \eqref{e.16}, satisfies problem \eqref{e.17}.
\end{itemize}

With the help of this rescaling, we can link
$\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau-0)$ with\linebreak
$\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau+0)$ and,
correspondingly,
$\lim\limits_{\varepsilon\to0+}\partial_xu_\varepsilon(x,\tau-0)$
with
$\lim\limits_{\varepsilon\to0+}\partial_xu_\varepsilon(x,\tau+0)$.
In each of the three domains, we apply the original version of the Aubin--Lions lemma.

\subsection{Rescaling on $\displaystyle
\boldsymbol{t\in \Bigl[0,\tau-\frac\varepsilon2\Bigr]}$}
\label{sec.5.1} Here we are going to rescale system \eqref{e.08}. On
$\displaystyle \Bigl\{0\leqslant t\leqslant\tau-\frac{\varepsilon}2\Bigr\}$ we
take $\displaystyle \widehat{t}:=
\frac{t\tau}{\displaystyle \Bigl(\tau-\frac{\varepsilon}2\Bigr)}\in[0,\tau]$ and
\[
\widehat{c}_{\varepsilon,n}(\widehat{t}):=
c_{\varepsilon,n}\left(\frac{\displaystyle \Bigl(\tau-\frac{\varepsilon}2\Bigr)}{\tau}\,\widehat{t}\right).
\]
With $\displaystyle
dt=\frac{\displaystyle \Bigl(\tau-\frac{\varepsilon}2\Bigr)}{\tau}\,d\widehat{t}$, system
\eqref{e.08} is rewritten in the form:
\begin{subequations}
\label{e.12}%
\begin{align}
&\widehat{c}_{\varepsilon,n}^{\,\,\prime}(\widehat{t})=
-\frac{\displaystyle \Bigl(\tau-\frac{\varepsilon}2\Bigr)}{\tau}\frac{\lambda_n^2}{1+\lambda_n^2}\,\widehat{c}_{\varepsilon,n}(\widehat{t}),\quad
\widehat{t}\in(0,\tau),\quad  n\in\mathbb{N},
\\
&\widehat{c}_{\varepsilon,n}(0)=g_n.
\end{align}
\end{subequations}
We introduce a function
\[
\widehat{u}_\varepsilon(x,\widehat{t})=\sum\limits_{{n}=1}^\infty
\widehat{c}_{\varepsilon,n}(\widehat{t})\varphi_n(x)
\]
as a solution of the rescaled equation
\begin{subequations}
\label{e.13}%
\begin{align}
&\partial_{\widehat{t}}\,
\widehat{u}_\varepsilon=\frac{\displaystyle \Bigl(\tau-\frac{\varepsilon}2\Bigr)}{\tau}
\partial_{xx}^2\widehat{u}_\varepsilon+
\partial_{xx\widehat{t}}^3\,\widehat{u}_\varepsilon
\end{align}
satisfying the respective initial and homogeneous boundary conditions
\begin{align}
&\widehat{u}_\varepsilon(x,0)=g(x),
\\
&\widehat{u}_\varepsilon(0,\widehat{t}\,)=\widehat{u}_\varepsilon(1,\widehat{t}\,)=0.
\end{align}
\end{subequations}
\begin{lemma}
\label{lemma2} The following estimate holds true:
\begin{align}
&\sup\limits_{\widehat{t}\in(0,\tau)} \Vert\widehat{u}_\varepsilon
(\cdot,\widehat{t})\Vert_{W^{2,2}(0,1)}^2+
\sup\limits_{\widehat{t}\in(0,\tau)}
\Vert\partial_{\widehat{t}}\,\widehat{u}_\varepsilon
(\cdot,\widehat{t})\Vert_{W_0^{1,2}(0,1)}^2\leqslant C\Vert
g\Vert_{W^{2,2}(0,1)}^2. \nonumber
\end{align}
\end{lemma}
\begin{proof}
The proof of this lemma is similar to the deduction of the first and the
second energy estimates, see Sections \ref{sec.2.1} and
\ref{sec.2.2}. After rescaling, it is feasible to
estimate $\partial_{x\widehat{t}}^2\,\widehat{u}_\varepsilon$ and
$\partial_{xx\widehat{t}}^3\,\widehat{u}_\varepsilon$ in
$L^\infty(0,\tau;L^2(0,1))$. But in the present paper, there is no
need in the latter estimate.
\end{proof}
\noindent The family of the functions $\widehat{u}_\varepsilon$ is
compact in $C([0,\tau];W_0^{1,2}(0,1))$. This is guaranteed by
Proposition \ref{proposition.1}, Lemma \ref{lemma2} and the
original version of the Aubin--Lions lemma, since the space
\[
\left\{v\in L^{\infty}(0,\tau;W^{2,2}(0,1)\cap W^{1,2}_0(0,1)):\,
\partial_tv\in L^\infty(0,\tau;W_0^{1,2}(0,1))\right\}
\]
is compactly embedded in $C([0,\tau];W_0^{1,2}(0,1))$.

