# The generalized combined effect for one dimensional wave equations with
semilinear terms including product type

Ryuki Kido 111 Master course, Mathematical Institute, Tohoku University, Aoba,
Sendai 980-8578, Japan. email<EMAIL_ADDRESS>(Kido),
<EMAIL_ADDRESS>(Takamatsu)., Takiko Sasaki 222 Department of
Mathematical Engineering, Faculty of Engineering, Musashino University, 3-3-3
Ariake, Koto-ku, Tokyo 135-8181, Japan./ Mathematical Institute, Tohoku
University, Aoba, Sendai 980-8578, Japan. e-mail<EMAIL_ADDRESS>, Shu Takamatsu 11footnotemark: 1 ,
Hiroyuki Takamura 333Mathematical Institute, Tohoku University, Aoba, Sendai
980-8578, Japan. e-mail<EMAIL_ADDRESS>
(Dedicated to Professor Takayoshi Ogawa on his sixtieth birthday
$\begin{array}[]{ll}\mbox{\footnotesize{\bf Keywords:}}&\mbox{\footnotesize
semilinear wave equation, one dimension, classical solution,}\\\
&\mbox{\footnotesize lifespan, combined effect}\\\ \mbox{\footnotesize{\bf
MSC2020:}}&\mbox{\footnotesize primary 35L71, secondary 35B44}\\\ \end{array}$
)

###### Abstract

We are interested in the so-called “combined effect” of two different kinds of
nonlinear terms for semilinear wave equations in one space dimension.
Recently, the first result with the same formulation as in the higher
dimensional case has been obtained if and only if the total integral of the
initial speed is zero, namely Huygens’ principle holds. In this paper, we
extend the nonlinear term to the general form including the product type. Such
model equations are extremely meaningful only in one space dimension because
the most cases in higher dimensions possess the global-in-time existence of a
classical solution in the general theory for nonlinear wave equations. It is
also remarkable that our results on the lifespan estimates are partially
better than those of the general theory. This fact tells us that there is a
possibility to improve the general theory which was expected complete more
than 30 years ago.

##  1 Introduction

Let us consider the initial value problems;

$\left\\{\begin{array}[]{ll}\displaystyle
u_{tt}-u_{xx}=A|u_{t}|^{p}|u|^{q}+B|u|^{r}&\mbox{in}\quad{\bf
R}\times(0,T),\\\ u(x,0)={\varepsilon}f(x),\
u_{t}(x,0)={\varepsilon}g(x),&x\in{\bf R},\end{array}\right.$ (1.1)

where $p,q,r>1$, $A,B\geq 0$ and $T>0$. We assume that $f$ and $g$ are given
smooth functions of compact support and a parameter ${\varepsilon}>0$ is
“small enough”. We are interested in the lifespan $T({\varepsilon})$, the
maximal existence time, of classical solutions of (1.1). Our results in this
paper are the following estimates for $A>0$ and $B>0$;

$T({\varepsilon})\sim\min\\{C{\varepsilon}^{-(p+q-1)},C{\varepsilon}^{-(r-1)/2}\\}\qquad\mbox{if}\
\int_{{\bf R}}g(x)dx\not=0\\\ $ (1.2)

and

$T({\varepsilon})\sim\left\\{\begin{array}[]{l}C{\varepsilon}^{-(p+q)(r-1)/(r+1)}\\\
\quad\mbox{for}\ \displaystyle\frac{r+1}{2}\leq p+q\leq r,\\\
\min\\{C{\varepsilon}^{-(p+q-1)},C{\varepsilon}^{-r(r-1)/(r+1)}\\}\\\
\quad\mbox{otherwise}\\\ \end{array}\right.\mbox{if}\ \displaystyle\int_{{\bf
R}}g(x)dx=0.$ (1.3)

Here we denote the fact that there are positive constants, $C_{1}$ and
$C_{2}$, independent of ${\varepsilon}$ satisfying $A({\varepsilon},C_{1})\leq
T({\varepsilon})\leq A({\varepsilon},C_{2})$ by $T({\varepsilon})\sim
A({\varepsilon},C)$. We note that (1.2) and (1.3) are already established in
the special setting $q=0$ by Morisawa, Sasaki and Takamura [17], but it is a
non-trivial business to extend it to (1.1) due to the first term of product
type for which different estimates from $q=0$ are required in the proof. Also
we note that the case of $p=1$, or $q=1$, is excluded because there is no hope
to construct a classical solution due to lack of the differentiability. If we
replace $|u_{t}|^{p}|u|^{q}$ with $u_{t}|u|^{q}$ for $p=1$ and $q>1$,
$|u_{t}|^{p}u$ for $p>1$ and $q=1$, $u_{t}u$ for $p=q=1$, then we may have the
similar result at least of the existence part, but our method in this paper
cannot be applicable directly for such terms.

First we note that it was conjectured that

$T({\varepsilon})\sim C{\varepsilon}^{-(p+q-1)}\quad\mbox{for $A>0$ and
$B=0$.}$

This was verified by Zhou [23] for the upper bound with integer $p,q$
satisfying $p\geq 1,q\geq 0,p+q\geq 2$, and by Li,Yu and Zhou [12, 13] for the
lower bound with integer $p,q$ satisfying $p+q\geq 2$ including more general
but smooth terms. Note that [23] is a preprint version of Zhou [22] in which
only the case of $q=0$ is considered. But it is easy to apply its argument to
the case of $q>0$ by making use of

$|u_{t}|^{p}|u|^{q}=(p/(p+q))^{p}\left|\left(|u|^{(p+q)/p}\right)_{t}\right|^{p}$
(1.4)

as in [23]. For the sake of completeness of this paper, we shall repeat its
proof in Appendix below.

On the other hand, Zhou [21] obtained

$T({\varepsilon})\sim\left\\{\begin{array}[]{ll}C{\varepsilon}^{-(r-1)/2}&\mbox{if}\
\displaystyle\int_{{\bf R}}g(x)dx\neq 0,\\\
C{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{if}\ \displaystyle\int_{{\bf
R}}g(x)dx=0\end{array}\right.\mbox{for $A=0$ and $B>0$}.$

Therefore (1.2) and (1.3) are quite natural as taking the minimum of both
results except for the first case in (1.3), in which, we have

$\begin{array}[]{c}\displaystyle
C{\varepsilon}^{-(p+q)(r-1)/(r+1)}\leq\min\\{C{\varepsilon}^{-(p+q-1)},C{\varepsilon}^{-r(r-1)/(r+1)}\\}\\\
\displaystyle\quad\mbox{for}\ \displaystyle\frac{r+1}{2}\leq p+q\leq
r.\end{array}$ (1.5)

We shall call this special phenomenon by “generalized combined effect” of two
nonlinearities. The original combined effect, which means the case of $q=0$,
was first observed by Han and Zhou [2] which targets to show the optimality of
the result of Katayama [7] on the lower bound of the lifespan of classical
solutions of nonlinear wave equations with a nonlinear term $u_{t}^{3}+u^{4}$
in two space dimensions including more general nonlinear terms. It is known
that $T({\varepsilon})\sim\exp\left(C{\varepsilon}^{-2}\right)$ for the
nonlinear term $u_{t}^{3}$ and $T({\varepsilon})=\infty$ for the nonlinear
term $u^{4}$, but Katayama [7] obtained only a much worse estimate than their
minimum as $T({\varepsilon})\geq c{\varepsilon}^{-18}$. Surprisingly, more
than ten years later, Han and Zhou [2] showed that this result is optimal as
$T({\varepsilon})\leq C{\varepsilon}^{-18}$. They also considered (1.1) with
$q=0$ for all space dimensions $n$ bigger than 1 and obtain the upper bound of
the lifespan. Its counter part, the lower bound of the lifespan, was obtained
by Hidano, Wang and Yokoyama [3] for $n=2,3$. See the introduction of [3] for
the precise results and references. We note that the first case in (1.3) with
$q=0$ coincides with the lifespan estimate for the combined effect in [2, 3]
if one sets $n=1$ formally. Indeed, [2] and [3] showed that

$T({\varepsilon})\sim C{\varepsilon}^{-2p(r-1)/\\{2(r+1)-(n-1)p(r-1)\\}}$
(1.6)

holds for $n=2,3$ provided

$(r-1)\\{(n-1)p-2\\}<4,\ 2\leq p\leq r\leq 2p-1,\ r>\frac{2}{n-1}.$ (1.7)

Later, Dai, Fang and Wang [1] improved the lower bound of lifespan for the
critical case in [3]. They also show that $T({\varepsilon})<\infty$ for all
$p,r>1$ in case of $n=1$, i.e. (1.1) with $q=0$. For the non-Euclidean setting
of the results above, see Liu and Wang [14] for example, in which the
application to semilinear damped wave equations is included.

Finally we strongly remark that our estimates in (1.2) and (1.3) are better
than those of the general theory by Li, Yu and Zhou [12, 13] in case of

$\frac{r+1}{2}<p+q<r\quad\mbox{and}\quad\int_{{\bf R}}g(x)dx=0$

with integer $p,q,r\geq 2$. Because our result on the lower bound of the
lifespan can be established also for the smooth terms as
$u_{tt}-u_{xx}=u_{t}^{p}u^{q}+u^{r}$. The typical example is
$(p,q,r)=(2,2,6)$. This fact shows a possibility to improve the general
theory. For details, see the last half of the next section. We note that this
kind of observations in Morisawa, Sasaki and Takamura [17] has an error by
wrong citation in the third case in (2.24) in [17]. This paper corrects it. We
also note that, even for the original combined effect of $q=0$, the integer
points satisfying (1.7) are $(p,r)=(2,3),(3,3),(3,4)$ for $n=2$ and
$(p,r)=(2,2)$ for $n=3$, but (1.6) with $p=r$ agrees with the case of $A=0$
and $B>0$. See Introduction of Imai, Kato, Takamura and Wakasa [4] for
references on the case of $A=0$ and $B>0$. Hence one can say that only the
lifespan estimates with $(p,r)=(2,3),(3,4)$ for $n=2$ are essentially in the
combined effect case. If $q\neq 0$, $p$ is replaced with $p+q$ in the results
above. Therefore it has less meaningful to consider (1.1) in higher space
dimensions, $n\geq 2$, if we discuss the optimality of the general theory.

Of course, some special structure of the nonlinear terms such as “null
condition” guarantees the global-in-time existence. See Nakamura [16], Luli,
Yang and Yu [15], Zha [19, 20] for examples in this direction. But we are
interested in the optimality of the general theory. The details are discussed
at the end of Section 2 below. This work is initiated by series of papers,
Kitamura [8], Kitamura, Morisawa and Takamura [9, 10], Kitamura, Takamura and
Wakasa [11], in which the weighted nonlinear terms are considered for the
purpose to be a trigger to extend the general theory to the one for non-
autonomous equations.

This paper is organized as follows. In the next section, the preliminaries are
introduced. Moreover, (1.2) and (1.3) are divided into four theorems, and we
compare our results with those of the general theory. Sections 3 is devoted to
the proof of the existence part of (1.2). Sections 4 and 5 are devoted to the
proof of the existence part of (1.3). Their main strategy is the iteration
method in the weighted $L^{\infty}$ space due to Morisawa, Sasaki and Takamura
[17] which is originally introduced by John [5]. Finally, we prove the blow-up
part of (1.2) and (1.3) by following essentially, Han and Zhou [2] for the
generalized combined effect, and the iteration argument in [17] for other
cases.

##  2 Preliminaries and main results

Throughout this paper, we assume that the initial data $(f,g)\in
C_{0}^{2}({\bf R})\times C^{1}_{0}({\bf R})$ satisfies

$\mbox{\rm supp }f,\ \mbox{supp }g\subset\\{x\in{\bf R}:|x|\leq R\\},\quad
R\geq 1.$ (2.1)

Let $u$ be a classical solution of (1.1) in the time interval $[0,T]$. Then
the support condition of the initial data, (2.1), implies that

$\mbox{supp}\ u(x,t)\subset\\{(x,t)\in{\bf R}\times[0,T]:|x|\leq t+R\\}.$
(2.2)

For example, see Appendix of John [6] for this fact.

It is well-known that $u$ satisfies the following integral equation.

