# Stability of planar rarefaction waves for scalar viscous conservation law
under periodic perturbations

Feimin Huang1,2 Qian Yuan1,∗
Dedicated to the 80th Birthday of Ling Hsiao 1Academy of Mathematics and
Systems Science, CAS, Beijing 100190, China  2 School of Mathematical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The large time behavior of the solutions to a multi-dimensional viscous
conservation law is considered in this paper. It is shown that the solution
time-asymptotically tends to the planar rarefaction wave if the initial
perturbations are multi-dimensional periodic. The time-decay rate is also
obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in
the domain ${\mathbb{R}}\times\mathbb{T}^{n-1}(n\geq 2)$, where
$\mathbb{T}^{n-1}$ is the $n-1$-dimensional torus.

*Corresponding Author

Feimin Huang is partially supported by NSFC Grant No. 11371349 and 11688101.

Qian Yuan is partially supported by the China Postdoctoral Science Foundation
funded project (2019M660831).

AMS Subject Classification: 35B65, 35L65, 35Q35

Keywords: Planar rarefaction wave, periodic perturbation, viscous conservation
law

##  1\. introduction

We are concerned with a scalar viscous conservation law, which reads in
${\mathbb{R}}^{n}$ as

${}\partial_{t}u(x,t)+\sum\limits_{i=1}^{n}\partial_{i}\left(f_{i}(u(x,t))\right)=\triangle
u(x,t),\quad t>0,x\in\mathbb{R}^{n},$ (1.1)

where $u(x,t)\in{\mathbb{R}},$ $\partial_{i}:=\partial_{x_{i}}~{}(i=1,2,..,n)$
and $\triangle=\sum\limits_{i=1}^{n}\partial_{i}^{2};$ the fluxes
$f_{i}(u)~{}(i=1,2,..,n)$ are smooth and $f_{1}^{\prime\prime}(u)\geq a_{0}$
for some constant $a_{0}>0$.

It is well known [11, 21] that the one-dimensional (1-d) conservation law,
i.e. the right hand side of (1.1) is zero and $n=1,$ has rich wave phenomena
including shocks and rarefaction waves. A centered rarefaction wave
$u^{R}(x_{1},t)$ is an entropy solution of the following Riemann problem

$\begin{cases}\partial_{t}u^{R}(x_{1},t)+\partial_{1}f_{1}(u^{R}(x_{1},t))=0,\\\
u^{R}(x_{1},0)=\left\\{\begin{array}[]{ll}\bar{u}_{l},~{}x_{1}<0,\\\
\bar{u}_{r},~{}x_{1}>0,\end{array}\right.\end{cases}$ (1.2)

with $\bar{u}_{l}<\bar{u}_{r}$, and has an explicit formula as

$u^{R}(x_{1},t)=\begin{cases}\bar{u}_{l},&x_{1}\leq
f^{\prime}_{1}(\bar{u}_{l})t,\\\
(f_{1}^{\prime})^{-1}(\frac{x_{1}}{t}),\quad&f^{\prime}_{1}(\bar{u}_{l})t<x_{1}\leq
f^{\prime}_{1}(\bar{u}_{r})t,\\\
\bar{u}_{r},&x_{1}>f^{\prime}_{1}(\bar{u}_{r})t.\end{cases}$ (1.3)

It is shown in [7, 12, 24, 10] that the rarefaction wave given in (1.3) is
asymptotically stable for both the inviscid and viscous conservation laws in
$L^{2}$ framework provided that the initial perturbations are $L^{2}$
integrable. The analysis for the solutions of conservation laws with periodic
initial data is important, cf. [11, 3, 1, 2, 25, 26, 27]. In particular, the
large time behavior of the nonlinear waves under periodic perturbations for
viscous conservation laws is firstly investigated in [25] with the aid of
maximum principle. The 1-d rarefaction wave $u^{R}(x_{1},t)$ becomes the
planar rarefaction wave in the multi-dimensional (m-d) case. The main purpose
of this paper is to use the energy method to extend the work of [25] to the
m-d case, i.e., to show whether the planar rarefaction wave is stable or not
under m-d periodic perturbations. Besides, we refer to [5, 7, 8, 9, 16, 17, 4,
13, 14, 6, 15] and the references therein for the other interesting works
about the stability of shocks, rarefaction waves and contact discontinuities.

Now we formulate the main result. Since the centered rarefaction wave given in
(1.3) is a weak solution and only Lipschitz continuous with respect to
$x_{1},$ we need to construct a viscous version of rarefaction wave
$\tilde{u}^{R}(x_{1},t)$ to replace the original one. Following [17, 24],
$\tilde{u}^{R}(x_{1},t)$ can be constructed as a smooth solution of the
viscous conservation law (1.1), i.e.,

$\begin{cases}\partial_{t}\tilde{u}^{R}+\partial_{1}(f_{1}(\tilde{u}^{R}))=\partial_{1}^{2}\tilde{u}^{R},&\\\
\tilde{u}^{R}(x_{1},0)=\frac{\overline{u}_{l}+\overline{u}_{r}}{2}+\frac{\overline{u}_{r}-\overline{u}_{l}}{2}\frac{e^{x_{1}}-e^{-x_{1}}}{e^{x_{1}}+e^{-x_{1}}}.&\end{cases}$
(1.4)

It is straightforward to check that

$\lim_{x_{1}\to-\infty}\tilde{u}^{R}(x_{1},0)=\overline{u}_{l}<\overline{u}_{r}=\lim_{x_{1}\to+\infty}\tilde{u}^{R}(x_{1},0).$

Consider the scalar equation (1.1) with the following initial data

$u(x,0)=u_{0}(x)=\tilde{u}^{R}(x_{1},0)+w_{0}(x),\qquad x\in\mathbb{R}^{n},$
(1.5)

where $w_{0}(x)$ is a m-d periodic function defined on the n-d torus
$\mathbb{T}^{n}=\prod\limits_{i=1}^{n}[0,1].$ Without loss of generality (by
adding the average constant onto $\overline{u}_{l}$ and $\overline{u}_{r},$
respectively), one can assume that

$\int_{\mathbb{T}^{n}}w_{0}(x)dx=0.$ (1.6)

We are ready to state the main theorem.

######  Theorem 1.1.

Assume that the periodic perturbation $w_{0}(x)\in
H^{\left[\frac{n}{2}\right]+2}(\mathbb{T}^{n})$ and satisfies Equation 1.6.

Then there exists a unique global smooth solution $u$ of Equation 1.1,
Equation 1.5 satisfying

$\left\lVert
u(x,t)-\tilde{u}^{R}(x_{1},t)\right\rVert_{L^{\infty}({\mathbb{R}}^{n})}\leq
C(1+t)^{-\frac{1}{2}},\quad t>0,$ (1.7)

where the constant $C>0$ is independent of $t.$

######  Remark 1.2.

It is shown in [10] that any viscous rarefaction waves connecting same end
states are time-asymptotically equivalent in the $L^{\infty}({\mathbb{R}})$
space with the rate $t^{-\frac{1}{2}}.$ We remark that the stability result
Equation 1.7 also holds true for more general initial values
$\tilde{u}^{R}(x_{1},0)+v_{0}(x_{1})+w_{0}(x)$ instead of Equation 1.5, where
$v_{0}(x_{1})$ is any 1-d function in $L^{1}({\mathbb{R}})\cap
H^{1}({\mathbb{R}}).$

######  Remark 1.3.

Theorem 1.1 is the first work concerning the m-d periodic perturbations around
the nonlinear waves for conservation laws, which shows that the oscillations
in all directions around the planar rarefaction wave decay to zero with the
rate $t^{-\frac{1}{2}},$ even though the initial perturbations keep
oscillating at infinity $\left\lvert x\right\rvert\rightarrow+\infty$. In
other words, the oscillations are eliminated due to the genuine nonlinearity
of the equation.

Let us outline the proof of 1.1. Motivated by [24] and [10] in which the
planar rarefaction waves are shown to be stable under m-d perturbations for
the scalar equation Equation 1.1, we want to use the energy method to prove
1.1. However, the initial periodic perturbation $w_{0}(x)$ is not integrable
on ${\mathbb{R}}^{n},$ and has no any limit at far fields. Thus, the effective
energy method developed in the previous articles can not be applied here
directly. To overcome this difficulty, we construct a suitable ansatz
$\tilde{u}(x,t)$ which contains the oscillations in the $x_{1}$ direction so
that the oscillations in the difference between the solution $u$ and the
ansatz $\tilde{u}$ is eliminated in the $x_{1}$ direction, i.e.,
$(u-\tilde{u})(x,t)\in L^{1}_{x_{1}}({\mathbb{R}}),$ This is the key point in
our proof. Then the stability with time-decay rates Equation 1.7 can be
obtained by the $L^{p}$ energy method developed in [10].

To prove 1.1, we establish a Gagliardo-Nirenberg type inequality in the domain
$\Omega:={\mathbb{R}}\times\mathbb{T}^{n-1}$, which is the second novelty in
this paper. The Gagliardo-Nirenberg (GN) inequality is very useful in the
field of PDEs and usually holds in the whole space $\mathbb{R}^{n}$ or in the
domain with zero Dirichlet boundary condition. Let us recall the GN inequality
which reads as, for any $1\leq p\leq+\infty$ and integer $0\leq j<m$,

$\left\lVert\nabla_{x}^{j}u\right\rVert_{L^{p}(\mathbb{R}^{n})}\leq
C\left\lVert\nabla_{x}^{m}u\right\rVert_{L^{r}(\mathbb{R}^{n})}^{\theta}\left\lVert
u\right\rVert_{L^{q}(\mathbb{R}^{n})}^{1-\theta},$ (1.8)

where
$\frac{1}{p}=\frac{j}{n}+\left(\frac{1}{r}-\frac{m}{n}\right)\theta+\frac{1}{q}\left(1-\theta\right)$,
$\frac{j}{m}\leq\theta\leq 1$ and $C$ is a constant independent of $u$.
However, the $n$-d GN inequality Equation 1.8 does not hold in the domain
$\Omega$ generally owing to the following counterexample.

Counterexample:  It is noted that any 1-d function $0\neq f(x_{1})\in
C_{c}^{\infty}({\mathbb{R}})$ is periodic in the $x_{i}$ direction for
$i=2,\cdots,n$ (one cannot exclude the 1-d case since the initial
perturbations satisfying Equation 1.6 include the 1-d periodic functions).
Then corresponding to the case $j=0,p=\frac{n}{n-1},\theta=m=r=1$ in Equation
1.8, which is exactly the $n$-d Sobolev inequality, one can let
$\zeta_{d}(x_{1})=f(d^{-1}x_{1}),$ $d>0,$ and a direct computation implies
that

$\left\lVert\zeta_{d}\right\rVert_{L^{\frac{n}{n-1}}(\Omega)}=C_{n,d}\left\lVert\nabla_{x}\zeta_{d}\right\rVert_{L^{1}(\Omega)},$
(1.9)

where $C_{n,d}:=d^{\frac{n-1}{n}}\left\lVert
f\right\rVert_{L^{\frac{n}{n-1}}({\mathbb{R}})}/\left\lVert
f^{\prime}\right\rVert_{L^{1}({\mathbb{R}})}$. Since $C_{n,d}\to+\infty$ as
$d\rightarrow+\infty$, we conclude that in general the $n$-d GN inequality
(1.8) is not true in $\Omega$ without additional conditions.

Instead, based on a function-decomposition in 3.3 below, we establish a
Gagliardo-Nirenberg type inequality on the domain $\Omega$.

######  Theorem 1.4 (GN type inequality on
$\Omega={\mathbb{R}}\times\mathbb{T}^{n-1}$).

Let $u\in L^{q}(\Omega)$ and $\nabla^{m}u\in L^{r}(\Omega)$ where $1\leq
q,r\leq+\infty$ and $m\geq 1,$ and $u$ is periodic in the $x_{i}$ direction
for $i=2,\cdots,n.$ Then there exists a decomposition
$u(x)=\sum\limits_{k=0}^{n-1}u^{(k)}(x)$ such that each $u^{(k)}$ satisfies
the $k+1$-dimensional GN inequality Equation 1.8, i.e.,

$\left\lVert\nabla^{j}u^{(k)}\right\rVert_{L^{p}(\Omega)}\leq
C\left\lVert\nabla^{m}u\right\rVert_{L^{r}(\Omega)}^{\theta_{k}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{1-\theta_{k}},$ (1.10)

for any $0\leq j<m$ and $1\leq p\leq+\infty$ satisfying
$\frac{1}{p}=\frac{j}{k+1}+\left(\frac{1}{r}-\frac{m}{k+1}\right)\theta_{k}+\frac{1}{q}\left(1-\theta_{k}\right)$
and $\frac{j}{m}\leq\theta_{k}\leq 1,$ with the following exceptional cases,

  * 1)

if $j=0,rm<k+1$ and $q=+\infty,$ additional assumption that $u\to 0$ as
$\left\lvert x_{1}\right\rvert\to+\infty$ is required;

  * 2)

if $1<r<+\infty$ and $m-j-\frac{k+1}{r}$ is a non-negative integer, additional
assumption that $\theta_{k}<1$ is required.