\subsection{Rescaling on
$\displaystyle \boldsymbol{t\in
\Bigl[\tau-\frac\varepsilon2,\tau+\frac\varepsilon2\Bigr]}$}
\label{sec.5.2} Let $\displaystyle
\bar{t}:=\frac{t-\tau}{\varepsilon}$ for $\displaystyle
t\in\left[\tau-\frac{\varepsilon}2,\tau+\frac{\varepsilon}2\right]$.
Note that $\displaystyle \bar{t}\in\left[-\frac12,\frac12\right]$,
$dt=\varepsilon d\bar{t}$, and $t=\tau+\varepsilon\bar{t}$.
Taking into account \eqref{e.03}, for
\[
{\overline{c}_{\varepsilon,n}(\bar{t}):=c_{\varepsilon,n}(\tau+\varepsilon\bar{t})}
\]
system \eqref{e.08} transforms into
\begin{subequations}
\label{e.14}%
\begin{align}
&\overline{c}_{\varepsilon,n}^{\,\,\prime}(\bar{t})=
-\frac{\lambda_n^2}{1+\lambda_n^2}\bigl(\varepsilon+\alpha
K(\bar{t})\bigr)\overline{c}_{\varepsilon,n}(\bar{t}),\quad
\bar{t}\in\Bigl(-\frac12,\frac12\Bigr),\quad n\in\mathbb{N},
\\
&\overline{c}_{\varepsilon,n}\Bigl(-\frac12\Bigr)=\widehat{c}_{\varepsilon,n}(\tau).
\end{align}
\end{subequations}
As in Sec. \ref{sec.5.1}, from \eqref{e.14} it follows that
\[
\bar{u}_\varepsilon(x,\bar{t})= \sum\limits_{{n}=1}^\infty
\overline{c}_{\varepsilon,n}(\bar{t})\varphi_n(x)
\]
satisfies equation \eqref{e.10} with initial and boundary conditions
\begin{subequations}
\begin{align}
& \label{e.15a}
\bar{u}_\varepsilon\Bigl(x,-\frac12\Bigr)=\widehat{u}_\varepsilon(x,\tau),
\\
&\label{e.15b}
\bar{u}_\varepsilon(0,\bar{t})=
\bar{u}_\varepsilon(1,\bar{t})=0.
\end{align}
\end{subequations}