$u(x,t)={\varepsilon}u^{0}(x,t)+L(A|u_{t}|^{p}|u|^{q}+B|u|^{r})(x,t),$ (2.3)

where $u^{0}$ is a solution of the free wave equation with the same initial
data,

$u^{0}(x,t):=\frac{1}{2}\\{f(x+t)+f(x-t)\\}+\frac{1}{2}\int_{x-t}^{x+t}g(y)dy,$
(2.4)

and a linear integral operator $L$ for a function $v=v(x,t)$ in Duhamel’s term
is defined by

$L(v)(x,t):=\frac{1}{2}\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}v(y,s)dy.$ (2.5)

Then, one can apply the time-derivative to (2.3) to obtain

$u_{t}(x,t)={\varepsilon}u_{t}^{0}(x,t)+L^{\prime}(A|u_{t}|^{p}|u|^{q}+B|u|^{r})(x,t)$
(2.6)

and

$u_{t}^{0}(x,t)=\frac{1}{2}\\{f^{\prime}(x+t)-f^{\prime}(x-t)+g(x+t)+g(x-t)\\},$
(2.7)

where $L^{\prime}$ for a function $v=v(x,t)$ is defined by

$L^{\prime}(v)(x,t):=\frac{1}{2}\int_{0}^{t}\\{v(x+t-s,s)+v(x-t+s,s)\\}ds.$
(2.8)

On the other hand, applying the space-derivative to (2.3), we have

$u_{x}(x,t)={\varepsilon}u_{x}^{0}(x,t)+\overline{L^{\prime}}(A|u_{t}|^{p}|u|^{q}+B|u|^{r})(x,t)$
(2.9)

and

$u_{x}^{0}(x,t)=\frac{1}{2}\\{f^{\prime}(x+t)+f^{\prime}(x-t)+g(x+t)-g(x-t)\\},$
(2.10)

where $\overline{L^{\prime}}$ for a function $v=v(x,t)$ is defined by

$\overline{L^{\prime}}(v)(x,t):=\frac{1}{2}\int_{0}^{t}\\{v(x+t-s,s)-v(x-t+s,s)\\}ds.$
(2.11)

Therefore, $u_{x}$ is expressed by $u$ and $u_{t}$. Moreover, one more space-
derivative to (2.6) yields that

$\begin{array}[]{ll}u_{tx}(x,t)=&\displaystyle{\varepsilon}u_{tx}^{0}(x,t)\\\
&\displaystyle+AL^{\prime}(p|u_{t}|^{p-2}u_{t}u_{tx}|u|^{q}+q|u|^{q-2}uu_{x}|u_{t}|^{p})(x,t)\\\
&\displaystyle+BL^{\prime}(r|u|^{r-2}uu_{x})(x,t)\end{array}$ (2.12)

and

$u_{tx}^{0}(x,t)=\frac{1}{2}\\{f^{\prime\prime}(x+t)-f^{\prime\prime}(x-t)+g^{\prime}(x+t)+g^{\prime}(x-t)\\}.$
(2.13)

Similarly, we have that

$\begin{array}[]{ll}u_{tt}(x,t)=&\displaystyle{\varepsilon}u_{tt}^{0}(x,t)+A|u_{t}(x,t)|^{p}|u(x,t)|^{q}+B|u(x,t)|^{r}\\\
&\displaystyle+A\overline{L^{\prime}}(p|u_{t}|^{p-2}u_{t}u_{tx}|u|^{q}+q|u|^{q-2}uu_{x}|u_{t}|^{p})(x,t)\\\
&\displaystyle+B\overline{L^{\prime}}(r|u|^{r-2}uu_{x})(x,t)\end{array}$

and

$u_{tt}^{0}(x,t)=\frac{1}{2}\\{f^{\prime\prime}(x+t)+f^{\prime\prime}(x-t)+g^{\prime}(x+t)-g^{\prime}(x-t)\\}.$

Therefore, $u_{tt}$ is expressed by $u,u_{t},u_{x},u_{tx}$ and so is $u_{xx}$,
because of

$\begin{array}[]{ll}u_{xx}(x,t)=&\displaystyle{\varepsilon}u_{xx}^{0}(x,t)\\\
&\displaystyle+A\overline{L^{\prime}}(p|u_{t}|^{p-2}u_{t}u_{tx}|u|^{q}+q|u|^{q-2}uu_{x}|u_{t}|^{p})(x,t)\\\
&\displaystyle+B\overline{L^{\prime}}(r|u|^{r-2}uu_{x})(x,t)\end{array}$

and

$u_{xx}^{0}(x,t)=u^{0}_{tt}(x,t).$

First, we note the following fact.

###### Proposition 2.1

Assume that $(f,g)\in C^{2}({\bf R})\times C^{1}({\bf R})$. Let $(u,w)$ be a
$C^{1}$ solution of a system of integral equations;

$\left\\{\begin{array}[]{l}u={\varepsilon}u^{0}+L(A|w|^{p}|u|^{q}+B|u|^{r}),\\\
w={\varepsilon}u_{t}^{0}+L^{\prime}(A|w|^{p}|u|^{q}+B|u|^{r})\end{array}\right.\mbox{in}\
{\bf R}\times[0,T]$ (2.14)

with some $T>0$. Then, $w\equiv u_{t}$ in ${\bf R}\times[0,T]$ holds and $u$
is a classical solution of (1.1) in ${\bf R}\times[0,T]$.

Proof. It is trivial that $w\equiv u_{t}$ by differentiating the first
equation with respect to $t$. The rest part is easy along with the
computations above in this section. $\Box$

Our results in (1.2) and (1.3) are divided into the following four theorems.

###### Theorem 2.1

Let $A>0$ and $B>0$. Assume (2.1) and

$\int_{{\bf R}}g(x)dx\neq 0.$ (2.15)

Then, there exists a positive constant
${\varepsilon}_{1}={\varepsilon}_{1}(f,g,p,q,A,B,R)>0$ such that a classical
solution $u\in C^{2}({\bf R}\times[0,T])$ of (1.1) exists as far as $T$
satisfies

$T\leq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\displaystyle\frac{r+1}{2},\\\ c{\varepsilon}^{-(r-1)/2}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q,\end{array}\right.$ (2.16)

where $0<{\varepsilon}\leq{\varepsilon}_{1}$, and $c$ is a positive constant
independent of ${\varepsilon}$.

###### Theorem 2.2

Let $A>0$ and $B>0$. Assume (2.1) and

$\int_{{\bf R}}g(x)dx=0.$ (2.17)

Then, there exists a positive constant
${\varepsilon}_{2}={\varepsilon}_{2}(f,g,p,q,A,B,R)>0$ such that a classical
solution $u\in C^{2}({\bf R}\times[0,T])$ of (1.1) exists as far as $T$
satisfies

$T\leq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\dfrac{r+1}{2},\\\ c{\varepsilon}^{-(p+q)(r-1)/(r+1)}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q\leq r,\\\
c{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{for}\ p+q\geq r,\end{array}\right.$
(2.18)

where $0<{\varepsilon}\leq{\varepsilon}_{2}$, and $c$ is a positive constant
independent of ${\varepsilon}$.

###### Theorem 2.3

Let $A>0$ and $B>0$. Assume (2.1) and

$\int_{{\bf R}}g(x)dx>0.$ (2.19)

Then, there exists a positive constant
${\varepsilon}_{3}={\varepsilon}_{3}(f,g,p,q,A,B,R)>0$ such that any classical
solution of (1.1) in the time interval $[0,T]$ cannot exist as far as $T$
satisfies

$T\geq\left\\{\begin{array}[]{ll}C{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\displaystyle\frac{r+1}{2},\\\ C{\varepsilon}^{-(r-1)/2}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q,\end{array}\right.$ (2.20)

where $0<{\varepsilon}\leq{\varepsilon}_{3}$, and $C$ is a positive constant
independent of ${\varepsilon}$.

###### Theorem 2.4

Let $A>0$ and $B>0$. Assume (2.1) and

$\begin{array}[]{l}f(x)\geq 0(\not\equiv 0),\ g(x)\equiv 0,\\\ f(x)\geq f_{0}\
\mbox{and}\ -f^{\prime}(x)\geq f_{0}\ \mbox{for}\ x\in(-R/2,0),\end{array}$
(2.21)

with some positive constant $f_{0}$. Then, there exists a positive constant
${\varepsilon}_{4}={\varepsilon}_{4}(f,p,q,A,B,R)>0$ such that any classical
solution of (1.1) in the time interval $[0,T]$ cannot exist as far as $T$
satisfies

$T\geq\left\\{\begin{array}[]{ll}C{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\dfrac{r+1}{2},\\\ C{\varepsilon}^{-(p+q)(r-1)/(r+1)}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q\leq r,\\\
C{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{for}\ p+q\geq r,\end{array}\right.$
(2.22)

where $0<{\varepsilon}\leq{\varepsilon}_{4}$, and $C$ is a positive constant
independent of ${\varepsilon}$.

###### Remark 2.1

It is trivial that Theorem 2.1 and Theorem 2.3 imply (1.2). On the other hand,
we have that

$p+q-1=\frac{r(r-1)}{r+1}\quad\Longleftrightarrow\quad
p+q=\frac{r^{2}+1}{r+1}$

and

$\frac{r+1}{2}<\frac{r^{2}+1}{r+1}<r.$

Moreover, we see that

$p+q-1\leq\frac{(p+q)(r-1)}{r+1}\quad\Longleftrightarrow\quad
p+q\leq\frac{r+1}{2}.$

Therefore Theorem 2.2 and Theorem 2.4 imply (1.3).

The proofs of four theorems above appear in the following sections. Form now
on, we shall compare our results with those of the general theory by Li, Yu
and Zhou [12, 13], in which the following problem of general form is
considered:

$\left\\{\begin{array}[]{ll}\displaystyle
u_{tt}-u_{xx}=F(u,Du,\partial_{x}Du)&\mbox{in}\quad{\bf R}\times(0,\infty),\\\
u(x,0)={\varepsilon}f(x),\ u_{t}(x,0)={\varepsilon}g(x),&x\in{\bf
R},\end{array}\right.$ (2.23)

where we denote $D:=(\partial_{t},\partial_{x})$ and $F\in C^{\infty}({\bf
R}^{5})$ satisfies

$F(\lambda)=O(|\lambda|^{1+\alpha})\quad\mbox{with $\alpha\in{\bf N}$ near
$\lambda=0$}.$

(2.23) requires $f,g\in C_{0}^{\infty}({\bf R})$. Then, the lifespan of the
classical solution of (2.23) defined by $\widetilde{T}({\varepsilon})$ has
estimates from below as

$\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-\alpha/2}&\mbox{in
general},\\\ c{\varepsilon}^{-\alpha(1+\alpha)/(2+\alpha)}&\mbox{if}\
\displaystyle\int_{{\bf R}}g(x)dx=0,\\\
c{\varepsilon}^{-\min\\{\beta_{0}/2,\alpha\\}}&\mbox{if
$\partial_{u}^{\beta}F(0)=0$ for
$1+\alpha\leq\forall\beta\leq\beta_{0}$}.\end{array}\right.$ (2.24)

This is the result of the general theory. If one applies it to our problem
(1.1) with

$F(u,Du,\partial_{x}Du)=u_{t}^{p}u^{q}+u^{r}\quad\mbox{with}\ p,q,r\in{\bf
N},$ (2.25)

one has the following estimates in each cases.

  * •

When $p+q<r$,

then, we have to set $\alpha=p+q-1$ and $\beta_{0}=r-1$ which yield that

$\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(p+q-1)/2}&\mbox{in
general},\\\ c{\varepsilon}^{-(p+q)(p+q-1)/(p+q+1)}&\mbox{if}\
\displaystyle\int_{{\bf R}}g(x)dx=0,\\\
c{\varepsilon}^{-\min\\{(r-1)/2,p+q-1\\}}&\begin{array}[]{l}\mbox{if
$\partial_{u}^{\beta}F(0)=0$}\\\ \mbox{for $p+q\leq\forall\beta\leq
r-1$}.\end{array}\end{array}\right.$

We note that the third case is available for (2.25). Therefore, for
$p+q\leq(r+1)/2$, we obtain that

$\widetilde{T}({\varepsilon})\geq c{\varepsilon}^{-(p+q-1)}$

whatever the value of $\displaystyle\int_{{\bf R}}g(x)dx$ is. On the other
hand, for $(r+1)/2<p+q$, i.e.

$\frac{r-1}{2}<p+q-1,$

we obtain

$\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(r-1)/2}&\mbox{if}\
\displaystyle\int_{{\bf R}}g(x)dx\neq 0,\\\
c{\varepsilon}^{-\min\\{(r-1)/2,(p+q)(p+q-1)/(p+q+1)\\}}&\mbox{if}\
\displaystyle\int_{{\bf R}}g(x)dx=0.\\\ \end{array}\right.$

  * •

When $p+q\geq r$,

then, similarly to the case above, we have to set $\alpha=r-1$, which yields
that

$\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(r-1)/2}&\mbox{in
general},\\\ c{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{if}\ \displaystyle\int_{{\bf
R}}g(x)dx=0,\\\ c{\varepsilon}^{-\min\\{\beta_{0}/2,(r-1)\\}}&\mbox{if
$\partial_{u}^{\beta}F(0)=0$ for
$r\leq\forall\beta\leq\beta_{0}$}.\end{array}\right.$

We note that the third case does not hold for (2.25) by
$\partial_{u}^{r}F(0)\neq 0$.