Hence, it holds that

$\left\lVert\nabla^{j}u\right\rVert_{L^{p}(\Omega)}\leq
C\sum_{k=0}^{n-1}\left\lVert\nabla^{m}u\right\rVert_{L^{r}(\Omega)}^{\theta_{k}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{1-\theta_{k}},$ (1.11)

where the constant $C>0$ is independent of $u.$

######  Remark 1.5.

It is noted that the first term for $k=0$ on the right-hand side (RHS) of
Equation 1.11 is necessary since $u$ can be a 1-d function defined on
${\mathbb{R}}.$ For example, considering the same function $\zeta_{d}(x_{1})$
as in the counterexample, when corresponding to the case $j=0,p=r=2$ and
$m=q=1$ in (1.11), a direct computation gives that

$\left\lVert\zeta_{d}\right\rVert_{L^{2}(\Omega)}=Cd^{\frac{1}{2}\left(3\theta-1\right)}\left\lVert\nabla\zeta_{d}\right\rVert^{\theta}_{L^{2}(\Omega)}\left\lVert\zeta_{d}\right\rVert^{1-\theta}_{L^{1}(\Omega)}\quad\text{for
any }0\leq\theta\leq 1,$ (1.12)

where $C>0$ is independent of $d.$ The estimate Equation 1.11 can be true only
for $\theta=\theta_{0}=\frac{1}{3}$. Otherwise, one can let $d\to 0+$ if
$\theta<\frac{1}{3}$ and $d\to+\infty$ if $\theta>\frac{1}{3},$ respectively
to get the contradiction.

######  Remark 1.6.

An interesting interpolation inequality involving $\nabla^{2}u$ in the domain
$\mathbb{R}\times\mathbb{T}^{2}$ was established in [23].

The rest of the paper is organized as follows. Some preliminaries on the
viscous rarefaction waves and the construction of the ansatz are given in
Section 2. Section 3 is devoted to the proof of 1.4 and an interpolation
inequality, 3.5. In Section 4, we show the desired a priori estimates, 4.2;
thereafter, the proof of 1.1 is completed.

##  2\. Preliminaries and Ansatz

Some properties of the viscous rarefaction wave are listed as follows.

######  Lemma 2.1.

The viscous rarefaction wave $\tilde{u}^{R}(x_{1},t)$ solving Equation 1.4
satisfies

$\lim\limits_{t\rightarrow+\infty}\left\lVert\left(\tilde{u}^{R}-u^{R}\right)(\cdot,t)\right\rVert_{L^{\infty}({\mathbb{R}})}=0.$
(2.1)

Moreover, for any $t>0$ it holds that

$\displaystyle
0<\partial_{1}\tilde{u}^{R}(x_{1},t)\leq\min\left\\{\frac{C}{t},~{}\max_{x_{1}\in{\mathbb{R}}}\left(u_{0}^{R}\right)^{\prime}\right\\}\qquad\forall
x_{1}\in{\mathbb{R}},$ (2.2)
$\displaystyle\left\lVert\left(\tilde{u}^{R}(\cdot,t)-\overline{u}_{l}\right)\left(\tilde{u}^{R}(\cdot,t)-\overline{u}_{r}\right)\right\rVert_{L^{1}({\mathbb{R}})}\leq
C(1+t),$ (2.3)
$\displaystyle\left\lVert\partial_{1}\tilde{u}^{R}(\cdot,t)\right\rVert_{L^{p}({\mathbb{R}})}\leq
Ct^{-1+\frac{1}{p}},\qquad p\in[1,+\infty),$ (2.4)

where the constant $C>0$ is independent of $t.$

###### Proof.

The $L^{\infty}$ time-asymptotically equivalence Equation 2.1 has been proved
in [7, 10]. By the maximum principles for Equation 1.4 and the equation of the
derivative $v=\partial_{1}\tilde{u}^{R},$

$\partial_{t}v-\partial_{1}^{2}v+f_{1}^{\prime\prime}(\tilde{u}^{R})v^{2}+f_{1}^{\prime}(\tilde{u}^{R})\partial_{1}v=0,$
(2.5)

one can get that $\sup\limits_{x_{1}\in{\mathbb{R}},t\geq
0}\left\lvert\tilde{u}^{R}(x_{1},t)\right\rvert\leq\left\lVert
u_{0}^{R}\right\rVert_{L^{\infty}({\mathbb{R}})}$ and
$0<\partial_{1}\tilde{u}^{R}(x_{1},t)\leq\max\limits_{x_{1}\in{\mathbb{R}}}\left(u_{0}^{R}\right)^{\prime}(x_{1}).$
Besides, it was shown in [19] that the solution $\tilde{u}^{R}$ of the 1-d
convex conservation law Equation 1.4 satisfies the well-known Oleǐnik entropy
condition, i.e., $\partial_{1}\tilde{u}^{R}(x_{1},t)\leq\frac{C}{t}$ for all
$x_{1}\in{\mathbb{R}},$ where $C>0$ is independent of $t.$

Since $\overline{u}_{l}<\tilde{u}^{R}<\overline{u}_{r}$ and both
$\tilde{u}^{R}$ and $\partial_{1}\tilde{u}^{R}$ are uniformly bounded, it
follows from Equation 1.4 that

$\frac{d}{dt}\left[\int_{-\infty}^{0}\left(\tilde{u}^{R}(x_{1},t)-\overline{u}_{l}\right)dx_{1}+\int_{0}^{+\infty}\left(\overline{u}_{r}-\tilde{u}^{R}(x_{1},t)\right)dx_{1}\right]\leq
C,\quad t>0,$

which implies that

$\int_{-\infty}^{0}\left(\tilde{u}^{R}(x_{1},t)-\overline{u}_{l}\right)dx_{1}+\int_{0}^{+\infty}\left(\overline{u}_{r}-\tilde{u}^{R}(x_{1},t)\right)dx_{1}\leq
C(1+t),\quad t>0,$

and thus Equation 2.3 holds true. And Equation 2.4 can follow from

$\left\lVert\partial_{1}\tilde{u}^{R}\right\rVert_{L^{p}({\mathbb{R}})}^{p}\leq\left\lVert\partial_{1}\tilde{u}^{R}\right\rVert_{L^{\infty}({\mathbb{R}})}^{p-1}\int_{\mathbb{R}}\partial_{1}\tilde{u}^{R}(x_{1},t)dx_{1}\leq
Ct^{-p+1}.$

∎

Now we construct the ansatz. Set

$g(x_{1},t):=\frac{\tilde{u}^{R}(x_{1},t)-\overline{u}_{l}}{\overline{u}_{r}-\overline{u}_{l}},\quad
x_{1}\in{\mathbb{R}},~{}t\geq 0,$ (2.6)

which is smooth, belongs to the interval $(0,1)$ and satisfies
$\partial_{1}g(x_{1},t)>0$ for any $x_{1}\in{\mathbb{R}},t\geq 0.$

Let $u_{l}(x,t)$ and $u_{r}(x,t)$ denote the two periodic solutions of (1.1)
(see [20] for the global existence) with the respective periodic data

$u_{l}(x,0)=\overline{u}_{l}+w_{0}(x)\quad\text{ and}\quad
u_{r}(x,0)=\overline{u}_{r}+w_{0}(x),\quad x\in{\mathbb{R}}^{n}.$ (2.7)

And define

$w_{i}(x,t):=u_{i}(x,t)-\overline{u}_{i},\quad i=l\text{ or }r,$ (2.8)

which is a periodic function with zero average for any $t\geq 0.$ Then we have

######  Lemma 2.2.

If $w_{0}(x)\in H^{\left[\frac{n}{2}\right]+2}(\mathbb{T}^{n})$ satisfies
Equation 1.6, then for $i=l$ or $r,$ it holds that

$\left\lVert w_{i}(\cdot,t)\right\rVert_{W^{1,\infty}({\mathbb{R}}^{n})}\leq
C\left\lVert
w_{0}\right\rVert_{H^{\left[\frac{n}{2}\right]+2}(\mathbb{T}^{n})}e^{-2\alpha
t},\quad t\geq 0,$ (2.9)

where the constants $\alpha>0$ and $C>0$ are independent of $t.$

###### Proof.

The proof of 2.2 is based on basic $L^{2}$ energy estimates and the Poincáre
inequality on $\mathbb{T}^{n},$ which is standard and thus omitted. ∎

We are ready to construct the ansatz as follows:

$\tilde{u}(x,t):=u_{l}(x,t)\left(1-g(x_{1},t)\right)+u_{r}(x,t)g(x_{1},t),$
(2.10)

which is periodic in the $x_{i}$ direction for $i=2,\cdots,n.$ By direct
calculations, the source term induced by the ansatz $\tilde{u}(x,t)$ is given
as follows.

$\displaystyle h(x,t):=~{}$
$\displaystyle\partial_{t}\tilde{u}+\sum\limits_{i=1}^{n}\partial_{i}\left(f_{i}(\tilde{u}(x,t))\right)-\triangle\tilde{u}(x,t)$
$\displaystyle=~{}$
$\displaystyle\left(u_{r}-u_{l}\right)g(1-g)\sum\limits_{i=1}^{n}\left[\sigma_{i}(u_{l},\tilde{u})\partial_{i}w_{l}-\sigma_{i}(u_{r},\tilde{u})\partial_{i}w_{r}\right]$
(2.11)
$\displaystyle+\left(u_{r}-u_{l}\right)\sigma_{1}(\tilde{u}^{R},\tilde{u})\left(\tilde{u}-\tilde{u}^{R}\right)\partial_{1}g-2\partial_{1}\left(w_{r}-w_{l}\right)\partial_{1}g,$

where
$\sigma_{i}(u,v):=\int_{0}^{1}f_{i}^{\prime\prime}(v+\theta(u-v))d\theta,~{}i=1,\cdots,n.$
Thus, the source term $h(x,t)$ is also periodic in the $x_{i}$ direction for
$i=2,\cdots,n$.

######  Lemma 2.3.

Under the assumptions of 2.2, it holds that for any $p\in[1,+\infty],$

$\left\lVert h(\cdot,t)\right\rVert_{L^{p}(\Omega)}\leq C\left\lVert
w_{0}\right\rVert_{H^{\left[\frac{n}{2}\right]+2}(\mathbb{T}^{n})}e^{-\alpha
t},\quad t\geq 0,$ (2.12)

where $\alpha>0$ is the constant in 2.2, and $C>0$ is independent of $t.$

###### Proof.

By Equation 2.3, Equation 2.4 and 2.2, it is straightforward to check that
Equation 2.12 holds for $p=1$ and $+\infty$. And for $p\in(1,+\infty),$
Equation 2.12 follows from the interpolation $\left\lVert
h\right\rVert_{L^{p}}\leq\left\lVert
h\right\rVert_{L^{\infty}}^{1-\frac{1}{p}}\left\lVert
h\right\rVert_{L^{1}}^{\frac{1}{p}}.$ ∎

##  3\. Interpolation inequality on $\Omega=\mathbb{R}\times\mathbb{T}^{n-1}$

Although the $n$-d GN inequality (1.8) does not hold in the domain
$\Omega={\mathbb{R}}\times\mathbb{T}^{n-1}$ generally, it is true with an
additional condition, that is,

######  Lemma 3.1.

Let $u\in L^{q}(\Omega),$ and its derivatives of order $m,$ $\nabla^{m}u\in
L^{r}(\Omega),$ where $m\geq 1$ and $1\leq r,q\leq+\infty.$ And assume that
$u$ is periodic in the $x_{i}$ direction and satisfies

$\int_{\mathbb{T}}u(x)dx_{i}=0\quad\text{ for all }i=2,3,\cdots,n.$ (3.1)

Then the $n$-d GN inequality (1.8) holds true for $u$.

###### Proof.

Following [18, Lecture II], it suffices to prove the following two extreme
cases.