\begin{lemma}
\label{lemma3} The following estimate holds true:
\begin{align*}
&\sup\limits_{\bar{t}\in(-\frac12,\frac12)}
\Vert\bar{u}_\varepsilon(\cdot,\bar{t})\Vert_{W^{2,2}(0,1)}^2+
\sup\limits_{\bar{t}\in(-\frac12,\frac12)}
\Vert\partial_{\bar{t}}\bar{u}_\varepsilon
(\cdot,\bar{t})\Vert_{W_0^{1,2}(0,1)}^2\leqslant
\overline{C}_\alpha\Vert g\Vert_{W^{2,2}(0,1)}^2.
\end{align*}
\end{lemma}
\begin{proof}
 Justification of the lemma is based on Lemma \ref{lemma2} and is similar to the proof of the first and the second energy inequalities, see Secs. \ref{sec.2.1} and \ref{sec.2.2}.
\end{proof}
\noindent The family of functions $\bar{u}_\varepsilon$ is
compact in $\displaystyle C\Bigl(\Bigl[-\frac12,\frac12\Bigr];W_0^{1,2}(0,1)\Bigr)$. This is
guaranteed by Lemma \ref{lemma3} and the original version of the Aubin--Lions lemma, since
the space
\[
\Bigl\{v\in
L^{\infty}\Bigl(-\frac12,\frac12;W^{2,2}(0,1) \cap W^{1,2}_0(0,1)\Bigr)\!:\,
\partial_tv\in L^\infty\Bigl(-\frac12,\frac12;W_0^{1,2}(0,1)\Bigr)\Bigr\}
\]
is compactly embedded in $\displaystyle C\Bigl(\Bigl[-\frac12,\frac12\Bigr];W_0^{1,2}(0,1)\Bigr)$.

\subsection{Rescaling on $\displaystyle \boldsymbol{t\in \Bigl[\tau+\frac\varepsilon2,T\Bigr)}$}
\label{sec.5.3} For $\displaystyle t\in \Bigl[\tau+\frac{\varepsilon}2,T\Bigr)$ we set
$$\widetilde{t}:=\frac{-\varepsilon
T}{\displaystyle 2\Bigl(T-\tau-\frac{\varepsilon}2\Bigr)}+
\frac{(T-\tau)\,t}{\displaystyle \Bigl(T-\tau-\frac{\varepsilon}2\Bigr)}\in[\tau,T),$$ and,
correspondingly,
$$
\widetilde{c}_{\varepsilon,n}(\widetilde{t})=
c_{\varepsilon,n}\left(\frac{\displaystyle \Bigl(T-\tau-\frac{\varepsilon}2\Bigr)}{(T-\tau)}\,\widetilde{t}
+\frac{-\varepsilon T}{2(T-\tau)}\right), \quad
dt=\frac{(T-\tau-\frac{\varepsilon}2)}{(T-\tau)}\,d\widetilde{t}.$$
Finally, we rewrite \eqref{e.08} as
\begin{subequations}
\label{e.16}%
\begin{align}
&\widetilde{c}_{\varepsilon,n}^{\,\,\prime}(\widetilde{t})=
-\frac{\displaystyle \Bigl(T-\tau-\frac{\varepsilon}2\Bigr)}{(T-\tau)}\,
\frac{\lambda_n^2}{1+\lambda_n^2}\,\widetilde{c}_{\varepsilon,n}(\widetilde{t}),\quad
\widetilde{t}\in(\tau,T),\,\,n\in\mathbb{N},
\\
&\widetilde{c}_{\varepsilon,n}(\tau)=\overline{c}_{\varepsilon,n}\Bigl(\frac12+0\Bigr).
\end{align}
\end{subequations}
We introduce a solution
\[
\widetilde{u}_\varepsilon(x,\widetilde{t})=\sum\limits_{{n}=1}^\infty
\widetilde{c}_{\varepsilon,n}(\widetilde{t})\varphi_n(x)
\]
of the rescaled {problem}

\begin{subequations}
\label{e.17}
\begin{align}
&\partial_{\widetilde{t}}\,
\widetilde{u}_\varepsilon=\frac{\displaystyle \Bigl(T-\tau-\frac{\varepsilon}2\Bigr)}{(T-\tau)}\,
\partial_{xx}^2\widetilde{u}_\varepsilon+
\partial_{xx\widetilde{t}}^3\,
\widetilde{u}_\varepsilon,
\\
& \label{e.17b}
\widetilde{u}_\varepsilon(x,\tau)=\bar{u}_\varepsilon\Bigl(x,\frac12+0\Bigr),
\\
&\widetilde{u}_\varepsilon(0,\widetilde{t}\,)=\widetilde{u}_\varepsilon(1,\widetilde{t}\,)=0.
\end{align}
\end{subequations}