In conclusion, for the special nonlinear term in (2.25), the result of the
general theory is

$\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\displaystyle\frac{r+1}{2},\\\ c{\varepsilon}^{-(r-1)/2}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q\end{array}\right.\mbox{if}\ \int_{{\bf
R}}g(x)dx\not=0\\\ $

and

$\begin{array}[]{c}\widetilde{T}({\varepsilon})\geq\left\\{\begin{array}[]{ll}c{\varepsilon}^{-(p+q-1)}&\mbox{for}\
p+q\leq\displaystyle\frac{r+1}{2},\\\
c{\varepsilon}^{-\max\\{(r-1)/2,(p+q)(p+q-1)/(p+q+1)\\}}&\mbox{for}\
\displaystyle\frac{r+1}{2}\leq p+q\leq r,\\\
c{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{for}\ r\leq p+q\\\ \end{array}\right.\\\
\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx=0.\end{array}$

Therefore a part of our results,

$\begin{array}[]{l}T({\varepsilon})\sim C{\varepsilon}^{-(p+q)(r-1)/(r+1)}\\\
\quad\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx=0\ \mbox{and}\
\frac{r+1}{2}<p+q<r,\end{array}$ (2.26)

is better than the lower bound of $\widetilde{T}({\varepsilon})$ because of

$\frac{p+q-1}{p+q+1}<\frac{r-1}{r+1}.$

If one follows the proof in the following sections, one can find that it is
easy to see that our results on the lower bounds also hold for a special term
(2.25) by estimating the difference of nonlinear terms from above after
employing the mean value theorem. We note that we have infinitely many
examples of $(p,q,r)=(m,m,2m+1)$ as the inequality

$\frac{r+1}{2}=m+1<p+q=2m<r=2m+1$

holds for $m=2,3,4,\ldots$. This fact indicates that we still have a
possibility to improve the general theory in the sense that the optimal
results in (2.26) should be included at least.

##  3 Proof of Theorem 2.1

We basically employ the argument in Morisawa, Sasaki and Takamura [17] here.
According to Proposition 2.1, we shall construct a $C^{1}$ solution of (2.14).
Let $\\{(u_{j},w_{j})\\}_{j\in{\bf N}}$ be a sequence of $\\{C^{1}({\bf
R}\times[0,T])\\}^{2}$ defined by

$\left\\{\begin{array}[]{ll}u_{j+1}={\varepsilon}u^{0}+L(A|w_{j}|^{p}|u_{j}|^{q}+B|u_{j}|^{r}),&u_{1}={\varepsilon}u^{0},\\\
w_{j+1}={\varepsilon}u_{t}^{0}+L^{\prime}(A|w_{j}|^{p}|u_{j}|^{q}+B|u_{j}|^{r}),&w_{1}={\varepsilon}u_{t}^{0}.\end{array}\right.$
(3.1)

Then, in view of (2.9) and (2.12), $\left((u_{j})_{x},(w_{j})_{x}\right)$ has
to satisfy

$\left\\{\begin{array}[]{ll}(u_{j+1})_{x}&={\varepsilon}u^{0}_{x}+\overline{L^{\prime}}\left(A|w_{j}|^{p}|u_{j}|^{q}+B|u_{j}|^{r}\right),\\\
(u_{1})_{x}&={\varepsilon}u^{0}_{x},\\\
(w_{j+1})_{x}&={\varepsilon}u^{0}_{tx}+L^{\prime}\left(Ap|w_{j}|^{p-2}w_{j}(w_{j})_{x}|u_{j}|^{q}\right)\\\
&\quad+L^{\prime}\left(Aq|u_{j}|^{q-2}u_{j}(u_{j})_{x}|w_{j}|^{p}\right)\\\
&\quad+L^{\prime}\left(Br|u_{j}|^{r-2}u_{j}(u_{j})_{x}\right),\\\
(w_{1})_{x}&={\varepsilon}u^{0}_{tx},\end{array}\right.$ (3.2)

so that the function space in which $\\{(u_{j},w_{j})\\}$ converges is

$\begin{array}[]{ll}X:=&\\{(u,w)\in\\{C^{1}({\bf R}\times[0,T])\\}^{2}\ :\
\|(u,w)\|_{X}<\infty,\\\ &\quad\mbox{supp}\ (u,w)\subset\\{(x,t)\in{\bf
R}\times[0,T]\ :\ |x|\leq t+R\\}\\},\end{array}$

which is equipped with a norm

$\|(u,w)\|_{X}:=\|u\|_{1}+\|u_{x}\|_{1}+\|w\|_{2}+\|w_{x}\|_{2},$

where

$\begin{array}[]{l}\displaystyle\|u\|_{1}:=\sup_{(x,t)\in{\bf
R}\times[0,T]}|u(x,t)|,\\\ \displaystyle\|w\|_{2}:=\sup_{(x,t)\in{\bf
R}\times[0,T]}|(t-|x|+2R)w(x,t)|.\end{array}$

First we note that supp $(u_{j},w_{j})\subset\\{(x,t)\in{\bf R}\times[0,T]\ :\
|x|\leq t+R\\}$ implies supp $(u_{j+1},w_{j+1})\subset\\{(x,t)\in{\bf
R}\times[0,T]\ :\ |x|\leq t+R\\}$. It is easy to check this fact by assumption
on the initial data (2.1) and the definitions of
$L,\overline{L},L^{\prime},\overline{L^{\prime}}$ in the previous section.

The following lemma contains some useful a priori estimates.

###### Proposition 3.1

Let $(u,w)\in\\{C({\bf R}\times[0,T])\\}^{2}$ and supp
$(u,w)\subset\\{(x,t)\in{\bf R}\times[0,T]:|x|\leq t+R\\}$. Then there exists
a positive constant $C$ independent of $T$ and ${\varepsilon}$ such that

$\begin{array}[]{l}\|L(|w|^{p}|u|^{q})\|_{1}\leq
C\|w\|_{2}^{p}\|u\|_{1}^{q}(T+R),\ \|L(|u|^{r})\|_{1}\leq
C\|u\|_{1}^{r}(T+R)^{2},\\\ \|L^{\prime}(|w|^{p}|u|^{q})\|_{2}\leq
C\|w\|_{2}^{p}\|u\|_{1}^{q}(T+R),\ \|L^{\prime}(|u|^{r})\|_{2}\leq
C\|u\|_{1}^{r}(T+R)^{2},\\\ \|L^{\prime}(|w|^{p}|u|^{q})\|_{1}\leq
C\|w\|_{2}^{p}\|u\|_{1}^{q}(T+R),\ \|L^{\prime}(|u|^{r})\|_{1}\leq
C\|u\|_{1}^{r}(T+R)^{2}.\end{array}$ (3.3)

Proof. The proof of Proposition 3.1 is completely same as the one of
Proposition 3.1 in Morisawa, Sasaki and Takamura [17] because $\|u\|_{1}$ has
no weight, so that $\|w\|_{2}^{p}$ in [17] is simply replaced with
$\|w\|_{2}^{p}\|u\|_{1}^{q}$. $\Box$

Let us continue to prove Theorem 2.1. Set

$M:=\sum_{\alpha=0}^{2}\|f^{(\alpha)}\|_{L^{\infty}({\bf
R})}+\|g\|_{L^{1}({\bf R})}+\sum_{\beta=0}^{1}\|g^{(\beta)}\|_{L^{\infty}({\bf
R})}.$

The convergence of the sequence $\\{(u_{j},w_{j})\\}$.

First we note that $\|u_{1}\|_{1},\|w_{1}\|_{2}\leq M{\varepsilon}$ by (2.4)
and (2.7). Since (3.1) and (3.3) yield that

$\left\\{\begin{array}[]{ll}\|u_{j+1}\|_{1}&\leq
M{\varepsilon}+A\|L(|w_{j}|^{p}|u_{j}|^{q})\|_{1}+B\|L(|u_{j}|^{r}))\|_{1}\\\
&\leq
M{\varepsilon}+AC\|w_{j}\|_{2}^{p}\|u_{j}\|_{1}^{q}(T+R)+BC\|u_{j}\|^{r}_{1}(T+R)^{2},\\\
\|w_{j+1}\|_{2}&\leq
M{\varepsilon}+A\|L^{\prime}(|w_{j}|^{p}|u_{j}|^{q})\|_{2}+B\|L^{\prime}(|u_{j}|^{r})\|_{2}\\\
&\leq
M{\varepsilon}+AC\|w_{j}\|_{2}^{p}\|u_{j}\|_{1}^{q}(T+R)+BC\|u_{j}\|^{r}_{1}(T+R)^{2},\end{array}\right.$

the boundedness of $\\{(u_{j},w_{j})\\}$, i.e.

$\|u_{j}\|_{1},\|w_{j}\|_{2}\leq 3M{\varepsilon}\quad(j\in{\bf N}),$ (3.4)

follows from

$AC(3M{\varepsilon})^{p+q}(T+R),BC(3M{\varepsilon})^{r}(T+R)^{2}\leq
M{\varepsilon}.$ (3.5)

Assuming (3.5), one can estimate $(u_{j+1}-u_{j})$ and $(w_{j+1}-w_{j})$ as
follows. Making use of

$\begin{array}[]{l}|w_{j}|^{p}|u_{j}|^{q}-|w_{j-1}|^{p}|u_{j-1}|^{q}\\\
=(|w_{j}|^{p}-|w_{j-1}|^{p})|u_{j}|^{q}+|w_{j-1}|^{p}(|u_{j}|^{q}-|u_{j-1}|^{q})\end{array}$

and

$\left||u_{j}|^{r}-|u_{j-1}|^{r}\right|\leq
2^{r-1}r\left(|u_{j}|^{r-1}+|u_{j-1}|^{r-1}\right)|u_{j}-u_{j-1}|$

together with Proposition 3.1, we have that

$\begin{array}[]{l}\|u_{j+1}-u_{j}\|_{1}\\\
\leq\|L(A|w_{j}|^{p}|u_{j}|^{q}-A|w_{j-1}|^{p}|u_{j-1}|^{q}+B|u_{j}|^{r}-B|u_{j-1}|^{r})\|_{1}\\\
\leq
2^{p-1}pA\|L\left((|w_{j}|^{p-1}+|w_{j-1}|^{p-1})|w_{j}-w_{j-1}||u_{j}|^{q}\right)\|_{1}\\\
\quad+2^{q-1}qA\|L\left(|w_{j-1}|^{p}(|u_{j}|^{q-1}+|u_{j-1}|^{q-1})|u_{j}-u_{j-1}|\right)\|_{1}\\\
\quad+2^{r-1}rB\|L\left((|u_{j}|^{r-1}+|u_{j-1}|^{r-1})|u_{j}-u_{j-1}|\right)\|_{1}\\\
\leq
2^{p-1}pAC(T+R)(\|w_{j}\|_{2}^{p-1}+\|w_{j-1}\|_{2}^{p-1})\|u_{j}\|_{1}^{q}\|w_{j}-w_{j-1}\|_{2}\\\
\quad+2^{q-1}qAC(T+R)\|w_{j-1}\|_{2}^{p}(\|u_{j}\|_{1}^{q-1}+\|u_{j-1}\|_{1}^{q-1})\|u_{j}-u_{j-1}\|_{1}\\\
\quad+2^{r-1}rBC(T+R)^{2}(\|u_{j}\|_{1}^{r-1}+\|u_{j-1}\|_{1}^{r-1})\|u_{j}-u_{j-1}\|_{1}\\\
\leq 2^{p}pAC(3M{\varepsilon})^{p+q-1}(T+R)\|w_{j}-w_{j-1}\|_{2}\\\
\quad+2^{q}qAC(3M{\varepsilon})^{p+q-1}(T+R)\|u_{j}-u_{j-1}\|_{1}\\\
\quad+2^{r}rBC(3M{\varepsilon})^{r-1}(T+R)^{2}\|u_{j}-u_{j-1}\|_{1}\end{array}$

and similarly

$\begin{array}[]{ll}\|w_{j+1}-w_{j}\|_{2}&\leq
2^{p}pAC(3M{\varepsilon})^{p+q-1}(T+R)\|w_{j}-w_{j-1}\|_{2}\\\
&\quad+2^{q}qAC(3M{\varepsilon})^{p+q-1}(T+R)\|u_{j}-u_{j-1}\|_{1}\\\
&\quad+2^{r}rBC(3M{\varepsilon})^{r-1}(T+R)^{2}\|u_{j}-u_{j-1}\|_{1}.\end{array}$

Here we employ Hölder’s inequality to obtain

$\begin{array}[]{l}\|L(|w_{j}|^{p-1}|w_{j}-w_{j-1}|)|u_{j}|^{q}\|_{1}\\\
=\|L\left(\left||w_{j}|^{(p-1)/p}|w_{j}-w_{j-1}|^{1/p}\right|^{p}|u_{j}|^{q}\right)\|_{1}\\\
\leq
C(T+R)\||w_{j}|^{(p-1)/p}|w_{j}-w_{j-1}|^{1/p}\|_{2}^{p}\|u_{j}\|_{1}^{q}\\\
\leq
C(T+R)\|w_{j}\|_{2}^{p-1}\|w_{j}-w_{j-1}\|_{2}\|u_{j}\|_{1}^{q}\end{array}$

and so on. Therefore the convergence of $\\{u_{j}\\}$ follows from

$\begin{array}[]{l}\|u_{j+1}-u_{j}\|_{1}+\|w_{j+1}-w_{j}\|_{2}\\\
\displaystyle\leq\frac{1}{2}\left(\|u_{j}-u_{j-1}\|_{1}+\|w_{j}-w_{j-1}\|_{2}\right)\end{array}\quad(j\geq
2)$ (3.6)

provided (3.5) and

$2^{p+q}(p+q)AC(3M{\varepsilon})^{p+q-1}(T+R),2^{r}rBC(3M{\varepsilon})^{r-1}(T+R)^{2}\leq\frac{1}{8}$
(3.7)

are fulfilled.