  * 1)

For a.e. $x\in\Omega,$ it holds that

$\left\lvert
u(x)\right\rvert^{n}\leq\int_{\mathbb{R}}\left\lvert\partial_{1}u\right\rvert
dx_{1}\prod_{i=2}^{n}\int_{\mathbb{T}}\left\lvert\partial_{i}u\right\rvert
dx_{i}.$ (3.2)

  * 2)

For any $1\leq i\leq n,$ and $1\leq q<+\infty,1<r<+\infty,$ it holds that

$\int\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq
C\left(\int\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{2r}}\left(\int\left\lvert
u\right\rvert^{q}dx_{i}\right)^{\frac{p}{2q}},$ (3.3)

where $\frac{2}{p}=\frac{1}{r}+\frac{1}{q},$ and $``\int"$ indicates the
integral over ${\mathbb{R}}$ if $i=1$ or $\mathbb{T}$ if $i>1.$

Thanks to Equation 3.1, it is straightforward to show Equation 3.2. And
Equation 3.3 for $i=1$ has been verified in [18]. It remains to show this
inequality on $\mathbb{T}=[0,1]$ for $i>1.$ In fact, for any bounded interval
$I\subset{\mathbb{R}},$ it follows from [18, (2.7)] that

$\int_{I}\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq C\left\lvert
I\right\rvert^{1+p-\frac{p}{r}}\left(\int_{I}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{r}}+C\left\lvert
I\right\rvert^{-\left(1+p-\frac{p}{r}\right)}\left(\int_{I}\left\lvert
u\right\rvert^{q}dx_{i}\right)^{\frac{p}{q}},$ (3.4)

where $C>0$ is independent of $I.$ Since $u$ is periodic in the $x_{i}$
direction with period $1,$ the first term on the RHS of Equation 3.4 must be
greater than the second one, as long as the length of $I$ is equal to a large
integer $N.$ Similar to [18, (2.8)], one can cover $[0,N]$ by finite intervals
$I_{j}\subset[0,2N],$ at each of which, the first term on the RHS of Equation
3.4 (where $I=I_{j}$) is either greater than or equal to the second one. The
proof is to repeat that of L. Nirenberg’s in [18]. However, to make this paper
complete, we still give the details here. First let $k>0$ be large enough and
fixed, and consider $I=(0,\frac{N}{k})$ in Equation 3.4. If the first term on
the RHS is greater than the second one, set $I_{1}=(0,\frac{N}{k}),$ and it
holds that

$\int_{I_{1}}\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq
C\left(\frac{N}{k}\right)^{1+p-\frac{p}{r}}\left(\int_{I_{1}}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{r}},$
(3.5)

where the constant $C>0$ is independent of either $k$ or $N.$ Otherwise, if
the second term on the RHS of Equation 3.4 is greater, then due to the choice
of $N,$ one can extend the interval $(0,\frac{N}{k})$ to a larger one,
$I_{1}\subset[0,2N],$ until the two terms become equal. Then one has that

$\int_{I_{1}}\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq
C\left(\int_{I_{1}}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{2r}}\left(\int_{I_{1}}\left\lvert
u\right\rvert^{q}dx_{i}\right)^{\frac{p}{2q}},$ (3.6)

where the constant $C>0$ is independent of either $k$ or $N.$ Starting at the
end point of $I_{1},$ one can repeat this process to set $I_{2},I_{3},\cdots,$
until $[0,N]$ is covered. For the fixed $k,$ there must be finitely many such
intervals $I_{j},$ each of which is contained in $[0,2N].$ Then summing the
estimates Equation 3.5 and Equation 3.6 together yields that

$\displaystyle\int_{0}^{N}\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq~{}$
$\displaystyle
Ck\left(\frac{N}{k}\right)^{1+p-\frac{p}{r}}\left(\int_{0}^{N}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{r}}$
$\displaystyle+C\left(\int_{0}^{2N}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{2r}}\left(\int_{0}^{2N}\left\lvert
u\right\rvert^{q}dx_{i}\right)^{\frac{p}{2q}}.$

Letting $k\to+\infty,$ one has that

$\int_{0}^{N}\left\lvert\partial_{i}u\right\rvert^{p}dx_{i}\leq
C\left(\int_{0}^{2N}\left\lvert\partial_{i}^{2}u\right\rvert^{r}dx_{i}\right)^{\frac{p}{2r}}\left(\int_{0}^{2N}\left\lvert
u\right\rvert^{q}dx_{i}\right)^{\frac{p}{2q}},$

which yields Equation 3.3, since $u$ is periodic in the $x_{i}$ direction and
$\frac{p}{2r}+\frac{p}{2q}=1.$ ∎

######  Remark 3.2.

3.1 means that the GN inequality (1.8) holds under the additional condition
(3.1). Nevertheless, it is difficult to verify (3.1) for the perturbation
$u(x,t)-\tilde{u}(x,t)$ for any $x_{i}$ direction ($i=2,\cdots,n$) even though
it holds initially, i.e., $\int_{\mathbb{T}}w_{0}(x)dx_{i}=0~{}\forall
i=2,\cdots,n$, which is much stronger than the original condition (1.6).

To prove 1.1, we shall establish a GN type inequality on the domain
${\mathbb{R}}\times\mathbb{T}^{n-1}$ without any additional condition by
applying 3.1 and the following decomposition lemma.

For any $k\geq 1,$ set

$\displaystyle{\mathcal{A}}_{k}=\Big{\\{}$ $\displaystyle u(x)\text{ is
measurable on }{\mathbb{R}}\times\mathbb{T}^{k}\text{ with
}x_{1}\in{\mathbb{R}},\text{ and periodic in }x_{i}\in\mathbb{T},$ (3.7)
$\displaystyle\text{ satisfying }\int_{\mathbb{T}}u(x)dx_{i}\equiv 0\text{ for
all }i=2,\cdots,k+1\Big{\\}}.$

Then for any function $u\in{\mathcal{A}}_{k}$, the $k+1$-dimensional GN
inequality (1.8) holds true. We are ready to decompose general function $u$ in
terms of ${\mathcal{A}}_{k}$, i.e.,

######  Lemma 3.3 (Decomposition Lemma).

One can decompose $u(x)$ as follows,

$u(x)=\sum\limits_{k=0}^{n-1}u^{(k)}(x)\quad a.e.~{}x\in\Omega,$ (3.8)

where

$\displaystyle u^{(0)}(x)$
$\displaystyle=u^{(0)}(x_{1})=\int_{\mathbb{T}^{n-1}}u(x_{1},x_{2},\cdots,x_{n})dx_{2}\cdots
dx_{n},$ (3.9) $\displaystyle u^{(k)}(x)$ $\displaystyle=\sum\limits_{2\leq
i_{1}<\cdots<i_{k}\leq
n}u_{i_{1},\cdots,i_{k}}(x_{1},x_{i_{1}},\cdots,x_{i_{k}}),\quad
k=1,\cdots,n-1,$

where each
$u_{i_{1},\cdots,i_{k}}(x_{1},x_{i_{1}},\cdots,x_{i_{k}})\in{\mathcal{A}}_{k}.$
Moreover, for any $m\geq 0$ and $1\leq p\leq+\infty,$ it holds that

$\displaystyle\left\lVert\nabla_{x}^{m}u^{(0)}\right\rVert_{L^{p}(\Omega)}+\sum\limits_{k=1}^{n-1}\sum\limits_{2\leq
i_{1}<\cdots<i_{k}\leq
n}\left\lVert\nabla^{m}_{x}u_{i_{1},\cdots,i_{k}}\right\rVert_{L^{p}(\Omega)}\leq
C\left\lVert\nabla^{m}_{x}u\right\rVert_{L^{p}(\Omega)},$ (3.10)

where $C>0$ is independent of $u.$

######  Remark 3.4.

To avoid excessive words, we omit the assumptions in 3.3 that the integrals
Equation 3.9 and

$\int_{\mathbb{T}^{n-1-k}}u(x)\frac{dx_{2}\cdots dx_{n}}{dx_{i_{1}}\cdots
dx_{i_{k}}},\quad 1\leq k\leq n-1,\quad 2\leq i_{1}<\cdots<i_{k}\leq n,$

should exist, here and hereafter we use
$\int\limits_{\mathbb{T}^{n-1-k}}(\cdot)\frac{dx_{2}\cdots
dx_{n}}{dx_{i_{1}}\cdots dx_{i_{k}}}$ to denote the integral that is
integrated with respect to $x_{2},\cdots,x_{n}$ except for
$x_{i_{1}},\cdots,x_{i_{k}}.$

###### Proof.

Starting with $u^{(0)}$ defined in Equation 3.9, set

$u_{i}(x_{1},x_{i})=\int_{\mathbb{T}^{n-2}}\left(u-u^{(0)}\right)\frac{dx_{2}\cdots
dx_{n}}{dx_{i}},\quad i=2,\cdots,n.$ (3.11)

It is straightforward to check that $u_{i}(x_{1},x_{i})\in{\mathcal{A}}_{1}$.
Define $u^{(1)}(x):=\sum\limits_{i=2}^{n}u_{i}$. Assume that for all
$l=1,2,\cdots,k-1$ with $k\geq 2$ and $2\leq j_{1}<\cdots<j_{l}\leq n,$ all
the functions
$u_{j_{1},\cdots,j_{l}}=u_{j_{1},\cdots,j_{l}}(x_{1},x_{j_{1}},\cdots,x_{j_{l}})\in{\mathcal{A}}_{l}$
and thus $u^{(l)}:=\sum\limits_{2\leq j_{1}<\cdots<j_{l}\leq
n}u_{j_{1},\cdots,j_{l}}$ have been well-defined. Then for any fixed sequence
$2\leq i_{1}<\cdots<i_{k}\leq n,$ set

$\displaystyle u_{i_{1},\cdots,i_{k}}(x_{1},x_{i_{1}},\cdots,x_{i_{k}})$
$\displaystyle:=\int_{\mathbb{T}^{n-1-k}}\Big{(}u-\sum\limits_{l=0}^{k-1}u^{(l)}\Big{)}\frac{dx_{2}\cdots
dx_{n}}{dx_{i_{1}}\cdots dx_{i_{k}}}$ (3.12)
$\displaystyle=\int_{\mathbb{T}^{n-1-k}}\Big{(}u-\sum\limits_{l=0}^{k-2}u^{(l)}-\sum\limits_{2\leq
j_{1}<\cdots<j_{k-1}\leq n}u_{j_{1},\cdots,j_{k-1}}\Big{)}\frac{dx_{2}\cdots
dx_{n}}{dx_{i_{1}}\cdots dx_{i_{k}}}.$

It is noted that for $r=1,\cdots,k,$ one has

$\int_{\mathbb{T}}u_{j_{1},\cdots,j_{k-1}}dx_{i_{r}}=\begin{cases}0&\text{ if
}i_{r}\in\\{j_{1},\cdots,j_{k-1}\\},\\\ u_{j_{1},\cdots,j_{k-1}}\qquad&\text{
if }i_{r}\notin\\{j_{1},\cdots,j_{k-1}\\}.\end{cases}$ (3.13)

Then it holds that

$\displaystyle\int_{\mathbb{T}}u_{i_{1},\cdots,i_{k}}dx_{i_{r}}=~{}$
$\displaystyle\int_{\mathbb{T}^{n-k}}\Big{(}u-\sum\limits_{l=0}^{k-2}u^{(l)}\Big{)}\frac{dx_{2}\cdots
dx_{n}}{dx_{i_{1}}\cdots dx_{i_{r-1}}dx_{i_{r+1}}\cdots dx_{i_{k}}}$
$\displaystyle-\sum\limits_{2\leq j_{1}<\cdots<j_{k-1}\leq
n}\int_{\mathbb{T}^{n-k}}u_{j_{1},\cdots,j_{k-1}}\frac{dx_{2}\cdots
dx_{n}}{dx_{i_{1}}\cdots dx_{i_{r-1}}dx_{i_{r+1}}\cdots dx_{i_{k}}}$
$\displaystyle=~{}$ $\displaystyle
u_{i_{1},\cdots,i_{r-1},i_{r+1},\cdots,i_{k}}-u_{i_{1},\cdots,i_{r-1},i_{r+1},\cdots,i_{k}}$
$\displaystyle=~{}$ $\displaystyle 0,$

which means $u_{i_{1},\cdots,i_{k}}\in{\mathcal{A}}_{k}$ and then the
decomposition (3.8) for function $u$ holds. It remains to show Equation 3.10.
It follows from Equation 3.9 and Minkowski inequality that

$\begin{array}[]{ll}\left\lVert\nabla_{x}^{m}u^{(0)}\right\rVert_{L^{p}({\mathbb{R}})}&\leq\left\lVert\left\lVert\nabla_{x}^{m}u\right\rVert_{L^{1}(\mathbb{T}^{n-1};dx_{2}\cdots
dx_{n})}\right\rVert_{L^{p}({\mathbb{R}};dx_{1})}\\\
&\leq\left\lVert\left\lVert\nabla_{x}^{m}u\right\rVert_{L^{p}({\mathbb{R}};dx_{1})}\right\rVert_{L^{1}(\mathbb{T}^{n-1};dx_{2}\cdots
dx_{n})}\\\
&\leq\left\lVert\nabla^{m}_{x}u\right\rVert_{L^{p}(\Omega)},\end{array}$

where the last inequality is derived from the Hölder inequality. Similarly,
one can obtain from Equation 3.11 that for any $2\leq i\leq n,$