\begin{lemma}
\label{lemma4} The following estimate holds true:
\begin{align*}
&\sup\limits_{\widetilde{t}\in(\tau,T)}
\Vert\widetilde{u}_\varepsilon
(\cdot,\widetilde{t})\Vert_{W^{2,2}(0,1)}^2+
\sup\limits_{\widetilde{t}\in(\tau,T)}
\Vert\partial_{\widetilde{t}}\,\widetilde{u}_\varepsilon
(\cdot,\widetilde{t})\Vert_{W_0^{1,2}(0,1)}^2\leqslant
\widetilde{C}_\alpha \Vert g\Vert_{W_0^{2,2}(0,1)}^2.
  \end{align*}
\end{lemma}
\begin{proof}
 Justification of the lemma is based on Lemma \ref{lemma2} and is similar to the proof of the first and the second energy inequalities, see Secs. \ref{sec.2.1} and \ref{sec.2.2}.
\end{proof}
The family of the functions $\widetilde{u}_\varepsilon$ is compact
in $C([\tau,T];W_0^{1,2}(0,1))$. This is guarante\-ed by Lemma
\ref{lemma4} and the original version of the Aubin--Lions lemma, since the space
\[
\left\{v\in L^{\infty}(\tau,T;W_0^{2,2}(0,1)):\,
\partial_tv\in L^\infty(\tau,T;W_0^{1,2}(0,1))\right\}
\]
is compactly embedded in $C([\tau,T];W_0^{1,2}(0,1))$.

\subsection{Matching conditions}
Summarizing Secs. \ref{sec.5.1}--\ref{sec.5.3}, we state that the sequences $\{\widehat{u}_\varepsilon\}_{\varepsilon \to 0+}$,
$\{\bar{u}_\varepsilon\}_{\varepsilon \to 0+}$, and $\{\widetilde{u}_\varepsilon\}_{\varepsilon \to 0+}$
are compact in the following spaces, respectively:
\[
C([0,\tau];W_0^{1,2}(0,1)),\quad C\Bigl(\Bigl[-\frac12,\frac12\Bigr];W_0^{1,2}(0,1)\Bigr),\quad \text{and }
C([\tau,T];W_0^{1,2}(0,1)).
\]
Moreover, we conclude that the limit of the sequence $\{\widehat{u}_\varepsilon\}_{\varepsilon \to 0+}$ satisfies the system \eqref{e.09b}--\eqref{e.09d} on $(0,1)\times (0,\tau)$ in the strong sense (see Remark \ref{remark1} and Sec. \ref{sec.3.5}) and the limit of the sequence $\{\widetilde{u}_\varepsilon\}_{\varepsilon \to 0+}$ satisfies the system \eqref{e.09b}--\eqref{e.09c} on $(0,1)\times (\tau,T)$ and the limit of the sequence $\{\bar{u}_\varepsilon\}_{\varepsilon \to 0+}$ satisfies
the system \eqref{e.11b}--\eqref{e.11c} on $\Pi$, at least, in the weak sense (in the sense of distributions) so far. From initial conditions
\eqref{e.15a} and \eqref{e.17b}, the matching conditions
\[
\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau-0)=
\lim\limits_{\varepsilon\to0+}
\widehat{u}_\varepsilon(x,\tau-0)=\lim\limits_{\varepsilon\to0+}
\bar{u}_\varepsilon\Bigl(x,-\frac12+0\Bigr)
\]
and
\[
\lim\limits_{\varepsilon\to0+}
\bar{u}_\varepsilon\Bigl(x,\frac12-0\Bigr)=
\lim\limits_{\varepsilon\to0+}\widetilde{u}_\varepsilon(x,\tau+0)=
\lim\limits_{\varepsilon\to0+}u_\varepsilon(x,\tau+0)
\]
follow, i.e., equalities \eqref{e.11d} and \eqref{e.11f} hold true.