The convergence of the sequence $\\{\left((u_{j})_{x},(w_{j})_{x}\right)\\}$.

First we note that $\|(u_{1})_{x}\|_{1},\|(w_{1})_{x}\|_{2}\leq
M{\varepsilon}$ by (2.10) and (2.13). Assume that (3.5) and (3.7) are
fulfilled. Since (3.2) and (3.3) yield that

$\begin{array}[]{ll}\|(u_{j+1})_{x}\|_{1}&\leq
M{\varepsilon}+A\|\overline{L^{\prime}}\left(|w_{j}|^{p}|u_{j}|^{q}|\right)\|_{1}+B\|\overline{L^{\prime}}\left(|u_{j}|^{r}|\right)\|_{1}\\\
&\leq
M{\varepsilon}+AC(T+R)\|w_{j}\|_{2}^{p}\|u_{j}\|_{1}^{q}+BC(T+R)^{2}\|u_{j}\|_{1}^{r}\\\
&\leq
M{\varepsilon}+AC(3M{\varepsilon})^{p+q}(T+R)+BC(3M{\varepsilon})^{r}(T+R)^{2}\end{array}$

because of a trivial property $|\overline{L^{\prime}}(v)|\leq L^{\prime}(|v|)$
and

$\begin{array}[]{ll}\|(w_{j+1})_{x}\|_{2}&\leq
M{\varepsilon}+pA\|L^{\prime}\left(|w_{j}|^{p-1}|(w_{j})_{x}||u_{j}|^{q}\right)\|_{2}\\\
&\quad+qA\|L^{\prime}\left(|w_{j}|^{p}|u_{j}|^{q-1}|(u_{j})_{x}|\right)\|_{2}\\\
&\quad+rB\|L^{\prime}\left(|u_{j}|^{r-1}|(u_{j})_{x}|\right)\|_{2}\\\ &\leq
M{\varepsilon}+pAC(T+R)\|w_{j}\|_{2}^{p-1}\|(w_{j})_{x}\|_{2}\|u_{j}\|_{1}^{q}\\\
&\quad+qAC(T+R)\|w_{j}\|_{2}^{p}\|u_{j}\|_{1}^{q-1}\|(u_{j})_{x}\|_{1}\\\
&\quad+rBC(T+R)^{2}\|u_{j}\|_{1}^{r-1}\|(u_{j})_{x}\|_{1}\\\ &\leq
M{\varepsilon}+pAC(3M{\varepsilon})^{p+q-1}(T+R)\|(w_{j})_{x}\|_{2}\\\
&\quad+qAC(3M{\varepsilon})^{p+q-1}(T+R)\|(u_{j})_{x}\|_{1}\\\
&\quad+rBC(3M{\varepsilon})^{r-1}(T+R)^{2}\|(u_{j})_{x}\|_{1}.\end{array}$

The boundedness of $\\{\left((u_{j})_{x},(w_{j})_{x}\right)\\}$, i.e.

$\|(u_{j})_{x}\|_{1},\|(w_{j})_{x}\|_{2}\leq 3M{\varepsilon}\quad(j\in{\bf
N}),$ (3.8)

follows from

$(p+q)AC(3M{\varepsilon})^{p+q}(T+R),rBC(3M{\varepsilon})^{r}(T+R)^{2}\leq
M{\varepsilon}.$ (3.9)

Assuming (3.9), one can estimate $\\{(u_{j+1})_{x}-(u_{j})_{x}\\}$ and
$\\{(w_{j+1})_{x}-(w_{j})_{x}\\}$ as follows. It is easy to see that

$\begin{array}[]{ll}\|(u_{j+1})_{x}-(u_{j})_{x}\|_{1}&\leq
A\|\overline{L^{\prime}}(|w_{j}|^{p}|u_{j}|^{q}-|w_{j-1}|^{p}|u_{j-1}|^{q})\|_{1}\\\
&\quad+B\|\overline{L^{\prime}}(|u_{j}|^{r}-|u_{j-1}|^{r})\|_{1},\end{array}$

which can be handled like $(w_{j+1}-w_{j})$ as before, so that we have that

$\begin{array}[]{ll}\|(u_{j+1})_{x}-(u_{j})_{x}\|_{1}&\leq
2^{p}pAC(3M{\varepsilon})^{p+q-1}(T+R)\|w_{j}-w_{j-1}\|_{2}\\\
&\quad+2^{q}qAC(3M{\varepsilon})^{p+q-1}(T+R)\|u_{j}-u_{j-1}\|_{1}\\\
&\quad+2^{r}rBC(3M{\varepsilon})^{r-1}(T+R)^{2}\|u_{j}-u_{j-1}\|_{1}\end{array}$

because of $|\overline{L^{\prime}}(v)|\leq L^{\prime}(|v|)$, which implies
that

$\|(u_{j+1})_{x}-(u_{j})_{x}\|_{1}=O\left(\frac{1}{2^{j}}\right)$ (3.10)

as $j\rightarrow\infty$ due to (3.6).

On the other hand, we have that

$\begin{array}[]{l}\|(w_{j+1})_{x}-(w_{j})_{x}\|_{2}\\\ \leq
pA\|L^{\prime}(|w_{j}|^{p-2}w_{j}(w_{j})_{x}|u_{j}|^{q}-|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_{x})|u_{j-1}|^{q}\|_{2}\\\
\quad+qA\|L^{\prime}(|w_{j}|^{p}|u_{j}|^{q-2}u_{j}(u_{j})_{x}-|w_{j-1}|^{p}|u_{j-1}|^{q-2}u_{j-1}(u_{j-1})_{x})\|_{2}\\\
\quad+rB\|L^{\prime}(|u_{j}|^{r-2}u_{j}(u_{j})_{x}-|u_{j-1}|^{r-2}u_{j-1}(u_{j-1})_{x})\|_{2}.\end{array}$

The first term on the right hand side of this inequality is divided into three
pieces according to

$\begin{array}[]{l}|w_{j}|^{p-2}w_{j}(w_{j})_{x}|u_{j}|^{q}-|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_{x}|u_{j-1}|^{q}\\\
=(|w_{j}|^{p-2}w_{j}-|w_{j-1}|^{p-2}w_{j-1})(w_{j})_{x}|u_{j}|^{q}\\\
\quad+|w_{j-1}|^{p-2}w_{j-1}((w_{j})_{x}-(w_{j-1})_{x})|u_{j}|^{q}\\\
\quad+|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_{x}(|u_{j}|^{q}-|u_{j-1}|^{q}).\end{array}$

Since one can employ the estimate

$\begin{array}[]{l}\left||w_{j}|^{p-2}w_{j}-|w_{j-1}|^{p-2}w_{j-1}\right|\\\
\leq\left\\{\begin{array}[]{ll}(p-1)2^{p-2}(|w_{j}|^{p-2}+|w_{j-1}|^{p-2})|w_{j}-w_{j-1}|&\mbox{for}\
p\geq 2,\\\ 2|w_{j}-w_{j-1}|^{p-1}&\mbox{for}\
1<p<2,\end{array}\right.\end{array}$

and the same one in which $w$ is replaced with $u$, we obtain that

$\begin{array}[]{l}\|(w_{j+1})_{x}-(w_{j})_{x}\|_{2}\\\ \leq
pAC(T+R)\|(w_{j})_{x}\|_{2}\|u_{j}\|_{1}^{q}\times\\\
\qquad\times\left\\{\begin{array}[]{ll}(p-1)2^{p-2}(\|w_{j}\|_{2}^{p-2}+\|w_{j-1}\|_{2}^{p-2})\|w_{j}-w_{j-1}\|_{2}&\mbox{for}\
p\geq 2,\\\ 2\|w_{j}-w_{j-1}\|_{2}^{p-1}&\mbox{for}\
1<p<2\end{array}\right.\\\
\quad+pAC(T+R)\|w_{j-1}\|_{2}^{p-1}\|(w_{j})_{x}-(w_{j-1})_{x}\|_{2}\|u_{j}\|_{1}^{q}\\\
\quad+2^{q}pqAC(T+R)\|w_{j-1}\|_{2}^{p-1}\|(w_{j-1})_{x}\|_{2}\times\\\
\qquad\times(\|u_{j}\|_{1}^{q-1}+\|u_{j-1}\|_{1}^{q-1})\|u_{j}-u_{j-1}\|_{1}\\\
\quad+qAC(T+R)\|(u_{j})_{x}\|_{1}\|w_{j}\|_{2}^{p}\times\\\
\qquad\times\left\\{\begin{array}[]{ll}(q-1)2^{q-2}(\|u_{j}\|_{1}^{q-2}+\|u_{j-1}\|_{1}^{q-2})\|u_{j}-u_{j-1}\|_{1}&\mbox{for}\
q\geq 2,\\\ 2\|u_{j}-u_{j-1}\|_{1}^{q-1}&\mbox{for}\
1<q<2\end{array}\right.\\\
\quad+qAC(T+R)\|u_{j-1}\|_{1}^{q-1}\|(u_{j})_{x}-(u_{j-1})_{x}\|_{1}\|w_{j}\|_{2}^{p}\\\
\quad+2^{p}pqAC(T+R)\|u_{j-1}\|_{1}^{q-1}\|(u_{j-1})_{x}\|_{1}\times\\\
\qquad\times(\|w_{j}\|_{2}^{p-1}+\|w_{j-1}\|_{2}^{p-1})\|w_{j}-w_{j-1}\|_{2}\\\
\quad+rBC(T+R)^{2}\|(u_{j})_{x}\|_{1}\times\\\
\qquad\times\left\\{\begin{array}[]{ll}(r-1)2^{r-2}(\|u_{j}\|_{1}^{r-2}+\|u_{j-1}\|_{1}^{r-2})\|u_{j}-u_{j-1}\|_{1}&\mbox{for}\
r\geq 2,\\\ 2\|u_{j}-u_{j-1}\|_{1}^{r-1}&\mbox{for}\
1<r<2\end{array}\right.\\\
\quad+rBC(T+R)^{2}\|u_{j-1}\|_{1}^{r-1}\|(u_{j})_{x}-(u_{j-1})_{x}\|_{1}.\end{array}$

Hence it follows from (3.6) and (3.10) that

$\begin{array}[]{ll}\|(w_{j+1})_{x}-(w_{j})_{x}\|_{2}&\leq
pAC(3M{\varepsilon})^{p+q-1}(T+R)\|(w_{j})_{x}-(w_{j-1})_{x}\|_{2}\\\
&\quad\displaystyle+O\left(\frac{1}{2^{j\min\\{p-1,q-1,r-1,1\\}}}\right)\end{array}$

as $j\rightarrow\infty$. Therefore we obtain the convergence of
$\\{\left((u_{j})_{x},(w_{j})_{x}\right)\\}$ provided

$pAC(3M{\varepsilon})^{p+q-1}(T+R)\leq\frac{1}{2}.$ (3.11)

Continuation of the proof.