$\left\lVert\nabla_{x}^{m}u_{i}\right\rVert\leq\left\lVert\nabla_{x}^{m}\left(u-u^{(0)}\right)\right\rVert_{L^{p}(\Omega)}\leq
2\left\lVert\nabla^{m}_{x}u\right\rVert_{L^{p}(\Omega)}.$

The remaining functions $u_{i_{1},\cdots,i_{k}}$ defined in Equation 3.12 can
be proved in the same way, and then the proof of Lemma 3.3 is completed. ∎

###### Proof of 1.4.

We first decompose $u(x)=\sum\limits_{k=0}^{n-1}u^{(k)}(x)$ as in 3.3. Then it
follows from Theorem 3.1 and Lemma 3.3 that $u^{(0)}\in
L^{q}({\mathbb{R}}),\nabla^{m}u^{(0)}\in L^{r}({\mathbb{R}}),$ and it
satisfies the $1$-d GN inequality (1.8); each $u_{i_{1},\cdots,i_{k}}\in
L^{q}(\Omega),\nabla^{m}u_{i_{1},\cdots,i_{k}}\in L^{r}(\Omega)$ and it
satisfies the $(k+1)$-d GN inequality (1.8). Hence,

$\displaystyle\left\lVert\nabla^{j}u^{(0)}\right\rVert_{L^{p}({\mathbb{R}})}\leq~{}$
$\displaystyle
C\left\lVert\nabla^{m}u^{(0)}\right\rVert_{L^{r}({\mathbb{R}})}^{\theta_{0}}\left\lVert
u^{(0)}\right\rVert_{L^{q}({\mathbb{R}})}^{1-\theta_{0}}\leq
C\left\lVert\nabla^{m}u\right\rVert_{L^{r}(\Omega)}^{\theta_{0}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{1-\theta_{0}};$
$\displaystyle\left\lVert\nabla^{j}u^{(k)}\right\rVert_{L^{p}(\Omega)}\leq~{}$
$\displaystyle\sum\limits_{2\leq i_{1}<\cdots<i_{k}\leq
n}\left\lVert\nabla^{j}u_{i_{1},\cdots,i_{k}}\right\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq~{}$ $\displaystyle C\sum\limits_{2\leq
i_{1}<\cdots<i_{k}\leq
n}\left\lVert\nabla^{m}u_{i_{1},\cdots,i_{k}}\right\rVert_{L^{r}(\Omega)}^{\theta_{k}}\left\lVert
u_{i_{1},\cdots,i_{k}}\right\rVert_{L^{q}(\Omega)}^{1-\theta_{k}}$
$\displaystyle\leq~{}$ $\displaystyle
C\left\lVert\nabla^{m}u\right\rVert_{L^{r}(\Omega)}^{\theta_{k}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{1-\theta_{k}},\qquad 1\leq k\leq n-1,$

where the indices $\theta_{0},\cdots,\theta_{n-1}$ are that introduced in 1.4.
The proof is finished.

∎

In order to use the $L^{p}$ energy method developed in [10], we shall
establish the following interpolation inequality by 1.4.

######  Corollary 3.5 (Interpolation Inequality in $\Omega$).

For any $2\leq p<\infty$ and $1\leq q\leq p,$ it holds that

$\displaystyle\left\lVert u\right\rVert_{L^{p}(\Omega)}\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla(\left\lvert
u\right\rvert^{\frac{p}{2}})\right\rVert^{\frac{2\gamma_{k}}{1+\gamma_{k}p}}_{L^{2}(\Omega)}\left\lVert
u\right\rVert^{\frac{1}{1+\gamma_{k}p}}_{L^{q}(\Omega)},$ (3.14)

where $\gamma_{k}=\frac{k+1}{2}\left(\frac{1}{q}-\frac{1}{p}\right)$ and the
constant $C=C(p,q,n)>0$ is independent of $u.$

###### Proof.

For the function $v(x)=\left\lvert u(x)\right\rvert^{\frac{p}{2}},$ it follows
from 1.4 that

$\displaystyle\left\lVert v\right\rVert_{L^{2}(\Omega)}\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla
v\right\rVert_{L^{2}(\Omega)}^{\theta_{k}}\left\lVert
v\right\rVert_{L^{2r_{k}}(\Omega)}^{1-\theta_{k}},$

where
$\frac{1}{2}=\left(\frac{1}{2}-\frac{1}{k+1}\right)\theta_{k}+\frac{1}{2r_{k}}\left(1-\theta_{k}\right)$
with $\frac{1}{2}\leq r_{k}\leq 1$ and $0\leq\theta_{k}\leq 1$ for
$k=0,1,\cdots,n-1.$ This yields that

$\left\lVert u\right\rVert_{L^{p}(\Omega)}\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla\left\lvert
u\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{\frac{2}{p}\theta_{k}}\left\lVert
u\right\rVert_{L^{pr_{k}}(\Omega)}^{1-\theta_{k}}.$ (3.15)

If $2\leq p\leq 2q,$ choosing $r_{k}=\frac{q}{p}\geq\frac{1}{2}$ for
$k=0,\cdots,n-1$ can finish the proof. If $p>2q,$ let $\frac{1}{2}<r_{k}<1,$
which will be determined later. Since $q<pr_{k}<p,$ it follows from
interpolation that

$\left\lVert u\right\rVert_{L^{pr_{k}}(\Omega)}\leq\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{1-\rho_{k}}\left\lVert
u\right\rVert_{L^{p}(\Omega)}^{\rho_{k}},$ (3.16)

where $\rho_{k}=\frac{pr_{k}-q}{(p-q)r_{k}}\in(0,1).$ Plugging Equation 3.16
into Equation 3.15 yields that

$\left\lVert u\right\rVert_{L^{p}(\Omega)}\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla\left\lvert
u\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{\frac{2}{p}\theta_{k}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{(1-\rho_{k})(1-\theta_{k})}\left\lVert
u\right\rVert_{L^{p}(\Omega)}^{\rho_{k}(1-\theta_{k})}.$ (3.17)

For $k=0,1,\cdots,n-1$ with $n\geq 2$ and
$r_{k}\in\left(\frac{1}{2},1\right),$ one can obtain by simple calculations
that

$\rho_{k}(1-\theta_{k})=\frac{p}{p-q}\times\frac{r_{k}-\frac{q}{p}}{\frac{k+1}{2}-\frac{k-1}{2}r_{k}}\quad\in\left(\frac{2(p-2q)}{(k+3)(p-q)},1\right).$
(3.18)

Thus, for any $\delta\in\left(\frac{2(p-2q)}{3(p-q)},1\right),$ there exist
$r_{0},r_{1},\cdots,r_{n-1}\in\left(\frac{1}{2},1\right)$ such that
$\rho_{k}(1-\theta_{k})=\delta$ for any $k=0,1,\cdots,n-1.$ Then Equation 3.17
yields that

$\left\lVert u\right\rVert_{L^{p}(\Omega)}\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla\left\lvert
u\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{\frac{2\theta_{k}}{p(1-\delta)}}\left\lVert
u\right\rVert_{L^{q}(\Omega)}^{\frac{(1-\rho_{k})(1-\theta_{k})}{1-\delta}}.$

A direct calculation implies that
$\frac{2\theta_{k}}{p(1-\delta)}=\frac{2\gamma_{k}}{1+\gamma_{k}p}$ and
$\frac{(1-\rho_{k})(1-\theta_{k})}{1-\delta}=\frac{1}{1+\gamma_{k}p}$ for all
$k=0,\cdots,n-1,$ where
$\gamma_{k}=\frac{k+1}{2}\left(\frac{1}{q}-\frac{1}{p}\right)$, and the proof
is completed. ∎

######  Lemma 3.6 ([10] Lemma 2.2).

For any $2\leq p<\infty,$ it holds that

$\left\lVert\partial_{i}u\right\rVert_{L^{p}(\Omega)}\leq
C\left\lVert\partial_{i}\left(\left\lvert\partial_{i}u\right\rvert^{\frac{p}{2}}\right)\right\rVert^{\frac{2}{p+2}}_{L^{2}(\Omega)}\left\lVert
u\right\rVert^{\frac{2}{p+2}}_{L^{p}(\Omega)},\quad i=1,\cdots,n,$ (3.19)

where the constant $C>0$ is independent of $u.$

###### Proof.

The inequality (3.19) has been established in [10] for the whole space
${\mathbb{R}}^{n},$ and it is still true in the domain $\Omega$ with the aid
of integration by parts and the Hölder inequality as in [10]. ∎

##  4\. A priori estimates and proof

Denote the perturbation by $\phi(x,t):=u(x,t)-\tilde{u}(x,t),$ which is
periodic in the $x_{i}$ direction for $i=2,\cdots,n,$ and satisfies

$\displaystyle\partial_{t}\phi+\sum_{i=1}^{n}\partial_{i}\left[f_{i}(\tilde{u}+\phi)-f_{i}(\tilde{u})\right]$
$\displaystyle=\triangle\phi-h,$ (4.1) $\displaystyle\phi(x,0)$
$\displaystyle=0.$ (4.2)

######  Remark 4.1.

In fact, given the initial data
$u_{0}(x)=\tilde{u}^{R}(x_{1},0)+v_{0}(x_{1})+w_{0}(x)$ stated in 1.2, the
initial condition Equation 4.2 turns to $\phi(x,0)=v_{0}(x_{1})\in
L^{1}({\mathbb{R}})\cap H^{1}({\mathbb{R}})$ instead, which makes no
difference in the following proof.

We shall prove the global existence and large time behavior of solution
$\phi(x,t)$ to the Cauchy problem (4.1) and (4.2). The global existence can be
established by obtaining the a priori estimates Equation 4.3 and Equation 4.4
below, since the local existence of the solution $\phi$ to Equation 4.1 with
the initial data in $L^{1}(\Omega)\cap H^{1}(\Omega)$ is standard.

######  Theorem 4.2 (A priori estimates).

Assume that $\phi(x,t)$ is the unique smooth solution to Equation 4.1,Equation
4.2 for any $t\in[0,T]$, then it holds that

$\displaystyle\left\lVert\phi(t)\right\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq C_{p}(1+t)^{-\frac{1}{2}+\frac{1}{2p}}\qquad\forall
p\in[1,+\infty),$ (4.3)
$\displaystyle\left\lVert\nabla\phi(t)\right\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq C_{p}(1+t)^{-1+\frac{1}{2p}},\qquad\forall
p\in[2,+\infty),$ (4.4)

where the constants $C>0$ and $C_{p}>0$ are independent of $t$.

###### Proof.

The uniform bound of $\left\lVert\phi(t)\right\rVert_{L^{\infty}(\Omega)}$
follows from the maximum principle easily. And following the $L^{p}$ energy
method as in [10], we first prove (4.3).

Step 1.  We first show the $L^{1}$ estimates. Given $\delta>0,$ let
$S_{\delta}(\eta)$ be a $C^{2}$ convex approximation to the function
$\left\lvert\eta\right\rvert,$ e.g.