\subsection{Equation \eqref{e.11b} on $\boldsymbol\Pi$} \label{sec.5.5} Problem \eqref{e.11b}--\eqref{e.11d} (for $\bar{u}$) has the form similar to the formulation of problem \eqref{e.02} and 
$$\bar{u}\Bigl(\cdot,-\frac{1}{2}+0\Bigr) =u(\cdot,\tau-0)\in W^{2,2}(0,1)\cap W^{1,2}_0(0,1).$$ 
Therefore $\bar{u}$ is unique and meets the same regularity requirements, as the solution $u_\varepsilon$ of \eqref{e.02} in Definition \ref{definition1}. In particular, equation \eqref{e.11b} holds a.e. on $\Pi$ and initial condition \eqref{e.11d} holds in the sense of the limiting relation \eqref{matching_01}. Theorem \ref{theorem.2} is fully proved.

\section{Proof of Theorem \ref{theorem.1} (completion)}
\label{sec.6} The initial-boundary value problem for equation \eqref{e.09b} on $(0,1)\times(\tau,T)$ supple\-mented with the boundary conditions \eqref{e.09c} and the initial conditions \eqref{e.11f} has the form similar to the formulation of problem \eqref{e.02} and 
$$u(\cdot,\tau+0)=\bar{u}\Bigl(\cdot,\frac{1}{2}-0\Bigr) \in W^{2,2}(0,1)\cap W^{1,2}_0(0,1),$$ due to the arguments in Sec. \ref{sec.5.5}. Therefore $u$ is unique and meets the same regularity requirements, as the solution $u_\varepsilon$ of \eqref{e.02} in Definition \ref{definition1}. In particular, equation \eqref{e.09b} holds a.e. on $(0,1)\times (0,\tau)$ and initial condition \eqref{e.11f} holds in the sense of the limiting relation \eqref{matching_02}.

Proof of Theorem \ref{theorem.1} is complete.

\section{Remark on the strong convergence}

In the end of the article, let us make an additional remark on strong convergence leading to some generalization of the established results.
\begin{remark} \label{remark2}
Note that the strong convergence properties for $\{u_\varepsilon\}_{\varepsilon\to 0+}$ and $\{\bar{u}_\varepsilon\}_{\varepsilon\to 0+}$ (see the limiting relations \eqref{e.09a} and \eqref{e.11a}) are redundant for the derivation of the limit model \eqref{e.09b}--\eqref{e.09d}, \eqref{e.11b}--\eqref{e.11f}, since the original model \eqref{e.02} is linear and therefore the mere properties of weak convergence would have been sufficient for the limiting transitions in \eqref{e.02}, in fact.

On the other hand, presence of strong convergence manifests that problem \eqref{e.02} can be regarded to as a `good' approximation of the limit problem \eqref{e.09b}--\eqref{e.09d}, \eqref{e.11b}--\eqref{e.11f}, in a sense. Besides, as an example, precisely following the arguments in the article, we can naturally generalize the research onto the case of the semilinear equation
$$\partial_tu_\varepsilon
=\partial_{xxt}^3u_\varepsilon+\left(1+\alpha
K_\varepsilon^\tau(t)\right)\partial_{xx}^2u_\varepsilon -f(u_\varepsilon),$$
where $f$ is a smooth given sublinear function.
\end{remark}

\section*{Acknowledgements} \indent
The authors are very grateful to Professor Stanislav N. Antontsev
(Lavren\-tyev Institute of Hydrodynamics, Novosibirsk, Russia) for many fruitful
discuss\-ions and to the anonymo\-us referee for valuable remarks and recommendations
that helped improve the earlier version of the paper.

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