The convergence of the sequence $\\{(u_{j},w_{j})\\}$ to $(u,w)$ in the closed
subspace of $X$ satisfying

$\|u\|_{1},\|(u_{x})\|_{1},\|w\|_{2},\|(w)_{x}\|_{2}\leq 3M{\varepsilon}$

is established by (3.5), (3.7), (3.9), and (3.11), which follow from

$C_{0}{\varepsilon}^{p+q-1}(T+R)\leq 1\quad\mbox{and}\quad
C_{0}{\varepsilon}^{r-1}(T+R)^{2}\leq 1,$

where

$\begin{array}[]{ll}C_{0}:=\max&\\{3^{p+q}ACM^{p+q-1},3^{r}BCM^{r-1},\\\
&\quad 2^{p+q+3}3^{p+q-1}(p+q)ACM^{p+q-1},2^{r+3}3^{r-1}rBCM^{r-1},\\\ &\quad
3^{p+q}(p+q)ACM^{p+q-1},3^{r}rBCM^{r-1},\\\ &\quad 2\cdot
3^{p+q-1}pACM^{p+q-1}\\}.\end{array}$

Therefore the statement of Theorem 2.1 is established with

$\left\\{\begin{array}[]{l}\displaystyle
c=\frac{1}{2}\min\\{C_{0}^{-1},C_{0}^{-1/2}\\},\\\
{\varepsilon}_{1}:=\min\\{(2C_{0}R)^{-1/(p+q-1)},(2^{2}C_{0}R^{2})^{-1/(r-1)}\\}\end{array}\right.$

because

$R\leq\frac{1}{2}\min\\{C_{0}^{-1}{\varepsilon}^{-(p+q-1)},C_{0}^{-1/2}{\varepsilon}^{-(r-1)/2}\\}$

holds for $0<{\varepsilon}\leq{\varepsilon}_{1}$. $\Box$

##  4 Proof of Theorem 2.2

In this section also, we basically employ the argument in Morisawa, Sasaki and
Takamura [17]. But the different estimates for the product term are required
here. First we note that the strong Huygens’ principle

$u^{0}(x,t)\equiv 0\quad\mbox{in}\ D$ (4.1)

holds in this case of (2.17), where

$D:=\\{(x,t)\in{\bf R}\times[0,\infty)\ :\ t-|x|\geq R\\}.$

This is almost trivial if one takes a look on the representation of $u^{0}$ in
(2.4) and the support condition on the data in (2.1). But one can see it also
by Proposition 2.2 in Kitamura, Morisawa and Takamura [9] for the details. So,
our unknown functions are $U:=u-{\varepsilon}u^{0}$ and
$W:=w-{\varepsilon}u_{t}^{0}$ in (2.14). Let $\\{(U_{j},W_{j})\\}_{j\in{\bf
N}}$ be a sequence of $\\{C^{1}({\bf R}\times[0,T])\\}^{2}$ defined by

$\left\\{\begin{array}[]{ll}U_{j+1}=L(A|W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}+B|U_{j}+{\varepsilon}u^{0}|^{r}),&U_{1}=0,\\\
W_{j+1}=L^{\prime}(A|W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}+B|U_{j}+{\varepsilon}u^{0}|^{r}),&W_{1}=0.\end{array}\right.$
(4.2)

Then, $\left\\{\left((U_{j})_{x},(W_{j})_{x}\right)\right\\}$ has to satisfy

$\left\\{\begin{array}[]{ll}(U_{j+1})_{x}&=\overline{L^{\prime}}\left(A|W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}+B|U_{j}+{\varepsilon}u^{0}|^{r}\right),\\\
(U_{1})_{x}&=0,\\\
(W_{j+1})_{x}&=L^{\prime}\left(Ap|W_{j}+{\varepsilon}u_{t}^{0}|^{p-2}(W_{j}+{\varepsilon}u_{t}^{0})((W_{j})_{x}+{\varepsilon}u_{tx}^{0})|U_{j}+{\varepsilon}u^{0}|^{q}\right)\\\
&\quad+L^{\prime}\left(Aq|W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q-2}(U_{j}+{\varepsilon}u^{0})((U_{j})_{x}+{\varepsilon}u_{x}^{0})\right)\\\
&\quad+L^{\prime}\left(Br|U_{j}+{\varepsilon}u^{0}|^{r-2}(U_{j}+{\varepsilon}u^{0})((U_{j})_{x}+{\varepsilon}u_{x}^{0})\right),\\\
(W_{1})_{x}&=0,\end{array}\right.$ (4.3)

so that the function space in which $\\{(U_{j},W_{j})\\}$ converges is

$\begin{array}[]{ll}Y:=&\\{(U,W)\in\\{C^{1}({\bf R}\times[0,T])\\}^{2}\ :\
\|(U,W)\|_{Y}<\infty,\\\ &\quad\mbox{supp}\ (U,W)\subset\\{|x|\leq
t+R\\}\\}\end{array}$

which is equipped with a norm

$\|(U,W)\|_{Y}:=\|U\|_{3}+\|U_{x}\|_{3}+\|W\|_{4}+\|W_{x}\|_{4},$

where

$\begin{array}[]{l}\displaystyle\|U\|_{3}:=\sup_{(x,t)\in{\bf
R}\times[0,T]}(t+|x|+R)^{-1}|U(x,t)|,\\\
\displaystyle\|W\|_{4}:=\sup_{(x,t)\in{\bf
R}\times[0,T]}\\{\chi_{D}(x,t)+(1-\chi_{D}(x,t))(t+|x|+R)^{-1}\\}|W(x,t)|,\end{array}$

and $\chi_{D}$ is a characteristic function of $D$. Similarly to the proof of
Theorem 2.1, we note that supp $(U_{j},W_{j})\in\\{(x,t)\in{\bf R}\times[0,T]\
:\ |x|\leq t+R\\}$ implies supp $(U_{j+1},W_{j+1})\in\\{(x,t)\in{\bf
R}\times[0,T]\ :\ |x|\leq t+R\\}$.

The following lemmas are a priori estimates in this case.

###### Proposition 4.1

Let $(U,W)\in\\{C({\bf R}\times[0,T])\\}^{2}$ with

$\mbox{\rm supp}\ (U,W)\subset\\{(x,t)\in{\bf R}\times[0,T]:|x|\leq t+R\\}$

and $U^{0}\in C({\bf R}\times[0,T])$ with

$\mbox{\em supp}\ U^{0}\subset\\{(x,t)\in{\bf
R}\times[0,T]:(t-|x|)_{+}\leq|x|\leq t+R\\}.$

Then there exists a positive constant $E$ independent of $T$ such that

$\left\\{\begin{array}[]{ll}\|L(|U^{0}|^{q-m}|W|^{m})\|_{3}&\leq
E\|U^{0}\|_{\infty}^{q-m}\|W\|_{4}^{m}(T+R)^{m},\\\
\|L(|U^{0}|^{p-m}|U|^{m})\|_{3}&\leq
E\|U^{0}\|_{\infty}^{p-m}\|U\|_{3}^{m}(T+R)^{m},\\\
\|L^{\prime}(|U^{0}|^{q-m}|W|^{m})\|_{4}&\leq
E\|U^{0}\|_{\infty}^{q-m}\|W\|_{4}^{m}(T+R)^{m},\\\
\|L^{\prime}(|U^{0}|^{p-m}|U|^{m})\|_{4}&\leq
E\|U^{0}\|_{\infty}^{p-m}\|U\|_{3}^{m}(T+R)^{m},\\\
\|L^{\prime}(|U^{0}|^{q-m}|W|^{m})\|_{3}&\leq
E\|U^{0}\|_{\infty}^{q-m}\|W\|_{4}^{m}(T+R)^{m},\\\
\|L^{\prime}(|U^{0}|^{p-m}|U|^{m})\|_{3}&\leq
E\|U^{0}\|_{\infty}^{p-m}\|U\|_{3}^{m}(T+R)^{m},\end{array}\right.$ (4.4)

where $p-m,q-m>0\ (m=0,1,2)$ and the norm $\|\cdot\|_{\infty}$ is defined by

$\|U^{0}\|_{\infty}:=\sup_{(x,t)\in{\bf R}\times[0,T]}|U^{0}(x,t)|.$

Proof. This lemma is exactly same as Proposition 5.1 in Morisawa, Sasaki and
Takamura [17]. $\Box$

We note that $U^{0}$ in the theorem above will be replaced with $u^{0}$ or
$u_{t}^{0}$ and their spatial derivatives later due to (4.1).

###### Proposition 4.2

Let $(U,W)\in\\{C({\bf R}\times[0,T])\\}^{2}$ with

$\mbox{\rm supp}\ (U,W)\subset\\{(x,t)\in{\bf R}\times[0,T]:|x|\leq t+R\\}.$

Then there exists a positive constant $C$ independent of $T$ such that

$\left\\{\begin{array}[]{ll}\|L(|W|^{p}|U|^{q})\|_{3}&\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}(T+R)^{p+q},\\\ \|L(|U|^{r})\|_{3}&\leq
C\|U\|_{3}^{r}(T+R)^{r+1},\\\ \|L^{\prime}(|W|^{p}|U|^{q})\|_{4}&\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}(T+R)^{p+q},\\\ \|L^{\prime}(|U|^{r})\|_{4}&\leq
C\|U\|_{3}^{r}(T+R)^{r+1},\\\ \|L^{\prime}(|W|^{p}|U|^{q})\|_{3}&\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}(T+R)^{p+q},\\\ \|L^{\prime}(|U|^{r})\|_{3}&\leq
C\|U\|_{3}^{r}(T+R)^{r+1}.\end{array}\right.$ (4.5)

The proof of Proposition 4.2 is established in the next section.

Let us proceed the proof of Theorem 2.2. Set

$\begin{array}[]{ll}N:=&\displaystyle
2^{p+q-1}\sum_{\gamma=0}^{1}AE(\|u_{t}^{0}\|_{\infty}^{p-\gamma}\|u_{tx}^{0}\|_{\infty}^{\gamma}\|u_{0}\|_{\infty}^{q}+\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-\gamma}\|u_{x}^{0}\|_{\infty}^{\gamma})\\\
&\displaystyle+\sum_{\gamma=0}^{1}2^{r-\gamma}BE\|u^{0}\|_{\infty}^{r-\gamma}\|u_{x}^{0}\|^{\gamma},\end{array}$

where $E$ is the one in (4.4). We note that

$\left\\{\begin{array}[]{lll}\|u^{0}\|_{\infty}&\leq\|f\|_{L^{\infty}(\mathbf{R})}+\|g\|_{L^{1}(\mathbf{R})}&<\infty,\\\
\|u_{t}^{0}\|_{\infty},\
\|u^{0}_{x}\|_{\infty}&\leq\|f^{\prime}\|_{L^{\infty}(\mathbf{R})}+\|g\|_{L^{\infty}(\mathbf{R})}&<\infty,\\\
\|u^{0}_{tx}\|_{\infty}&\leq\|f^{\prime\prime}\|_{L^{\infty}(\mathbf{R})}+\|g^{\prime}\|_{L^{\infty}(\mathbf{R})}&<\infty.\end{array}\right.$

Assume that

$0<{\varepsilon}\leq 1.$

The four quantities, ${\varepsilon}_{2i}\ (i=1,2,3,4)$, are defined in the
following;

$\begin{array}[]{ll}{\varepsilon}_{21}:=\min&\left[\left\\{2^{p+q}AC(5N)^{p+q}N^{-1}(2R)^{p+q}\right\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\right.\\\
&\quad\\{2^{p+q}AE\|u_{t}^{0}\|_{\infty}^{p}(5N)^{q}N^{-1}(2R)^{q}\\}^{-1/[(q-1)\min\\{p+q,r\\}+p]},\\\
&\quad\\{2^{p+q}AE\|u^{0}\|_{\infty}^{q}(5N)^{p}N^{-1}(2R)^{p}\\}^{-1/[(p-1)\min\\{p+q,r\\}+q]},\\\
&\quad\left.\left\\{2^{r}BC(5N)^{r}N^{-1}(2R)^{r+1}\right\\}^{-1/[\min\\{p+q,r\\}(r-1)]}\right]\end{array}$
(4.6)

and

${\varepsilon}_{22}:=\min\\{{\varepsilon}_{221},{\varepsilon}_{222},{\varepsilon}_{223}\\},$
(4.7)

where

$\begin{array}[]{l}{\varepsilon}_{221}:=\min\left[\left\\{2^{q+5}3^{p-1}pAC(5N)^{p+q-1}(2R)^{p+q}\right\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\right.\\\
\quad\\{2^{q+4}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q}N(2R)^{q+1}\\}^{-1/(p-1)},\\\
\quad\\{2^{q+4}3^{p-1}pAE\|u^{0}\|_{\infty}^{q}(5N)^{p-1}(2R)^{p}\\}^{-1/[\min\\{p+q,r\\}(p-1)]},\\\
\quad\left.\\{2^{q+4}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}(2R)\\}^{-1/(p+q-1)}\right]\end{array}$

and

$\begin{array}[]{l}{\varepsilon}_{222}:=\min\left[\\{2^{p+4}3^{q}qAC(5N)^{p+q-1}(2R)^{p+q}\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\right.\\\
\quad\\{2^{p+3}3^{q}qAE\|u^{0}\|_{\infty}^{q-1}(5N)^{p}(2R)^{p+1}\\}^{-1/[p\min\\{p+q,r\\}+q-1]},\\\
\quad\\{2^{p+4}3^{q}qAE\|u_{t}\|_{\infty}^{p}(5N)^{q-1}(2R)^{q}\\}^{-1/[(q-1)\min\\{p+q,r\\}+p]},\\\
\quad\left.\\{2^{p+3}3^{q}qAE\|u_{t}^{0}\|^{p}\|u^{0}\|_{\infty}^{q-1}(2R)\\}^{-1(p+q-1)}\right],\\\
{\varepsilon}_{223}:=\min\left[\\{2^{3}3^{r}rBC(5N)^{r-1}(2R)^{r+1}\\}^{-1/(r-1)},\right.\\\
\quad\left.\quad\\{2^{3}3^{r}BE\|u^{0}\|_{\infty}^{r-1}(2R)\\}^{-1/(r-1)}\right].\end{array}$

Moreover, we set

${\varepsilon}_{23}:=\min\\{{\varepsilon}_{21},{\varepsilon}_{231},{\varepsilon}_{232},{\varepsilon}_{233},{\varepsilon}_{234}\\},$
(4.8)

where ${\varepsilon}_{21}$ is the one in (4.6) and ${\varepsilon}_{23i}\
(i=1,2,3,4)$ are defined by