$S_{\delta}(\eta)=\begin{cases}-\eta,\quad&\eta\leq-\delta;\\\
-\frac{\eta^{4}}{8\delta^{3}}+\frac{3\eta^{2}}{4\delta}+\frac{3\delta}{8},\quad&-\delta<\eta\leq\delta;\\\
\eta,\quad&\eta>\delta.\end{cases}$

Multiplying $S_{\delta}^{\prime}(\phi)$ on both sides of Equation 4.1 yields
that

$\displaystyle\partial_{t}S_{\delta}(\phi)+S_{\delta}^{\prime\prime}(\phi)\left\lvert\nabla\phi\right\rvert^{2}+\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left(f_{1}^{\prime}(\tilde{u}+\eta)-f_{1}^{\prime}(\tilde{u})\right)d\eta~{}\partial_{1}\tilde{u}^{R}$
(4.5) $\displaystyle={}$
$\displaystyle\sum\limits_{i=1}^{n}\partial_{i}\\{\cdots\\}-\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left(f_{1}^{\prime}(\tilde{u}+\eta)-f_{1}^{\prime}(\tilde{u})\right)d\eta~{}\partial_{1}\left(\tilde{u}-\tilde{u}^{R}\right)$
$\displaystyle-\sum\limits_{i=2}^{n}\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left(f_{i}^{\prime}(\tilde{u}+\eta)-f_{i}^{\prime}(\tilde{u})\right)d\eta~{}\partial_{i}\tilde{u}-S_{\delta}^{\prime}(\phi)h,$

where

$\displaystyle\\{\cdots\\}=S_{\delta}^{\prime}(\phi)\partial_{i}\phi-
S_{\delta}^{\prime}(\phi)\left(f_{i}(\tilde{u}+\phi)-f_{i}(\tilde{u})\right)+\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left(f_{i}(\tilde{u}+\eta)-f_{i}(\tilde{u})\right)d\eta.$

Since $S_{\delta}^{\prime\prime}\geq
0,f_{1}^{\prime\prime}>0,\partial_{1}\tilde{u}^{R}>0$ and
$\left\lvert\phi\right\rvert\leq S_{\delta}(\phi),$ integrating Equation 4.5
over $\Omega$, together with Lemmas 2.2 and 2.3, gives that

$\displaystyle\frac{d}{dt}\int_{\Omega}S_{\delta}(\phi)dx\leq~{}$
$\displaystyle Ce^{-\alpha
t}\int_{\Omega}\sum\limits_{i=1}^{n}\left\lvert\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left(f_{i}^{\prime}(\tilde{u}+\eta)-f_{i}^{\prime}(\tilde{u})\right)d\eta\right\rvert
dx+C\left\lVert h\right\rVert_{L^{1}(\Omega)}$ $\displaystyle\leq~{}$
$\displaystyle Ce^{-\alpha
t}\int_{\Omega}\left\lvert\int_{0}^{\phi}S_{\delta}^{\prime\prime}(\eta)\left\lvert\eta\right\rvert
d\eta\right\rvert dx+Ce^{-\alpha t}$ $\displaystyle\leq~{}$ $\displaystyle
Ce^{-\alpha t}\left\lVert\phi\right\rVert_{L^{1}(\Omega)}+Ce^{-\alpha t}$
$\displaystyle\leq~{}$ $\displaystyle Ce^{-\alpha
t}\int_{\Omega}S_{\delta}(\phi)dx+Ce^{-\alpha t},$

where $C>0$ is independent of $\delta.$ Then by the Gronwall inequality and
letting $\delta\rightarrow 0+,$ one has that

$\left\lVert\phi(t)\right\rVert_{L^{1}(\Omega)}\leq C.$ (4.6)

For $p\in[2,+\infty),$ multiplying Equation 4.1 by
$\left\lvert\phi\right\rvert^{p-2}\phi$ yields that

$\displaystyle\frac{1}{p}\partial_{t}\left\lvert\phi\right\rvert^{p}+\frac{4(p-1)}{p^{2}}\left\lvert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rvert^{2}+(p-1)\int_{0}^{\phi}\left(f^{\prime}_{1}(\tilde{u}+\eta)-f^{\prime}_{1}(\tilde{u})\right)|\eta|^{p-2}\mathrm{d}\eta~{}\partial_{1}\tilde{u}^{R}$
(4.7) $\displaystyle={}$
$\displaystyle\sum\limits_{i=1}^{n}\partial_{i}\\{\cdots\\}-(p-1)\int_{0}^{\phi}(f^{\prime}_{1}(\tilde{u}+\eta)-f^{\prime}_{1}(\tilde{u}))|\eta|^{p-2}\mathrm{d}\eta~{}\partial_{1}\left(\tilde{u}-\tilde{u}^{R}\right)$
$\displaystyle-(p-1)\sum\limits_{i=2}^{n}\int_{0}^{\phi}(f^{\prime}_{i}(\tilde{u}+\eta)-f^{\prime}_{i}(\tilde{u}))|\eta|^{p-2}\mathrm{d}\eta~{}\partial_{i}\tilde{u}-\left\lvert\phi\right\rvert^{p-2}\phi
h,$

where

$\displaystyle\\{\cdots\\}=~{}$
$\displaystyle\left\lvert\phi\right\rvert^{p-2}\phi\partial_{i}\phi-(f_{i}(\tilde{u}+\phi)-f_{i}(\tilde{u}))\left\lvert\phi\right\rvert^{p-2}\phi$
$\displaystyle+(p-1)\int_{0}^{\phi}\left(f_{i}(\tilde{u}+\eta)-f_{i}(\tilde{u})\right)|\eta|^{p-2}\mathrm{d}\eta.$

Since

$(p-1)\int_{0}^{\phi}(f^{\prime}_{1}(\tilde{u}+\eta)-f^{\prime}_{1}(\tilde{u}))|\eta|^{p-2}\mathrm{d}\eta~{}\partial_{1}\tilde{u}^{R}\geq
C\partial_{1}\tilde{u}^{R}\left\lvert\phi\right\rvert^{p}\geq 0,$

then integrating Equation 4.7 over $\Omega$, together with Lemma 2.3, gives
that

$\displaystyle\frac{d}{dt}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}+\left\lVert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rVert^{2}_{L^{2}(\Omega)}+\left\lVert\left(\partial_{1}\tilde{u}^{R}\right)^{\frac{1}{p}}\phi\right\rVert_{L^{p}(\Omega)}^{p}$
$\displaystyle\leq~{}$ $\displaystyle Ce^{-\alpha
t}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}+C\left(\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}\right)^{1-\frac{1}{p}}\left\lVert
h\right\rVert_{L^{p}(\Omega)}$ $\displaystyle\leq~{}$ $\displaystyle
Ce^{-\alpha t}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}+Ce^{-\alpha t}.$
(4.8)

Multiplying Equation 4.8 by $(1+t)^{\beta}$, where
$\beta>\frac{n}{2}\left(p-1\right)$ is a constant, then integrating the
resulting equation over $(0,T)$ yields that

$\displaystyle(1+T)^{\beta}\left\lVert\phi(T)\right\rVert^{p}_{L^{p}(\Omega)}+\int_{0}^{T}(1+t)^{\beta}\left\lVert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rVert^{2}_{L^{2}(\Omega)}dt$
$\displaystyle\leq~{}$ $\displaystyle C\int_{0}^{T}e^{-\alpha
t}(1+t)^{\beta}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}dt+C+C\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}dt$
$\displaystyle\leq~{}$ $\displaystyle
C+C\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}dt.$
(4.9)

It follows from 3.5 and Equation 4.6 that

$\displaystyle
C\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\phi\right\rVert^{p}_{L^{p}(\Omega)}dt$
$\displaystyle\leq~{}$ $\displaystyle
C\sum\limits_{k=0}^{n-1}\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rVert^{\frac{2\gamma_{k}p}{1+\gamma_{k}p}}_{L^{2}(\Omega)}\left\lVert\phi\right\rVert^{\frac{p}{1+\gamma_{k}p}}_{L^{1}(\Omega)}dt$
$\displaystyle\leq~{}$
$\displaystyle\frac{1}{2}\int_{0}^{T}(1+t)^{\beta}\left\lVert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rVert^{2}_{L^{2}(\Omega)}dt+C\left(1+T\right)^{\beta-\gamma_{0}p},$
(4.10)

where $\gamma_{k}=\frac{k+1}{2}\left(1-\frac{1}{p}\right).$ This, together
with Equation 4.9, yields that

$\displaystyle(1+T)^{\beta}\left\lVert\phi(T)\right\rVert^{p}_{L^{p}(\Omega)}+\int_{0}^{T}(1+t)^{\beta}\left\lVert\nabla\left\lvert\phi\right\rvert^{\frac{p}{2}}\right\rVert^{2}_{L^{2}(\Omega)}dt\leq
C+C\left(1+T\right)^{\beta-\frac{p-1}{2}}.$

Thus Equation 4.3 holds for $p\in[2,+\infty).$ The case for $p\in(1,2)$
follows from the interpolation.

Step 2.  We now prove Equation 4.4. Let $\psi_{i}:=\partial_{i}\phi.$ Taking
the derivative on Equation 4.1 with respect to $x_{i}$ yields that

$\partial_{t}\psi_{i}+\sum_{j=1}^{n}\partial_{j}\left(f^{\prime}_{j}(\tilde{u})\psi_{i}\right)+\sum_{j=1}^{n}\partial_{j}\left[\left(f^{\prime}_{j}(\tilde{u}+\phi)-f^{\prime}_{j}(\tilde{u})\right)\left(\partial_{i}\tilde{u}+\psi_{i}\right)\right]=\triangle\psi_{i}-\partial_{i}h.$

Multiplying the result by $\left\lvert\psi_{i}\right\rvert^{p-2}\psi_{i},$ we
arrive at

$\displaystyle\frac{1}{p}\partial_{t}\left\lvert\psi_{i}\right\rvert^{p}+\frac{4(p-1)}{p^{2}}\left\lvert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rvert^{2}+\frac{p-1}{p}f_{1}^{\prime\prime}(\tilde{u})\partial_{1}\tilde{u}^{R}\left\lvert\psi_{i}\right\rvert^{p}$
(4.11) $\displaystyle={}$
$\displaystyle\sum_{j=1}^{n}\partial_{j}\\{\cdots\\}-\frac{p-1}{p}f_{1}^{\prime\prime}(\tilde{u})\partial_{1}\left(\tilde{u}-\tilde{u}^{R}\right)\left\lvert\psi_{i}\right\rvert^{p}-\frac{p-1}{p}\sum_{j=2}^{n}f_{j}^{\prime\prime}(\tilde{u})\partial_{j}\tilde{u}\left\lvert\psi_{i}\right\rvert^{p}$
$\displaystyle+(p-1)\sum_{j=1}^{n}\left(f^{\prime}_{j}(\tilde{u}+\phi)-f^{\prime}_{j}(\tilde{u})\right)\left(\partial_{i}\tilde{u}+\psi_{i}\right)\left\lvert\psi_{i}\right\rvert^{p-2}\partial_{j}\psi_{i}$
$\displaystyle-\partial_{i}\left(\left\lvert\psi_{i}\right\rvert^{p-2}\psi_{i}h\right)+(p-1)\left\lvert\psi_{i}\right\rvert^{p-2}\partial_{i}\psi_{i}h,$

where

$\displaystyle\\{\cdots\\}=\left\lvert\psi_{i}\right\rvert^{p-2}\psi_{i}\partial_{j}\psi_{i}-\frac{1}{p}f_{j}^{\prime}(\tilde{u})\left\lvert\psi_{i}\right\rvert^{p}-\left(f^{\prime}_{j}(\tilde{u}+\phi)-f^{\prime}_{j}(\tilde{u})\right)\left(\partial_{i}\tilde{u}+\psi_{i}\right)\left\lvert\psi_{i}\right\rvert^{p-2}\psi_{i}.$

Then integrating (4.11) over $\Omega$ and using 2.3 yield that

$\displaystyle\frac{d}{dt}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}+\left\lVert\left(\partial_{1}\tilde{u}^{R}\right)^{\frac{1}{p}}\psi_{i}\right\rVert^{p}_{L^{p}(\Omega)}$
(4.12) $\displaystyle\leq{}$ $\displaystyle Ce^{-\alpha
t}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+\underbrace{C\int_{\Omega}\left\lvert\partial_{i}\tilde{u}\right\rvert\left\lvert\phi\right\rvert\left\lvert\psi_{i}\right\rvert^{p-2}\left\lvert\nabla\psi_{i}\right\rvert
dx}_{I_{1}}$
$\displaystyle+\underbrace{C\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert\psi_{i}\right\rvert^{p-1}\left\lvert\nabla\psi_{i}\right\rvert
dx}_{I_{2}}+\underbrace{C\int_{\Omega}\left\lvert\psi_{i}\right\rvert^{p-2}\left\lvert\nabla\psi_{i}\right\rvert\left\lvert
h\right\rvert dx}_{I_{3}}.$

First, it follows from Lemma 2.3 that

$\displaystyle I_{3}$
$\displaystyle=C\int_{\Omega}\left\lvert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rvert\left(e^{-\alpha
t}\left\lvert\psi_{i}\right\rvert^{p}\right)^{\frac{p-2}{2p}}e^{\frac{p-2}{2p}\alpha
t}\left\lvert h\right\rvert dx$
$\displaystyle\leq\frac{1}{8}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}+Ce^{-\alpha
t}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+Ce^{-\alpha\left(\frac{p}{2}+1\right)t},$
(4.13)

where the Hölder inequality for $\frac{1}{2}+\frac{p-2}{2p}+\frac{1}{p}=1$ is
used. Following [10], one can claim that

$\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\left\lvert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rvert
dx\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\phi\right\rVert_{L^{2(k+1)}(\Omega)}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{\frac{p}{4}}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert^{\frac{3}{2}}_{L^{2}(\Omega)}.$
(4.14)

In fact, we first decompose
$v:=\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}=\sum\limits_{k=0}^{n-1}v^{(k)}$
as in 1.4, then

$\displaystyle\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\left\lvert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rvert
dx\leq\sum\limits_{k=0}^{n-1}\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert
v^{(k)}\right\rvert\left\lvert\nabla v\right\rvert dx$ $\displaystyle\leq~{}$
$\displaystyle\left\lVert\phi\right\rVert_{L^{2}(\Omega)}\left\lVert
v^{(0)}\right\rVert_{L^{\infty}({\mathbb{R}})}\left\lVert\nabla
v\right\rVert_{L^{2}(\Omega)}+\left\lVert\phi\right\rVert_{L^{4}(\Omega)}\left\lVert
v^{(1)}\right\rVert_{L^{4}(\Omega)}\left\lVert\nabla
v\right\rVert_{L^{2}(\Omega)}$
$\displaystyle+\sum\limits_{k=2}^{n-1}\left\lVert\phi\right\rVert_{L^{2(k+1)}(\Omega)}\left\lVert\left\lvert
v^{(k)}\right\rvert^{\frac{1}{2}}\right\rVert_{L^{4}(\Omega)}\left\lVert\left\lvert
v^{(k)}\right\rvert^{\frac{1}{2}}\right\rVert_{L^{\frac{4(k+1)}{k-1}}(\Omega)}\left\lVert\nabla
v\right\rVert_{L^{2}(\Omega)}$ $\displaystyle\leq~{}$
$\displaystyle\sum\limits_{k=0}^{n-1}\left\lVert\phi\right\rVert_{L^{2(k+1)}(\Omega)}\left\lVert
v\right\rVert_{L^{2}(\Omega)}^{\frac{1}{2}}\left\lVert\nabla
v\right\rVert^{\frac{3}{2}}_{L^{2}(\Omega)},$

which yields Equation 4.14. Then combining Equation 4.3 and Equation 4.14, the
term $I_{2}$ in Equation 4.12 satisfies that

$\displaystyle I_{2}$ $\displaystyle\leq
C(1+t)^{-\frac{1}{4}}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{\frac{p}{4}}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert^{\frac{3}{2}}_{L^{2}(\Omega)}$
$\displaystyle\leq
C(1+t)^{-1}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+\frac{1}{8}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert^{2}_{L^{2}(\Omega)}.$
(4.15)

For $I_{1},$ if $i\neq 1,$ it follows from the Hölder inequality and Equation
4.3 that

$\displaystyle I_{1}$ $\displaystyle\leq Ce^{-\alpha
t}\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert\psi_{i}\right\rvert^{\frac{p-2}{2}}\left\lvert\psi_{i}\right\rvert^{\frac{p-2}{2}}\left\lvert\nabla\psi_{i}\right\rvert
dx$ (4.16) $\displaystyle\leq Ce^{-\alpha
t}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}$
$\displaystyle\leq Ce^{-\alpha t}+Ce^{-\alpha
t}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+\frac{1}{8}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}.$

If $i=1,$ then

$\displaystyle I_{1}\leq{}$ $\displaystyle
C\int_{\Omega}\partial_{1}\tilde{u}^{R}\left\lvert\phi\right\rvert\left\lvert\psi_{1}\right\rvert^{p-2}\left\lvert\nabla\psi_{1}\right\rvert
dx+Ce^{-\alpha
t}\int_{\Omega}\left\lvert\phi\right\rvert\left\lvert\psi_{1}\right\rvert^{p-2}\left\lvert\nabla\psi_{1}\right\rvert
dx,$ (4.17)

where the second term on the RHS of Equation 4.17 can be estimated in the same
way as in Equation 4.16. For the first term, one can use 3.6 to obtain that

$\displaystyle\int_{\Omega}\partial_{1}\tilde{u}^{R}\left\lvert\phi\right\rvert\left\lvert\psi_{1}\right\rvert^{p-2}\left\lvert\nabla\psi_{1}\right\rvert
dx$ $\displaystyle\leq
C\left\lVert\partial_{1}\tilde{u}^{R}\right\rVert_{L^{\infty}(\Omega)}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{\frac{2p}{p+2}}\left\lVert\nabla\left\lvert\psi_{1}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{\frac{2p}{p+2}}$
$\displaystyle\leq
C\left\lVert\partial_{1}\tilde{u}^{R}\right\rVert_{L^{\infty}(\Omega)}^{\frac{p+2}{2}}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{p}+\frac{1}{8}\left\lVert\nabla\left\lvert\psi_{1}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}.$
(4.18)

Thus, collecting Equations 4.13, 4.15, 4.16, 4.17 and 4.18 and applying
Equation 2.2, one has that

$\frac{d}{dt}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}\leq
Ce^{-\alpha
t}+C(1+t)^{-1}\left\lVert\psi_{i}\right\rVert_{L^{p}(\Omega)}^{p}+C(1+t)^{-\frac{p+2}{2}}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{p}.$
(4.19)

Multiplying Equation 4.19 by $(1+t)^{\beta}$ with $\beta>p+1$ and then
integrating the result over $[0,T],$ one has that

$\displaystyle(1+t)^{\beta}\left\lVert\psi_{i}(t)\right\rVert_{L^{p}(\Omega)}^{p}+\int_{0}^{T}(1+t)^{\beta}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}dt$
$\displaystyle\leq~{}$ $\displaystyle
C+C\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\psi_{i}(t)\right\rVert_{L^{p}(\Omega)}^{p}dt+C\int_{0}^{T}(1+t)^{\beta-\frac{p+2}{2}}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{p}dt.$
(4.20)

It follows from 3.6 that

$\displaystyle\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\psi_{i}(t)\right\rVert_{L^{p}(\Omega)}^{p}dt$
$\displaystyle\leq~{}$ $\displaystyle
C\int_{0}^{T}(1+t)^{\beta-1}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{\frac{2p}{p+2}}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{\frac{2p}{p+2}}dt$
$\displaystyle\leq~{}$
$\displaystyle\frac{1}{2}\int_{0}^{T}(1+t)^{\beta}\left\lVert\nabla\left\lvert\psi_{i}\right\rvert^{\frac{p}{2}}\right\rVert_{L^{2}(\Omega)}^{2}dt+C\int_{0}^{T}(1+t)^{\beta-\frac{p+2}{2}}\left\lVert\phi\right\rVert_{L^{p}(\Omega)}^{p}dt.$

From Equation 4.3, one can get that

$\displaystyle\int_{0}^{T}(1+t)^{\beta-\frac{p+2}{2}}\left\lVert\phi(t)\right\rVert_{L^{p}(\Omega)}^{p}dt$
$\displaystyle\leq
C\int_{0}^{T}(1+t)^{\beta-\frac{p+2}{2}}(1+t)^{-\frac{p-1}{2}}dt$
$\displaystyle\leq C(1+T)^{\beta-p+\frac{1}{2}}.$

The proof of 4.2 is finished. ∎

###### Proof of 1.1.

It follows from Theorems 1.4 and 4.2 that

$\displaystyle\left\lVert\phi\right\rVert_{L^{\infty}(\Omega)}$
$\displaystyle\leq
C\sum\limits_{k=0}^{n-1}\left\lVert\nabla\phi\right\rVert_{L^{p_{k}}(\Omega)}^{\theta_{k}}\left\lVert\phi\right\rVert_{L^{q_{k}}(\Omega)}^{1-\theta_{k}}$
$\displaystyle\leq
C\sum\limits_{k=0}^{n-1}(1+t)^{-\left[\left(1-\frac{1}{2p_{k}}\right)\theta_{k}+\left(\frac{1}{2}-\frac{1}{2q_{k}}\right)(1-\theta_{k})\right]},$

where
$0=\left(\frac{1}{p_{k}}-\frac{1}{k+1}\right)\theta_{k}+\frac{1}{q_{k}}(1-\theta_{k}),$
$\max\\{k+1,2\\}<p_{k}<+\infty$ and $1\leq q_{k}<+\infty$ for
$k=0,1,\cdots,n-1$. A direct calculation yields that

$\displaystyle\left(1-\frac{1}{2p_{k}}\right)\theta_{k}+\left(\frac{1}{2}-\frac{1}{2q_{k}}\right)(1-\theta_{k})=\frac{1}{2}+\frac{k\theta_{k}}{2(k+1)}\geq\frac{1}{2},$

which implies that

$\left\lVert\phi\right\rVert_{L^{\infty}({\mathbb{R}}^{n})}=\left\lVert\phi\right\rVert_{L^{\infty}(\Omega)}\leq
C(1+t)^{-\frac{1}{2}}.$

Thus,

$\displaystyle\left\lVert
u-\tilde{u}^{R}\right\rVert_{L^{\infty}({\mathbb{R}}^{n})}\leq\left\lVert\phi\right\rVert_{L^{\infty}({\mathbb{R}}^{n})}+\left\lVert\tilde{u}-\tilde{u}^{R}\right\rVert_{L^{\infty}({\mathbb{R}}^{n})}\leq
C(1+t)^{-\frac{1}{2}}.$

∎

## References

  * [1] C. M. Dafermos, _Large time behavior of periodic solutions of hyperbolic systems of conservation laws_ , Journal of Differential Equations 121 (1995), no. 1, 183-202.
  * [2] C. M. Dafermos, _Long time behavior of periodic solutions to scalar conservation laws in several space dimensions_ , SIAM J. Math. Anal. 45 (2013), no. 4, 2064-2070.
  * [3] J. Glimm and P. Lax, _Decay of solutions of systems of nonlinear hyperbolic conservation laws_ , Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970.
  * [4] J. Goodman, _Nonlinear asymptotic stability of viscous shock profiles for conservation laws_ , Arch. Ration. Mech. Anal. 95 (4) (1986) 325-344.
  * [5] L. He and F. Huang, Nonlinear stability of large amplitude viscous shock wave for general viscous gas, Journal of Differential Equations, https://doi.org/10.1016/j.jde.2020.01.004.
  * [6] F. Huang, Z. Xin and T. Yang, _Contact discontinuities with general perturbation for gas motion_ ,  Adv. Math., 219 (2008), 1246–1297.
  * [7] A. M. Ilin and O. A. Oleinik, _Asymptotic behavior of solutions of the cauchy problem for some quasi-linear equations for large values of the time_ , Mat. Sb. (N.S.) 51 (93) (1960), 191-216.
  * [8] S. Kawashima and A. Matsumura, _Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion_ , Comm. Math. Phys. Vol. 101, No. 1 (1985), 97-127.
  * [9] S. Kawashima, A. Matsumura and K. Nishihara, _Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas_ , Proc. Japan Acad. Ser. A Math. Sci. Vol. 62, No. 7 (1986), 249-252.
  * [10] S. Kawashima, S. Nishibata and M. Nishikawa, _$L^{p}$ energy method for multi- dimensional viscous conservation laws and application to the stability of planar waves_, J. Hyperbolic Differ. Equations 01 (2004), no. 03, 581-603.
  * [11] P. Lax, _Hyperbolic systems of conservation laws II_ , Communications on Pure and Applied Mathematics 10 (1957), no. 4, 537-566.
  * [12] T. Liu, _Invariants and asymptotic behavior of solutions of a conservation law_ , Proceedings of the American Mathematical Society 71 (1978), no. 2, 227.
  * [13] T. Liu and Z. Xin, _Nonlinear Stability of Rarefaction Waves for Compressible Navier Stokes Equations_ , Communications in Mathematical Physics 118 (1988), 451-465.
  * [14] T. Liu and Z. Xin, _Pointwise decay to contact discontinuities for systems of viscous conservation laws_ , Asian J. Math. 1 (1997) 34-84.
  * [15] T. Liu and Y. Zeng, _Time-asymptotic behavior of wave propagation around a viscous shock profile_ , Comm. Math. Phys. 290 (2009), no. 1, 23-82.
  * [16] A. Matsumura and K. Nishihara, _On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas_ , Japan J. Appl. Math., 2 (1985), 17-25.
  * [17] A. Matsumura and K. Nishihara, _Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas_ , Japan Journal of Applied Mathematics 3 (1986), no. 1, 1-13.
  * [18] L. Nirenberg, _On elliptic partial differential equations_ , Ann. della Sc. Norm. Super. di Pisa, Cl. di Sci. 2 (1959), no. 13, 115-162.
  * [19] O. A. Oleinik, _Discontinuous solutions of non-linear differential equations_ , Uspehi Mat. Nauk (N.S.) 12 (1957), no. 3(75), 3-73.
  * [20] D. Serre, _ $L^{1}$-stability of nonlinear waves in scalar conservation laws_, Handbook of Differential Equations: Evolutionary Equations, (2002),473-553.
  * [21] J. Smoller, _Shock Waves and Reaction Diffusion Equations_ , vol. 258, Springer-Verlag, New York-Berlin, 1994.
  * [22] A. Szepessy, Z. Xin, _Nonlinear stability of viscous shock waves_ , Arch. Ration. Mech. Anal. 122 (1993) 53-103.
  * [23] T. Wang and Y. Wang, _Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation_ , Kinetic and Related Models, Vol. 12, No. 3, 2019, 637-679.
  * [24] Z. Xin, _Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions_ , Trans. Amer. Math. Soc. 319 (1990), no. 2, 805-820.
  * [25] Z. Xin, Q. Yuan and Y. Yuan, _Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-D convex scalar viscous conservation laws_ , arxiv:1902.09772 (2019), 1-42.
  * [26] Z. Xin, Q. Yuan and Y. Yuan, _Asymptotic stability of shock waves and rarefaction waves under periodic perturbations for 1-d convex scalar conservation laws_ , SIAM J. Math. Anal. 51 (2019), no. 4, 2971-2994.
  * [27] Q. Yuan and Y. Yuan, _On Riemann solutions under different initial periodic perturbations at two infinities for 1-d scalar convex conservation laws_ , J. Differ. Equ. (2019), 1-15.