$\begin{array}[]{l}{\varepsilon}_{231}:=\left[\\{2^{p+q-1}pAC(5N)^{p+q}N^{-1}(2R)^{p+q}\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\right.\\\
\quad\\{2^{p+q-1}qAC(5N)^{p+q}N^{-1}(2R)^{p+q}\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\\\
\quad\left.\\{2^{r-1}qAC(5N)^{r}N^{-1}(2R)^{r+1}\\}^{-1/[\min\\{p+q,r\\}(r-1)]}\right],\end{array}$
$\begin{array}[]{l}{\varepsilon}_{232}:=\min\big{[}\\\
\quad\\{2^{p+q-1}3^{2}pAE\|u_{tx}^{0}\|_{\infty}(5N)^{p+q-1}N^{-1}(2R)^{p+q-1}\\}^{-1/[(p+q-1)\min\\{p+q,r\\}+1]},\\\
\quad\\{2^{p+q-1}3^{2}pAE\|u^{0}\|_{\infty}^{q}(5N)^{p}N^{-1}(2R)^{p}\\}^{-1/(p\min\\{p+q,r\\}+q)},\\\
\quad\\{2^{p+q-1}3^{2}pAE\|u_{tx}^{0}\|_{\infty}\|u^{0}\|_{\infty}^{q}(5N)^{p-1}N^{-1}(2R)^{p-1}\\}^{-1/[(p-1)\min\\{p+q,r\\}+q+1]},\\\
\quad\\{2^{p+q-1}3^{2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q+1}N^{-1}(2R)^{q+1}\\}^{-1/[(q+1)\min\\{p+q,r\\}+p-1]},\\\
\quad\\{2^{p+q-1}3^{2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u_{tx}^{0}\|_{\infty}(5N)^{q}N^{-1}(2R)^{q}\\}^{-1/(q\min\\{p+q,r\\}+p)},\\\
\quad\left.\\{2^{p+q-1}3^{2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}5(2R)\\}^{-1/(\min\\{p+q,r\\}+p+q-1)}\right],\end{array}$
$\begin{array}[]{l}{\varepsilon}_{233}:=\min\big{[}\\\
\quad\\{2^{p+q-1}3^{2}qAE\|u_{tx}^{0}\|_{\infty}(5N)^{p+q-1}N^{-1}(2R)^{p+q-1}\\}^{-1/[(p+q-1)\min\\{p+q,r\\}+1]},\\\
\quad\\{2^{p+q-1}3^{2}qAE\|u^{0}\|_{\infty}^{q-1}(5N)^{p+1}N^{-1}(2R)^{p+1}\\}^{-1/[(p+1)\min\\{p+q,r\\}+q-1]},\\\
\quad\\{2^{p+q-1}3^{2}qAE\|u^{0}\|_{\infty}^{q-1}\|u_{x}^{0}\|_{\infty}(5N)^{p}N^{-1}(2R)^{p}\\}^{-1/(p\min\\{p+q,r\\}+q)},\\\
\quad\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}(5N)^{q}N^{-1}(2R)^{q}\\}^{-1/(q\min\\{p+q,r\\}+p)},\\\
\quad\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u_{x}^{0}\|_{\infty}(5N)^{q-1}N^{-1}(2R)^{q-1}\\}^{-1/[(q-1)\min\\{p+q,r\\}+p+1]},\\\
\quad\left.\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}5(2R)\\}^{-1/(\min\\{p+q,r\\}+p+q-1)}\right]\end{array}$

and

$\begin{array}[]{l}{\varepsilon}_{234}:=\min\big{[}\\\
\quad\\{2^{r-1}3rBE\|u_{x}^{0}\|_{\infty}(5N)^{r-1}N^{-1}(2R)^{r-1}\\}^{-1/[(r-1)\min\\{p+q,r\\}+1]},\\\
\quad\left.\\{2^{r-1}3rBE\|u^{0}\|_{\infty}^{r-1}5(2R)\\}^{-1/(\min\\{p+q,r\\}+r-1)}\right].\end{array}$

Finally, we also set

$\begin{array}[]{ll}{\varepsilon}_{24}:=\min&\left[\\{2^{p+q+2}pAC(5N)^{p+q-1}(2R)^{p+q}\\}^{-1/[\min\\{p+q,r\\}(p+q-1)]},\right.\\\
&\quad\\{2^{p+q+2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q}(2R)^{q+1}\\}^{-1/(q\min\\{p+q,r\\})},\\\
&\quad\\{2^{p+q+2}pAE\|u^{0}\|_{\infty}^{q}(5N)^{q}(2R)^{p-1}\\}^{-1/[(p-1)\min\\{p+q,r\\}]},\\\
&\quad\left.\\{2^{p+q+2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}(2R)\\}^{-1}\right].\end{array}$
(4.9)

The convergence of the sequence $\\{(U_{j},W_{j})\\}$.

It follows from (4.2), Proposition 4.1 and 4.2 that

$\begin{array}[]{ll}\|U_{j+1}\|_{3}&\leq
A\|L(|W_{j}+{\varepsilon}u^{0}_{t}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q})\|_{3}+B\|L(|U_{j}+{\varepsilon}u^{0}|^{r})\|_{3}\\\
&\leq
2^{p+q}A\|L\\{(|W_{j}|^{p}+|{\varepsilon}u^{0}_{t}|^{p})(|U_{j}|^{q}+|{\varepsilon}u^{0}|^{q})\\}\|_{3}\\\
&\quad+2^{r}B\|L(|U_{j}|^{r}+|{\varepsilon}u^{0}|^{r})\|_{3}\\\ &\leq
2^{p+q}A\left\\{\|L(|W_{j}|^{p}|U_{j}|^{q})\|_{3}+{\varepsilon}^{p}\|L(|u_{t}^{0}|^{p}|U_{j}|^{q})\|_{3}\right.\\\
&\qquad\left.+{\varepsilon}^{q}\|L(|u^{0}|^{q}|W_{j}|^{p})\|_{3}+{\varepsilon}^{p+q}\|L(|u^{0}_{t}|^{p}|u^{0}|^{q})\|_{3}\right\\}\\\
&\quad+2^{r}B\left\\{\|L(|U_{j}|^{r})\|_{3}+{\varepsilon}^{q}\|L(|u^{0}|^{r})\|_{3}\right\\}\\\
&\leq
2^{p+q}A\left\\{C\|W_{j}\|_{4}^{p}\|U_{j}\|^{q}(T+R)^{p+q}+E{\varepsilon}^{p}\|u_{t}^{0}\|_{\infty}^{p}\|U_{j}\|_{3}^{q}(T+R)^{q}\right.\\\
&\quad+\left.E{\varepsilon}^{q}\|u^{0}\|_{\infty}^{q}\|W_{j}\|_{4}^{p}(T+R)^{p}+E{\varepsilon}^{p+q}\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q}\right\\}\\\
&\quad+2^{r}B\left\\{C\|U_{j}\|_{3}^{r}(T+R)^{r+1}+E{\varepsilon}^{r}\|u^{0}\|_{\infty}^{r}\right\\}.\\\
&\leq 2^{p+q}AC\|W_{j}\|_{4}^{p}\|U_{j}\|_{3}^{q}(T+R)^{p+q}\\\
&\quad+2^{p+q}AE\|u_{t}^{0}\|_{\infty}^{p}{\varepsilon}^{p}\|U_{j}\|_{3}^{q}(T+R)^{q}\\\
&\quad+2^{p+q}AE\|u^{0}\|_{\infty}^{q}{\varepsilon}^{q}\|W_{j}\|_{4}^{p}(T+R)^{p}\\\
&\quad+2^{r}BC\|U_{j}\|_{3}^{r}(T+R)^{r+1}+N{\varepsilon}^{\min\\{p+q,r\\}}\end{array}$

and similarly

$\begin{array}[]{ll}\|W_{j+1}\|_{4}&\leq
A\|L^{\prime}(|W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q})\|_{4}+B\|L^{\prime}(|U_{j}+{\varepsilon}u_{0}|^{r})\|_{4}\\\
&\leq 2^{p+q}AC\|W_{j}\|_{4}^{p}\|U_{j}\|_{3}^{q}(T+R)^{p+q}\\\
&\quad+2^{p+q}AE\|u_{t}^{0}\|_{\infty}^{p}{\varepsilon}^{p}\|U_{j}\|_{3}^{q}(T+R)^{q}\\\
&\quad+2^{p+q}AE\|u^{0}\|_{\infty}^{q}{\varepsilon}^{q}\|W_{j}\|_{4}^{p}(T+R)^{p}\\\
&\quad+2^{r}BC\|U_{j}\|_{3}^{r}(T+R)^{r+1}+N{\varepsilon}^{\min\\{p+q,r\\}}\end{array}$

Hence the boundedness of $\\{(U_{j},W_{j})\\}$, i.e.

$\|U_{j}\|_{3},\|W_{j}\|_{4}\leq
5N{\varepsilon}^{\min\\{p+q,r\\}}\quad(j\in{\bf N}),$ (4.10)

follows from

$\left\\{\begin{array}[]{ll}2^{p+q}AC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q}(T+R)^{p+q}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}},\\\
2^{p+q}AE\|u_{t}^{0}\|_{\infty}^{p}{\varepsilon}^{p}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q}(T+R)^{q}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}},\\\
2^{p+q}AE\|u^{0}\|_{\infty}^{q}{\varepsilon}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p}(T+R)^{p}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}},\\\
2^{r}BC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{r}(T+R)^{r+1}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}}.\end{array}\right.$ (4.11)

Since (4.6) yields that

$\begin{array}[]{ll}R\leq
C_{1}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},\right.\\\
&\left.\quad{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)}\right\\}\end{array}$

for $0<{\varepsilon}\leq{\varepsilon}_{21}$, where

$\begin{array}[]{ll}\displaystyle
C_{1}:=\frac{1}{2}\min&\left[\\{2^{p+q}AC(5N)^{p+q}N^{-1}\\}^{-1/(p+q)},\right.\\\
&\quad\\{2^{p+q}AE\|u_{t}^{0}\|_{\infty}^{p}(5N)^{q}N^{-1}\\}^{-1/q},\\\
&\quad\\{2^{p+q}AE\|u^{0}\|_{\infty}^{q}(5N)^{p}N^{-1}\\}^{-1/p},\\\
&\left.\quad\\{2^{r}BC(5N)^{r}N^{-1}\\}^{-1/(r+1)}\right],\end{array}$ (4.12)

we find that (4.11) as well as (4.10) follows from

$\begin{array}[]{ll}T\leq
C_{1}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},\right.\\\
&\quad\left.{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)}\right\\}\end{array}$
(4.13)

for $0<{\varepsilon}\leq{\varepsilon}_{21}$.

Let us write down (4.13) in each cases.

  * •

When $p+q\leq(r+1)/2(<r)$,
$\min\\{p+q,r\\}=p+q$ and

$\frac{p+q-1}{p+q}-\frac{r-1}{r+1}=\frac{2(p+q)-(r+1)}{(p+q)(r+1)}\leq 0$

imply that (4.13) is equivalent to

$T\leq C_{1}{\varepsilon}^{-(p+q-1)}$

because of

$\frac{(q-1)(p+q)+p}{q}=\frac{(p-1)(p+q)-q}{p}=p+q-1.$

  * •

When $(r+1)/2\leq p+q\leq r$,
$\min\\{p+q,r\\}=p+q$ and

$\frac{p+q-1}{p+q}-\frac{r-1}{r+1}=\frac{2(p+q)-(r+1)}{(p+q)(r+1)}\geq 0$

imply that (4.13) is equivalent to

$T\leq C_{1}{\varepsilon}^{-(p+q)(r-1)/(r+1)}.$

  * •

When $r\leq p+q$,
$\min\\{p+q,r\\}=r$ implies that (4.13) is equivalent to

$T\leq C_{1}{\varepsilon}^{-r(r-1)/(r+1)}$

because of

$\frac{r(q-1)+p}{q}-\frac{r(r-1)}{r+1}=\frac{r(p+q-r)+qr+p-r}{q(r+1)}>0$

and similarly

$\frac{r(p-1)+q}{p}-\frac{r(r-1)}{r+1}>0.$

Next, assuming (4.13), one can estimate $(U_{j+1}-U_{j})$ and
$(W_{j+1}-W_{j})$ as follows. It follows from the inequality