# Impact of Network Delay and Decision Imperfections in IoT Assisted Cruise
Ship Evacuation
††thanks: This research was partially funded by the EU H2020 Program’s IoTAC
Research and Innovation Action under Grant No. 952684 at the Institute of
Theoretical and Applied Informatics, Polish Academy of Sciences (IITIS-PAN),
partially by the National Natural Science Foundation of China (NSFC) under
Grant No. 51979216 and the Natural Science Foundation of Hubei Province,
China, under Grants No.2021CFA001 and 20221j0059.

Yuting Ma  IITiS-PAN, Gliwice, Poland
& Wuhan University of Technology
Wuhan, China
<EMAIL_ADDRESS>Erol Gelenbe  IITiS-PAN, Gliwice, Poland
& Université Côte d’Azur, Nice, France
&Yaşar University, Bornova, Izmir, Turkey
<EMAIL_ADDRESS>Kezhong Liu  Wuhan University of Technology
Wuhan, China
<EMAIL_ADDRESS>
###### Abstract

Major challenges of assisting passengers to safely and quickly escape from
ships when an emergency occurs, include complex realistic features such as
human behavior uncertainty, dynamic human traversal times, and the computation
and communication delays in the systems that offer advice to users during an
emergency. In this paper, we present simulations that examine the influence of
these key features on evacuation performance in terms of evacuation time. The
approach is based on our previously proposed lookup table-based ship passenger
evacuation method, i.e., ANT. The simulation results we present show that
delays in the users’ reception of instructions significantly impair the
effectiveness of the evacuation service. In contrast, behavior uncertainty has
a weaker influence on the performance of the navigation method. In addition,
these effects also vary with the extent of the behavior uncertainty, the
dynamics of the traversal time distributions, and the delay in receiving
directions. These findings demonstrate the importance of carefully designing
evacuation systems for passenger ships in a way that takes into account all
realistic features of the ship’s indoor evacuation environment, including the
crucial role of information technology.

###### Index Terms:

Ship evacuation, Emergency navigation, WSN-assisted, Behavior uncertainty,
Dynamics, Communication delays

##  I Introduction

One of the main safety requirements for passenger ships is offering a reliable
and efficient emergency evacuation service to passengers. Therefore,
evacuation analysis is of primary importance even in the first stages of the
design of a vessel [1, 2, 3]. The International Maritime Organization (IMO)
has issued many Circulars about evacuation analysis and procedures for
passenger and ro-ro ships. For example, in 2016, IMO approved a new version of
the guidelines for evacuation analysis for new and existing passenger ships to
enhance the ability to evacuate passenger vessels in case of accidents [4].

However, much of the research in this area does not consider many realistic
features which may have a huge impact on both user safety and navigation
efficiency [5, 6, 7, 8], such as variations in traversal times across passages
and staircases due to vessel motion conditions, human behavior uncertainty
caused by missing or misunderstanding instructions due to panic, noise, and
overcrowding as illustrated in Figure 1, and delays in the arrival of correct
instructions due to computation and communication delays in heavily loaded
communication networks that monitor and interact with the physical world. As a
result, evacuation metrics such as evacuation time and passenger congestion
calculated by these methods are not necessarily in line with those in actual
situations. Thus the design of the evacuation system based on such analyses
may not ensure the needed safe and timely navigation of evacuees. Hence, this
paper investigates the influence of such key features on the performance of
the navigation method proposed in [9] under simulated emergency conditions.

Figure 1: Schematic representation of behavior uncertainty of a human user
during an emergency.

In this paper, we make some model simplifications and assume that the
inclination angle of the damaged ship changes at regular intervals, so that
the traversal times encountered across each individual corridor and staircase
accordingly change due to the variation of the inclination status. In
addition, some significant extensions are made in this paper with respect to
previous work [9] by introducing an error probability PoE. Then we change the
delay probability PoD and its extent SoD, to learn the variation in evacuation
performance. In addition, we explore the combined effects of both behavior
uncertainty and service delay on evacuation times.

##  II Related work

Global path-planning algorithms (e.g., Dijkstra, A*, RRT, Ant Colony
Optimization, Genetic algorithm, etc.,) have often been successfully used in
evacuation routing to minimize path length or to maximize the distance from
the hazard nodes to the exit node, based on 2D/3D maps of a built environment
[10, 11]. However, the resulting navigation paths provided by those methods
are not necessarily useful since evacuees are likely to encounter unexpected
hazards due to changing emergency conditions, such as the spread of a fire,
flooding of parts of the ship, or cascaded failures in emergency management
systems themselves. Some work has attempted to address the dynamics of danger
through the Expected Number of Oscillations (ENO) concept [12], that
quantifies the dynamics of the emergency and explores navigation paths that
have the smallest probability of changing repeatedly.

Furthermore, such global path planning methods require that the system be
completely specified regarding the parameters needed to choose optimal paths,
in contradiction with realistic situations where emergency guidance is needed.
Therefore, follow-up studies relax the requirement that all the information
about the hazards and exits is precisely known prior to path computation,
which enhances their applicability to address practical scenarios [13, 5].

Using local path-planning methods such as Artificial Potential Fields [14],
and the best direction for users based on conditions in their local
neighborhood [15, 16] together with techniques such as partial reversal, users
can avoid entering hazardous areas. However, these methods neglect the fact
that the evacuation system for passenger ships must offer safety guarantees,
and that the emergency navigation system must offer directions that guarantee
a time to the exit that does not exceed a specified upper bound (i.e., the
survival or capsizing time) even under worst-case circumstances (e.g., the
case where the roll angle reaches the 30∘) in the presence of dynamic hazards
and ship motion.

However our earlier work [9] proposed an emergency navigation system based on
condition which neglect the effect of uncertainty regarding the evacuees’
behavior due to panic or errors, and the possible lack of up-to-date
information regarding estimates of typical delays due to computtaional or
communication overload in the information processing system. Thus in the
present paper we specifically address the effect of delays in providing
accurate guidance to end users, as well as the possibility that human evacuees
may make mistakes in applying the guidance instructions due to panic or lack
of attention.

##  III Impact of delay in user navigation

The lookup tables in a static scenario guarantee that users at different
locations can identify the hazard-free direction along which they can arrive
at the exit with minimum delay, while not exceeding the worst-case time bound
from the current location to the exit. However, walking conditions are dynamic
in practice due to individual differences of the passengers, as well as other
factors such as lighting, possible smoke conditions, panic, and the ship’s
motion. These lookup tables then need to be recomputed due to the overall
system dynamics over time. Once there is a significant change in the tuples
contained in the lookup table at each node in the navigation graph, the
corresponding navigation direction needs to be updated for each of the users.
However, wireless sensor network (WSN, and other network and computational
server congestion effects may occur, so that updates of the navigation
direction can be delayed, or even occur with errors due to the use of
incomplete data in the decision algorithms [17, 18, 19]. Thus we analyze and
discuss in the following section the impact of these dynamics on the effective
delays that may be experienced by the users.

###  III-A Simulation setup

The simulated indoor environment is the second, third, and fourth floors of
the Yangtze Gold 7 Cruise, as shown in Figure 2, where 346 navigation nodes
are deployed in the simulated floors with one exit node. The number of
passageway segments and staircase segments is 600 and 5, respectively. In our
simulation settings, the worst-case traversal time
$\textit{T}_{\rm{W}}$($\overrightarrow{v_{i}v_{j}}$) across each segment
$\overrightarrow{v_{i}v_{j}}$ is calculated according to the worst-case
traversal speed, which is set to 0.067 m/s, and the typical traversal time
experienced in traversing each segment is set to a random value in the
interval
$\left[\textit{T}_{\rm{N}}(\overrightarrow{v_{i}v_{j}}),\textit{T}_{\rm{W}}(\overrightarrow{v_{i}v_{j}}\right]$.
$\textit{T}_{\rm{N}}$($\overrightarrow{v_{i}v_{j}}$) = 0.67m/s, which is set
according to the average movement speed of passengers on the passenger ships
where the walking condition remains horizontal. In addition, the deadline
$\textit{T}_{\rm{D}}$ for ship capsizing is calculated as follows:

$\textit{T}_{\rm{D}}=\textit{T}_{\rm{S}}-\textit{T}_{\rm{A}}-\textit{T}_{\rm{EL}}.$
(1)

where $\textit{T}_{\rm{S}}$ denotes the total survival time until the ship
will capsize, and $\textit{T}_{\rm{A}}$ is the awareness time beginning upon
initial notification of an emergency and ending when passengers accept the
situation and start to move based on the provided navigation direction.
$\textit{T}_{\rm{EL}}$ denotes the sum of embarkation time and the launching
time, i.e., the time required to provide for abandonment by the total number
of persons on board. According to the guidelines approved by the Maritime
Safety Committee (MSC), $\textit{T}_{S}$ equals to 60 minutes,
$\textit{T}_{\rm{A}}$ equals to 5 minutes, and $\textit{T}_{\rm{EL}}$ equals
to 25 minutes in our simulations. That is, $\textit{T}_{\rm{D}}$ equals to 30
minutes unless stated otherwise. In addition, the change interval of typical
traversal time across each segment is set to 5s.

Figure 2: The second, third, and fourth floors of the Yangtze Gold 7 Cruise.

###  III-B Simulation results

We first examine the manner in which performance is affected by the delay in
the computing and communication system which advises the users, and combine a
probabilistic and temporal representation of the imperfection with which the
system provides advice.

We assume that a delay in the communication and computing system can occur
with some probability PoD at any of the user nodes. Data about the system as a
whole is assumed to be updated in unit time steps, and we define SoD as being
the number of unit time steps regarding the delay with which the data is used.
Thus, for instance, $\textit{SoD}=5$ means that at some time t the
computational system will provide decisions for the users based on the data
that was available at time $\textit{t}-5$.

Thus in Figure 3 we consider results for the following parameters
$\textit{PoD}=0.1$ and $\textit{SoD}=1,~{}2,~{}3,~{}4,~{}5$. More
particularly, we show the relative difference in evacuation time for each user
with these five different delay values for all of the unique Node identifiers
$0$ to $345$ (assigned when the navigation network is constructed) in order to
visualize the simulation results. The relative difference in evacuation times
for each node is defined as follows:

$\Delta(i,PoD,SoD)=\frac{T(i,PoD,SoD)-T(i)}{T(i)},$ (2)

where $T(i)$ denotes the “ideal” evacuation time for a user at location (node)
$i$ all the way to the exit, when there is no delay, i.e. $PoD=0$, while
$T(i,PoD,SoD)$ is the corresponding value with probability PoD that there will
be a delay of value SoD. We clearly see that guiding evacuees with up-to-date
navigation suggestions has better performance with respect to the evacuation
time, and therefore the observed relative differences are positive in most
cases. But it is worth noticing that there are still a few cases where the
performance of guiding with up-to-date navigation directions is even worse
than that of guiding with outdated navigation suggestions. It is mainly
because the navigation based on up-to-date suggestions is not the optimal
scheme for addressing the dynamics of walking conditions.

In addition, we also simulate the average evacuation time with different delay
values for all the users located in the 346 different user nodes. Figure 4
plots the relative difference in average evacuation time defined as:

$\displaystyle\Delta_{AVG}(PoD,SoD)$
$\displaystyle~{}~{}=\frac{\sum_{i=0}^{345}T(i,PoD,SoD)-\sum_{i=0}^{345}T(i)}{\sum_{i=0}^{345}T(i)}.$
(3)

It is observed that the relative difference in average evacuation time
initially increassd with the delay SoD, but when SoD exceeds $3$, the relative
difference does not continue to increase. Thus it appears that for the
parameters we have chosen there is a criticality point for the worst
evacuation performance, and in our simulation, it is at $\textit{SoD}=3$.

We also carry out a group of simulations, where $150$ users are randomly
deployed at the user nodes, and Figure 5 shows the average evacuation time and
the relative difference of the $150$ users from $53$ runs of the simulation.
The observation from Figure 5 aligns with the findings presented in Figures 3
and 4.

We further evaluate the influence of the different possibilities of delay in
navigation service on evacuation time. Specifically, we conduct a group of
simulations, where the possibility of delay in navigation service at each node
is set to $0.0,~{}0.1,~{}0.2,~{}0.3,~{}0.4$, and $0.5$. The delay value $SoD$
is set to $1,~{}2$, and $3$. Figure 6 shows the relative difference in average
evacuation time for 346 users at different user nodes. We can see that the
relative difference in average evacuation time is increased with the
possibility of delay in navigation service and achieves its maximum when
$PoD=0.4$ for $SoD=1,~{}2,~{}3$.

Figure 3: Relative difference in evacuation time between guiding without delay
and guiding with different extents of delay. Figure 4: Relative difference in
average evacuation time between guiding without delay and guiding with
different extents of delay.

(a) Average evacuation time of the 150 users (b) Relative difference in the
average evacuation time of the 150 users

Figure 5: Average evacuation time and the relative difference of the 150
users.

(a) SoD = 1 (b) SoD = 2 (c) SoD = 3

Figure 6: Relative difference of average evacuation time of the 346 users for
PoD = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5 and SoD=1, 2, and 3, respectively.

##  IV Impact of behavior uncertainty

When an emergency occurs, the evacuees may move due to panic or difficulties
in reading instructions, along a random direction instead of the direction
provided by the evacuation system. In order to explore the resulting
performance, we represent this effect by an error probability and measure the
evacuation time accordingly. 346 users are inserted into the simulated network
in Section III. In addition, in this group of simulations, the typical
traversal time across each segment is static in order to focus on the impact
of the behavior uncertainty. The simulation parameters used in this section
are the same as those in Section III.

Figure 7 plots the relative difference in the evacuation time between escaping
according to the navigation directions totally and escaping with a certain
uncertainty probability. The relative difference is defined as follows:

$\delta(i,PoE)=\frac{\textit{T}(i,PoE)-\textit{T}(i)}{\textit{T}(i)}.$ (4)

We can see that the relative differences are positive in all cases, which
means failing to escape according to the provided navigation direction impairs
the capability of the evacuation scheme.

Furthermore, we measure the relative difference in average evacuation time for
346 users, which is calculated as follows:

$\delta_{AVG}(PoE)=\frac{\sum_{i=0}^{345}\textit{T}(i,PoE)-\sum_{i=0}^{345}\textit{T}(i)}{\sum_{i=0}^{345}\textit{T}(i)}.$
(5)

Figure 8 plots the relative difference in average evacuation time when
escaping with different probabilities of behavior uncertainty (i.e., the
probabilities of escaping along an error direction). While behavior
uncertainty has an effect on evacuation time, it is tiny. Moreover, we can see
that the performance deteriorates with the increase of the uncertainty
probability until it achieves 0.4 (i.e., $PoE=0.4$). That is, in our
simulation, a greater probability than 0.4 will not further jeopardize the
evacuation performance. Compared with the impact of the delay in navigation
service, the impact of behavior uncertainty is relatively slight, which means
the lookup table-based method is more resilient to behavior uncertainty than
navigation delay.

Figure 7: Evacuation time with different uncertainty probabilities and the
relative difference in evacuation time between escaping without behavior
uncertainty and with different uncertainty probabilities. Figure 8: Relative
difference in average evacuation time between escaping without behavior
uncertainty and with different uncertainty probabilities.

##  V Impact of both behavior uncertainty and delay in navigation service

In this experiment, we evaluate the combined influence of both the navigation
delay and behavior uncertainty on evacuation performance measured by
evacuation time.

The experiment is done in a scenario where the refreshing period is set to 5s,
the probability, as well as the delay value, is set as 0.4 and 3,
respectively, and the uncertainty probability is fixed at 0.4. Figure 9 plots
the relative difference in the evacuation time taken for all users to arrive
at the specified exit. We observe that there is a dramatic increment in the
evacuation time when we take the above-mentioned realistic features into
consideration. Therefore, from a practical perspective, it is of significant
value to consider the behavior uncertainty and the navigation delay to assist
users with a successful escape while doing path planning for passengers on
damaged ships.

Figure 9: Evacuation time and the relative difference in evacuation time.

##  VI Conclusions and Further Research

In passenger ships and other large vehicles and aircraft, reliable emergency
evacuation is required for ensuring passenger trust in the means of travel,
and for their safety and wellbeing. Thus over the last decade substantial
research has been conducted in designing technology-assisted means to provide
passengers with the best advice regarding evacuation procedures during
emergencies [20] and many of the proposed approaches use some form of
optimization [21].

However during emergencies, especially if the vessel is damaged, it becomes
very challenging both for the underlying ICT (Information and
Telecommunication Technologies) and for the passengers and staff to implement
and follow instructions that are based on prior optimization and established
routines. In addition, the ICT infrastructure may also be damaged and
disconnected, and the recommended evacuation routes and the communication
network may be congested. Thus the need to support user safety and evacuation
in such complex and dynamic environments with many rapidly changing variables
can also overwhelm the ICT-based emergency management system itself [22]. In
addition, path planning based on the updated emergency situation may be
accompanied by messages which flood the communication system about
environmental dynamics, aggravate the congestion, and result in packet losses
and longer end-to-end delays in communicating decisions.

Thus this work is the first study of the influence of such key side effects of
emergency evacuation in a practical ship evacuation scenario. Through
extensive simulations, this paper analyzes the effects on passenger evacuation
performance in ship emergencies, of realistic features including the behavior
uncertainty represented with different probabilities, the dynamics of
traversal time with change frequencies, and the different delays in the
arrival of navigation instructions to passengers. Furthermore, these
simulations use real-world parameters from the real passenger cruise ship
“Yangtze Gold 7” and its passenger evacuation system, and evaluate the effect
of delays in the information that reaches the human users during its passenger
evacuation.

Assuming that ongoing situational information is gathered by wireless sensor
networks [23, 24, 25], we have considered the effect of delayed decisions
which are computed and forwarded by a central decision ICT system, to all end
users as they pass through pre-determined “nodes” that guide them towards safe
exit points. As these delays increase, we see that the emergency exit times of
passengers are often substantially increased.

In future research we plan to take into account the delaying factors in
advance, so as to design novel decentralized emergency navigation systems for
guiding passengers to safety, that pre-locate advisory data to passengers in
key system intermediate nodes, and also combine centralized decisions together
with individual user decision aids with hand-held mobile devices [16].

##

## References

  * [1] C. Nasso, S. Bertagna, F. Mauro, A. Marinò, and V. Bucci, “Simplified and advanced approaches for evacuation analysis of passenger ships in the early stage of design,” _Brodogradnja: Teorija i praksa brodogradnje i pomorske tehnike_ , vol. 70, no. 3, pp. 43–59, 2019.
  * [2] X. Wang, Z. Liu, S. Loughney, Z. Yang, Y. Wang, and J. Wang, “Numerical analysis and staircase layout optimisation for a ro-ro passenger ship during emergency evacuation,” _Reliability Engineering & System Safety_, vol. 217, p. 108056, 2022.
  * [3] H. Arshad, J. Emblemsvåg, G. Li, and R. Ostnes, “Determinants, methods, and solutions of evacuation models for passenger ships: A systematic literature review,” _Ocean Engineering_ , vol. 263, p. 112371, 2022.
  * [4] I. M. Organization, “Guidelines for evacuation analysis for new and existing passenger ships,” _MSC. 1/Circ. 1238_ , 2016.
  * [5] E. Gelenbe and H. Bi, “Emergency navigation without an infrastructure,” _Sensors_ , vol. 14, no. 8, pp. 15 142–15 162, 2014. [Online]. Available: https://www.mdpi.com/1424-8220/14/8/15142
  * [6] Q. Xie, S. Guo, Y. Zhang, C. Wang, C. Ma, and Q. Li, “An integrated method for assessing passenger evacuation performance in ship fires,” _Ocean Engineering_ , vol. 262, p. 112256, 2022.
  * [7] S. Fang, Z. Liu, X. Wang, J. Wang, and Z. Yang, “Simulation of evacuation in an inclined passenger vessel based on an improved social force model,” _Safety science_ , vol. 148, p. 105675, 2022.
  * [8] X. Wang, G. Xia, J. Zhao, J. Wang, Z. Yang, S. Loughney, S. Fang, S. Zhang, Y. Xing, and Z. Liu, “A novel method for the risk assessment of human evacuation from cruise ships in maritime transportation,” _Reliability Engineering & System Safety_, vol. 230, p. 108887, 2023.
  * [9] Y. Ma, K. Liu, M. Chen, J. Ma, X. Zeng, K. Wang, and C. Liu, “Ant: Deadline-aware adaptive emergency navigation strategy for dynamic hazardous ship evacuation with wireless sensor networks,” _IEEE Access_ , vol. 8, pp. 135 758–135 769, 2020.
  * [10] A. Desmet, “A modelling approach to human navigation in constrained spaces,” _PhD Dissertation, Dept. of Electrical & Electronic Engineering, Imperial College London_, 2014.
  * [11] B. Wiandt, A. Kokuti, and V. Simon, “Application of collective movement in real life scenarios: Overview of current flocking solutions,” _Scalable Comput. Pract. Exp._ , vol. 16, no. 3, pp. 233–248, 2015.
  * [12] L. Wang, Y. He, W. Liu, N. Jing, J. Wang, and Y. Liu, “On oscillation-free emergency navigation via wireless sensor networks,” _IEEE Transactions on Mobile Computing_ , vol. 14, no. 10, pp. 2086–2100, 2014.
  * [13] A. Filippoupolitis et al., “Emergency response simulation using wireless sensor networks,” in _Proceedings of the 1st International Conference on Ambient Media and Systems_ , ser. Ambi-Sys ’08. Brussels, BEL: ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering), 2008.
  * [14] E. Gelenbe, N. Schmajuk, J. Staddon, and J. Reif, “Autonomous search by robots and animals: A survey,” _Robotics and Autonomous Systems_ , vol. 22, no. 1, pp. 23–34, 1997.
  * [15] Y.-C. Tseng, M.-S. Pan, and Y.-Y. Tsai, “Wireless sensor networks for emergency navigation,” _Computer_ , vol. 39, no. 7, pp. 55–62, 2006.
  * [16] A. Kokuti and E. Gelenbe, “Directional navigation improves opportunistic communication for emergencies,” _Sensors_ , vol. 14, no. 8, pp. 15 387–15 399, 2014. [Online]. Available: https://www.mdpi.com/1424-8220/14/8/15387
  * [17] E. Shi, J. Zhang, S. Chen, J. Zheng, Y. Zhang, D. W. K. Ng, and B. Ai, “Wireless energy transfer in ris-aided cell-free massive mimo systems: Opportunities and challenges,” _IEEE Communications Magazine_ , vol. 60, no. 3, pp. 26–32, 2022.
  * [18] J. Dong, J. Zhang, Y. Shi, and J. H. Wang, “Faster activity and data detection in massive random access: A multiarmed bandit approach,” _IEEE Internet of Things Journal_ , vol. 9, no. 15, pp. 13 664–13 678, 2022.
  * [19] Y. Ma, H. Tian, X. Liao, J. Zhang, W. Wang, K. Chen, and X. Jin, “Multi-objective congestion control,” in _Proceedings of the Seventeenth European Conference on Computer Systems_ , 2022, pp. 218–235.
  * [20] N. Dimakis, A. Filippoupolitis, and E. Gelenbe, “Distributed building evacuation simulator for smart emergency management,” _The Computer Journal_ , vol. 53, no. 9, pp. 1384–1400, 2010.
  * [21] E. Gelenbe, “Future research on cyber-physical emergency management systems,” _Future Internet_ , vol. 5, no. 3, pp. 336–354, 2013.
  * [22] A. Filippoupolitis, “An adaptive system for movement decision support in building evacuation,” in _Computer and Information Sciences: Proceedings of the 25th International Symposium on Computer and Information Sciences_. Springer, 2010, pp. 389–392.
  * [23] M.-S. Pan, C.-H. Tsai, and Y.-C. Tseng, “Emergency guiding and monitoring applications in indoor 3d environments by wireless sensor networks,” _International Journal of Sensor Networks_ , vol. 1, no. 1-2, pp. 2–10, 2006\.
  * [24] S. Li, A. Zhan, X. Wu, and G. Chen, “Ern: Emergence rescue navigation with wireless sensor networks,” in _2009 15th International Conference on Parallel and Distributed Systems_. IEEE, 2009, pp. 361–368.
  * [25] C. Wang, H. Lin, R. Zhang, and H. Jiang, “Send: A situation-aware emergency navigation algorithm with sensor networks,” _IEEE Transactions on Mobile Computing_ , vol. 16, no. 4, pp. 1149–1162, 2016.