$\begin{array}[]{l}\left||W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}-|W_{j-1}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}\right|\\\
\leq\left||W_{j}+{\varepsilon}u_{t}^{0}|^{p}-|W_{j-1}+{\varepsilon}u_{t}^{0}|^{p}\right||U_{j}+{\varepsilon}u^{0}|^{q}\\\
\quad+|W_{j-1}+{\varepsilon}u_{t}^{0}|^{p}\left||U_{j}+{\varepsilon}u^{0}|^{q}-|U_{j-1}+{\varepsilon}u^{0}|^{q}\right|\\\
\leq
2^{q}3^{p}p(|W_{j}|^{p-1}+|W_{j-1}|^{p-1}+{\varepsilon}^{p-1}|u_{t}^{0}|^{p-1})|W_{j}-W_{j-1}|(|U_{j}|^{q}+{\varepsilon}^{q}|u^{0}|^{q})\\\
\quad+2^{p}3^{q}q(|W_{j-1}|^{p}+{\varepsilon}^{p}|u_{t}^{0}|^{p})(|U_{j}|^{q-1}+|U_{j-1}|^{q-1}+{\varepsilon}^{q-1}|u|^{q-1})|U_{j}-U_{j-1}|,\end{array}$

that

$\begin{array}[]{l}\|U_{j+1}-U_{j}\|_{3}\\\ \leq
2^{q}3^{p-1}pA\|L\\{(|W_{j-1}|^{p-1}+|W_{j}|^{p-1}+{\varepsilon}^{p-1}|u^{0}_{t}|^{p-1})\times\\\
\qquad\times|W_{j}-W_{j-1}|(|U_{j}|^{q}+{\varepsilon}^{q}|u^{0}|^{q})\\}\|_{3}\\\
\quad+2^{p}3^{q-1}qA\|L\\{(|W_{j-1}|^{p}+{\varepsilon}^{p}|u_{t}^{0}|^{p})\times\\\
\qquad\times(|U_{j-1}|^{q-1}+|U_{j}|^{q-1}+{\varepsilon}^{q-1}|u^{0}|^{q-1})|U_{j}-U_{j-1}|\\}\|_{3}\\\
\quad+3^{r-1}rB\|L\\{(|U_{j-1}|^{r-1}+|U_{j}|^{r-1}+{\varepsilon}^{r-1}|u^{0}|^{r-1})|U_{j}-U_{j-1}|\\}\|_{3}.\end{array}$

Hence Propositions 4.1 and 4.2 yield that

$\begin{array}[]{l}\|U_{j+1}-U_{j}\|_{3}\\\ \leq
2^{q}3^{p-1}pA\|U_{j}\|_{3}^{q}\\{C\|W_{j-1}\|_{4}^{p-1}(T+R)^{p+q}+C\|W_{j}\|_{4}^{p-1}(T+R)^{p+q}\\\
\qquad+E{\varepsilon}^{p-1}\|u^{0}_{t}\|_{\infty}^{p-1}(T+R)^{q+1}\\}\|W_{j}-W_{j-1}\|_{3}\\\
\quad+2^{q}3^{p-1}pAE{\varepsilon}^{q}\|u^{0}\|_{\infty}^{q}\\{\|W_{j-1}\|_{4}^{p-1}(T+R)^{p}+\|W_{j}\|_{4}^{p-1}(T+R)^{p}\\\
\qquad+E{\varepsilon}^{p-1}\|u^{0}_{t}\|_{\infty}^{p-1}(T+R)\\}\|W_{j}-W_{j-1}\|_{3}\\\
\quad+2^{p}3^{q-1}qA\|W_{j-1}\|_{4}^{p}\\{C\|U_{j-1}\|_{3}^{q-1}(T+R)^{p+q}+C\|U_{j}\|_{3}^{q-1}(T+R)^{p+q}\\\
\qquad+E{\varepsilon}^{q-1}\|u^{0}\|_{\infty}^{q-1}(T+R)^{p+1}\\}\|U_{j}-U_{j-1}\|_{3}\\\
\quad+2^{p}3^{q-1}qAE{\varepsilon}^{p}\|u_{t}^{0}\|_{\infty}^{p}\\{\|U_{j-1}\|_{3}^{q-1}(T+R)^{q}+\|U_{j}\|_{3}^{q-1}(T+R)^{q}\\\
\qquad+E{\varepsilon}^{q-1}\|u^{0}\|_{\infty}^{q-1}(T+R)\\}\|U_{j}-U_{j-1}\|_{3}\\\
\quad+3^{r-1}rB\\{C\|U_{j-1}\|_{3}^{r-1}(T+R)^{r+1}+C\|U_{j}\|_{3}^{r-1}(T+R)^{r+1}\\\
\qquad+E{\varepsilon}^{r-1}\|u^{0}\|_{\infty}^{r-1}(T+R)\\}\|U_{j}-U_{j-1}\|_{3}.\end{array}$

Therefore (4.11) implies that

$\begin{array}[]{l}\|U_{j+1}-U_{j}\|_{3}\\\
\leq\\{2^{q+1}3^{p-1}pAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q}\\\
\qquad+2^{q}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N{\varepsilon}^{\min\\{p+q.r\\}})^{q}{\varepsilon}^{p-1}(T+R)^{q+1}\\}\|W_{j}-W_{j-1}\|_{3}\\\
\quad+\\{2^{q+1}3^{p-1}pAE\|u^{0}\|_{\infty}^{q}{\varepsilon}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p-1}(T+R)^{p}\\\
\qquad+2^{q}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}{\varepsilon}^{p-1}(T+R)\\}\|W_{j}-W_{j-1}\|_{3}\\\
\quad+\\{2^{p+1}3^{q-1}qAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q}\\\
\qquad+2^{p}3^{q-1}qAE\|u^{0}\|_{\infty}^{q-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p}{\varepsilon}^{q-1}(T+R)^{p+1}\\}\|U_{j}-U_{j-1}\|_{3}\\\
\quad+\\{2^{p+1}3^{q-1}qAE\|u_{t}^{0}\|_{\infty}^{p}{\varepsilon}^{p}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q-1}(T+R)^{q}\\\
\qquad+2^{p}3^{q-1}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}{\varepsilon}^{p+q-1}(T+R)\\}\|U_{j}-U_{j-1}\|_{3}\\\
\quad+\\{2\cdot
3^{r-1}rBC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{r-1}(T+R)^{r+1}\\\
\qquad+3^{r-1}rBE\|u^{0}\|_{\infty}^{r-1}{\varepsilon}^{r-1}(T+R)\\}\|U_{j}-U_{j-1}\|_{3}.\end{array}$

We note that $\|W_{j+1}-W_{j}\|_{4}$ has the same upper bound as
$\|U_{j+1}-U_{j}\|_{3}$ in view of Propositions 4.1 and 4.2. Here we employ
Hölder’s inequality like the one in the proof of Theorem 2.1.

Therefore the convergence of $\\{(U_{j},W_{j})\\}$ follows from

$\begin{array}[]{l}\|U_{j+1}-U_{j}\|_{3}+\|W_{j+1}-W_{j}\|_{4}\\\
\displaystyle\leq\frac{1}{2}\left(\|U_{j}-U_{j-1}\|_{3}+\|W_{j}-W_{j-1}\|_{4}\right)\end{array}\quad(j\geq
2)$ (4.14)

provided (4.11) and

$\begin{array}[]{l}\left.\begin{array}[]{l}2^{q+1}3^{p-1}pAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q},\\\
2^{q}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q}{\varepsilon}^{p-1}(T+R)^{q+1},\\\
2^{q+1}3^{p-1}pAE\|u^{0}\|_{\infty}^{q}{\varepsilon}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p-1}(T+R)^{p},\\\
2^{q}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}{\varepsilon}^{p+q-1}(T+R)\end{array}\right\\}\leq\displaystyle\frac{1}{16},\\\
\left.\begin{array}[]{l}2^{p+1}3^{q-1}qAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q},\\\
2^{p}3^{q-1}qAE\|u^{0}\|_{\infty}^{q-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p}{\varepsilon}^{q-1}(T+R)^{p+1},\\\
2^{p+1}3^{q-1}qAE\|u_{t}^{0}\|_{\infty}^{p}{\varepsilon}^{p}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q-1}(T+R)^{q},\\\
2^{p}3^{q-1}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}{\varepsilon}^{p+q-1}(T+R),\\\
2\cdot 3^{r-1}rBC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{r-1}(T+R)^{r+1},\\\
3^{r-1}rBE\|u^{0}\|_{\infty}^{r-1}{\varepsilon}^{r-1}(T+R)\end{array}\right\\}\leq\displaystyle\frac{1}{24}.\end{array}$
(4.15)

are fulfilled. Since (4.7) yields that

$\begin{array}[]{ll}R\leq
C_{2}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-(q\min\\{p+q,r\\}+p-1)/(q+1)},\right.\\\
&\quad{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-(p+q-1)},\\\
&\quad{\varepsilon}^{-(p\min\\{p+q,r\\}+q-1)/(p+1)},{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},\\\
&\quad\left.{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)},{\varepsilon}^{-(r-1)}\right\\},\end{array}$

for $0<{\varepsilon}\leq{\varepsilon}_{22}$, where

$\begin{array}[]{ll}C_{2}:=\displaystyle\frac{1}{2}\min&\left[\\{2^{q+5}3^{p-1}pAC(5N)^{p+q-1}\\}^{-1/(p+q)},\right.\\\
&\quad\\{2^{q+4}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q}\\}^{-1/(q+1)},\\\
&\quad\\{2^{q+5}3^{p-1}pAE\|u^{0}\|_{\infty}^{q}(5N)^{p-1}\\}^{-1/p},\\\
&\quad\\{2^{q+4}3^{p-1}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}\\}^{-1},\\\
&\quad\\{2^{p+4}3^{q}qAC(5N)^{p+q-1}\\}^{-1/(p+q)},\\\
&\quad\\{2^{p+3}3^{q}qAE\|u^{0}\|_{\infty}^{q-1}(5N)^{p}\\}^{-1/(p+1)},\\\
&\quad\\{2^{p+4}3^{q}qAE\|u_{t}^{0}\|_{\infty}^{p}(5N)^{q-1}\\}^{-1/q},\\\
&\quad\\{2^{p+3}3^{q}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}\\}^{-1},\\\
&\quad\\{2^{3}3^{r}rBC(5N)^{r-1}\\}^{-1/{r+1}},\\\
&\quad\left.\\{2^{3}3^{r}rBE\|u^{0}\|_{\infty}^{r-1}\\}^{-1}\right],\end{array}$

we find that (4.15) as well as (4.14) follows from

$\begin{array}[]{ll}T\leq
C_{2}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-(q\min\\{p+q,r\\}+p-1)/(q+1)},\right.\\\
&\quad{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-(p+q-1)},\\\
&\quad{\varepsilon}^{-(p\min\\{p+q,r\\}+q-1)/(p+1)},{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},\\\
&\quad\left.{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)},{\varepsilon}^{-(r-1)}\right\\}\end{array}$
(4.16)

for $0<{\varepsilon}\leq{\varepsilon}_{22}$.

Let us write down (4.16) in each cases.

  * •

When $p+q\leq(r+1)/2(<r)$,
$\min\\{p+q,r\\}=p+q$ and the same reason as the argument after (4.13) imply
that (4.16) is equivalent to

$T\leq C_{2}{\varepsilon}^{-(p+q-1)}$

because of

$\frac{q(p+q)+p-1}{q+1}=\frac{p(p+q)+q-1}{p+1}=p+q-1.$

  * •

When $(r+1)/2\leq p+q\leq r$,
similarly to the argument above, we find that (4.16) is equivalent to

$T\leq C_{2}{\varepsilon}^{-(p+q)(r-1)/(r+1)}.$

  * •

When $r\leq p+q$,
$\min\\{p+q,r\\}=r$ and the same reason as the argument after (4.13) imply
that (4.16) is equivalent to

$T\leq C_{2}{\varepsilon}^{-r(r-1)/(r+1)}$

because of

$\frac{qr+p-1}{q+1}-\frac{r(r-1)}{r+1}=\frac{r(p+q-r)+qr+p-1}{(q+1)(r+1)}>0$

and similarly

$\frac{pr+q-1}{p+1}-\frac{r(r-1)}{r+1}>0.$

The convergence of the sequence $\\{((U_{j})_{x},(W_{j})_{x})\\}$.

Assume (4.13) and (4.16). Then we have (4.10) and (4.14). Since it follows
from (4.3) that

$\begin{array}[]{ll}|(U_{j+1})_{x}|&\leq
2^{p+q}A\\{L^{\prime}(|W_{j}|^{p}|U_{j}|^{q})+{\varepsilon}^{p}L^{\prime}(|u_{t}|^{0}|^{p}|U_{j}|^{q})\\\
&\qquad+{\varepsilon}^{q}L^{\prime}(|W_{j}|^{p}|u^{0}|^{q})+{\varepsilon}^{p+q}L^{\prime}(|u^{0}_{t}|^{p}|u^{0}|^{q})\\}\\\
&\quad+2^{r}B\\{L^{\prime}(|U_{j}|^{r}+{\varepsilon}^{r}L^{\prime}(|u^{0}|^{r})\\}\end{array}$

and

$\begin{array}[]{ll}|(W_{j+1})_{x}|&\leq
2^{p+q-1}pA\\{L^{\prime}(|W_{j}|^{p-1}|(W_{j})_{x}||U_{j}|^{q})+{\varepsilon}L^{\prime}(|W_{j}|^{p-1}|u_{tx}^{0}||U_{j}|^{q})\\\
&\qquad+{\varepsilon}^{q}L^{\prime}(|W_{j}|^{p-1}|(W_{j})_{x}||u^{0}|^{q})+{\varepsilon}^{q+1}L^{\prime}(|W_{j}|^{p-1}|u_{tx}^{0}||u^{0}|^{q})\\\
&\qquad+{\varepsilon}^{p-1}L^{\prime}(|u_{t}^{0}|^{p-1}|(W_{j})_{x}||U_{j}|^{q})+{\varepsilon}^{p}L^{\prime}(|u_{t}^{0}|^{p-1}|u_{tx}^{0}||U_{j}|^{q})\\\
&\qquad+{\varepsilon}^{p+q-1}L^{\prime}(|u_{t}^{0}|^{p-1}|(W_{j})_{x}||u^{0}|^{q})+{\varepsilon}^{p+q}L^{\prime}(|u_{t}^{0}|^{p-1}|u_{tx}^{0}||u^{0}|^{q})\\}\\\
&\quad+2^{p+q-1}qA\\{L^{\prime}(|W_{j}|^{p}|U_{j}|^{q-1}|(U_{j})_{x}|)+{\varepsilon}L^{\prime}(|W_{j}|^{p}|U_{j}|^{q-1}|u_{x}^{0}|)\\\
&\qquad+{\varepsilon}^{q-1}L^{\prime}(|W_{j}|^{p}|u^{0}|^{q-1}|(U_{j})_{x}|)+{\varepsilon}^{q}L^{\prime}(|W_{j}|^{p}|u^{0}|^{q-1}|u_{x}^{0}|)\\\
&\qquad+{\varepsilon}^{p}L^{\prime}(|u_{t}^{0}|^{p}|U_{j}|^{q-1}|(U_{j})_{x}|)+{\varepsilon}^{p+1}L^{\prime}(|u_{t}^{0}|^{p}|U_{j}|^{q-1}|u_{x}^{0}|)\\\
&\qquad+{\varepsilon}^{p+q-1}L^{\prime}(|u_{t}^{0}|^{p}|u^{0}|^{q-1}|(U_{j})_{x}|)+{\varepsilon}^{p+q}L^{\prime}(|u_{t}^{0}|^{p}|u^{0}|^{q-1}|u_{x}^{0}|)\\}\\\
&\quad+2^{r-1}rB\\{L^{\prime}(|U_{j}|^{r-1}|(U_{j})_{x}|)+{\varepsilon}L^{\prime}(|U_{j}|^{r-1}|u_{x}^{0}|)\\\
&\qquad+{\varepsilon}^{r-1}L^{\prime}(|u^{0}|^{r-1}|(U_{j})_{x}|+{\varepsilon}^{r}L^{\prime}(|u^{0}|^{r-1}|u^{0}_{x}|))\\},\end{array}$

Proposition 4.1 and 4.2 yield that

$\begin{array}[]{ll}\|(U_{j+1})_{x}\|_{3}&\leq
2^{p+q}A\\{C\|W_{j}\|_{4}^{p}\|U_{j}\|_{3}^{q}(T+R)^{p+q}\\\
&\qquad+{\varepsilon}^{p}E\|u_{t}^{0}\|_{\infty}^{p}\|U_{j}\|_{3}^{q}(T+R)^{q}\\\
&\qquad+{\varepsilon}^{q}E\|W_{j}\|_{4}^{p}\|u^{0}\|_{\infty}^{q}(T+R)^{p}\\\
&\qquad+{\varepsilon}^{p+q}E\|u^{0}_{t}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q})\\}\\\
&\quad+2^{r}B\\{C\|U_{j}\|_{3}^{r}(T+R)^{r+1}+{\varepsilon}^{r}E\|u^{0}\|_{\infty}^{r})\\}\end{array}$

and

$\|(W_{j+1})_{x}\|_{4}\leq Z_{1}+Z_{2}+Z_{3},$

where $Z_{i}\ (i=1,2,3)$ are defined by

$\begin{array}[]{ll}Z_{1}:=&2^{p+q-1}pA\\{C\|W_{j}\|_{4}^{p-1}\|(W_{j})_{x}\|_{4}\|U_{j}\|_{3}^{q}(T+R)^{p+q}\\\
&+{\varepsilon}E\|u_{tx}^{0}\|_{\infty}\|W_{j}\|_{4}^{p-1}\|U_{j}\|_{3}^{q}(T+R)^{p+q-1}\\\
&+{\varepsilon}^{q}E\|u^{0}\|_{\infty}^{q}\|W_{j}\|_{4}^{p-1}\|(W_{j})_{x}\|_{4}(T+R)^{p}\\\
&+{\varepsilon}^{q+1}E\|u_{tx}^{0}\|_{\infty}\|u^{0}\|_{\infty}^{q}\|W_{j}\|_{4}^{p-1}(T+R)^{p-1}\\\
&+{\varepsilon}^{p-1}E\|u_{t}^{0}\|_{\infty}^{p-1}\|(W_{j})_{x}\|_{4}\|U_{j}\|_{3}^{q}(T+R)^{q+1}\\\
&+{\varepsilon}^{p}E\|u_{t}^{0}\|_{\infty}^{p-1}\|u_{tx}^{0}\|_{\infty}\|U_{j}\|_{3}^{q}(T+R)^{q}\\\
&+{\varepsilon}^{p+q-1}E\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}\|(W_{j})_{x}\|_{4}(T+R)\\\
&+{\varepsilon}^{p+q}E\|u_{t}^{0}\|_{\infty}^{p-1}\|u_{tx}^{0}\|_{\infty}\|u^{0}\|_{\infty}^{q}\\},\end{array}$
$\begin{array}[]{ll}Z_{2}:=&2^{p+q-1}qA\\{C\|W_{j}\|_{4}^{p}\|U_{j}\|_{3}^{q-1}\|(U_{j})_{x}\|_{3}(T+R)^{p+q}\\\
&+{\varepsilon}E\|u_{x}^{0}\|_{\infty}\|W_{j}\|_{4}^{p}\|U_{j}\|_{3}^{q-1}(T+R)^{p+q-1}\\\
&+{\varepsilon}^{q-1}E\|u^{0}\|_{\infty}^{q-1}\|W_{j}\|_{4}^{p}\|(U_{j})_{x}\|_{3}(T+R)^{p+1}\\\
&+{\varepsilon}^{q}E\|u^{0}\|_{\infty}^{q-1}\|u_{x}^{0}\|_{\infty}\|W_{j}\|_{4}^{p}(T+R)^{p}\\\
&+{\varepsilon}^{p}E\|u_{t}^{0}\|_{\infty}^{p}\|U_{j}\|_{3}^{q-1}\|(U_{j})_{x}\|_{3}(T+R)^{q}\\\
&+{\varepsilon}^{p+1}E\|u_{t}^{0}\|_{\infty}^{p}\|u_{x}^{0}\|_{\infty}\|U_{j}\|_{3}^{q-1}(T+R)^{q-1}\\\
&+{\varepsilon}^{p+q-1}E\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}\|(U_{j})_{x}\|_{3}(T+R)\\\
&+{\varepsilon}^{p+q}E\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}\|u_{x}^{0}\|_{\infty}\\},\end{array}$

and

$\begin{array}[]{ll}Z_{3}:=&2^{r-1}rB\\{C\|U_{j}\|_{3}^{r-1}\|(U_{j})_{x}\|_{3}(T+R)^{r+1}\\\
&+{\varepsilon}E\|u_{x}^{0}\|_{\infty}\|U_{j}\|_{3}^{r-1}(T+R)^{r-1}\\\
&+{\varepsilon}^{r-1}E\|u^{0}\|_{\infty}^{r-1}\|(U_{j})_{x}\|(T+R)\\\
&{\varepsilon}^{r}E\|u^{0}\|_{\infty}^{r-1}\|u^{0}_{x}\|_{\infty}\\}.\end{array}$

Hence the boundedness of
$\left\\{\left((U_{j})_{x},(W_{j})_{x}\right)\right\\}$, i.e.

$\|(U_{j})_{x}\|_{3},\|(W_{j})_{x}\|_{4}\leq
5N{\varepsilon}^{\min\\{p+q,r\\}}\quad(j\in\mathbf{N}),$ (4.17)

follows from (4.11) for the $(U_{j})_{x}$-component,

$\left\\{\begin{array}[]{ll}2^{p+q-1}pAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q}(T+R)^{p+q}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}},\\\
2^{p+q-1}qAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q}(T+R)^{p+q}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}},\\\
2^{r-1}rBC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{r}(T+R)^{r+1}&\leq
N{\varepsilon}^{\min\\{p+q,r\\}}\end{array}\right.$ (4.18)

and the following three groups of conditions for the $(W_{j})_{x}$-component;

$\begin{array}[]{l}\left.\begin{array}[]{l}{\varepsilon}E\|u_{tx}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q-1},\\\
{\varepsilon}^{q}E\|u^{0}\|_{\infty}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p}(T+R)^{p},\\\
{\varepsilon}^{q+1}E\|u_{tx}^{0}\|_{\infty}\|u^{0}\|_{\infty}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p-1}(T+R)^{p-1},\\\
{\varepsilon}^{p-1}E\|u_{t}^{0}\|_{\infty}^{p-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q+1}(T+R)^{q+1},\\\
{\varepsilon}^{p}E\|u_{t}^{0}\|_{\infty}^{p-1}\|u_{tx}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q}(T+R)^{q},\\\
{\varepsilon}^{p+q-1}E\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})(T+R)\end{array}\right\\}\\\
\displaystyle\leq\frac{2}{3}\cdot\frac{1}{6}\cdot\frac{1}{2^{p+q-1}pA}N{\varepsilon}^{\min\\{p+q,r\\}},\end{array}$
(4.19)
$\begin{array}[]{l}\left.\begin{array}[]{l}{\varepsilon}E\|u_{x}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q-1},\\\
{\varepsilon}^{q-1}E\|u^{0}\|_{\infty}^{q-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+1}(T+R)^{p+1},\\\
{\varepsilon}^{q}E\|u^{0}\|_{\infty}^{q-1}\|u_{x}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p}(T+R)^{p},\\\
{\varepsilon}^{p}E\|u_{t}^{0}\|_{\infty}^{p}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q}(T+R)^{q},\\\
{\varepsilon}^{p+1}E\|u_{t}^{0}\|_{\infty}^{p}\|u_{x}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q-1}(T+R)^{q-1},\\\
{\varepsilon}^{p+q-1}E\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})(T+R)\end{array}\right\\}\\\
\displaystyle\leq\frac{2}{3}\cdot\frac{1}{6}\cdot\frac{1}{2^{p+q-1}qA}N{\varepsilon}^{\min\\{p+q,r\\}}\par\end{array}$
(4.20)

and

$\begin{array}[]{l}\left.\begin{array}[]{l}{\varepsilon}E\|u_{x}^{0}\|_{\infty}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{r-1}(T+R)^{r-1},\\\
{\varepsilon}^{r-1}E\|u^{0}\|_{\infty}^{r-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})(T+R)\end{array}\right\\}\\\
\displaystyle\leq\frac{2}{3}\cdot\frac{1}{2}\cdot\frac{1}{2^{r-1}rB}N{\varepsilon}^{\min\\{p+q,r\\}},\end{array}$
(4.21)

Note that (4.8) yields

$\begin{array}[]{ll}R\leq
C_{3}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)},\right.\\\
&\quad{\varepsilon}^{-[(p+q-2)\min\\{p+q,r\\}+1]/(p+q-1)},{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},\\\
&\quad{\varepsilon}^{-[(p-2)\min\\{p+q,r\\}+q+1]/(p-1)},{\varepsilon}^{-[q\min\\{p+q,r\\}+p-1]/(q+1)},\\\
&\quad{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},{\varepsilon}^{-(p+q-1)},\\\
&\quad{\varepsilon}^{-[p\min\\{p+q,r\\}+q-1]/(p+1)},{\varepsilon}^{-[(q-2)\min\\{p+q,r\\}+p+1]/(q-1)}\\\
&\quad\left.{\varepsilon}^{-[(r-2)\min\\{p+q,r\\}+1]/(r-1)},{\varepsilon}^{-(r-1)}\right\\}\end{array}$

for $0<{\varepsilon}\leq{\varepsilon}_{23}$. Here we set

$C_{3}:=\frac{1}{2}\min\\{2C_{1},C_{31},C_{32},C_{33},C_{34}\\},$

where $C_{1}$ is the one in (4.12) and $C_{3i}\ (i=1,2,3,4)$ are defined by

$\begin{array}[]{ll}C_{31}:=\min&\left[\\{2^{p+q-1}pAC(5N)^{p+q}N^{-1}\\}^{-1/(p+q)},\right.\\\
&\quad\\{2^{p+q-1}qAC(5N)^{p+q}N^{-1}\\}^{-1/(p+q)},\\\
&\quad\left.\\{2^{r-1}rBC(5N)^{r}N^{-1}\\}^{-1/(r+1)}\right],\end{array}$
$\begin{array}[]{ll}C_{32}:=\min&\left[\\{2^{p+q-1}3^{2}pAE\|u_{tx}^{0}\|_{\infty}(5N)^{p+q-1}N^{-1}\\}^{-1/(p+q-1)},\right.\\\
&\quad\\{2^{p+q-1}3^{2}pAE\|u^{0}\|_{\infty}^{q}(5N)^{p}N^{-1}\\}^{-1/p},\\\
&\quad\\{2^{p+q-1}3^{2}pAE\|u_{tx}^{0}\|_{\infty}\|u^{0}\|_{\infty}^{q}(5N)^{p-1}N^{-1}\\}^{-1/(p-1)},\\\
&\quad\\{2^{p+q-1}3^{2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q+1}N^{-1}\\}^{-1/(q+1)},\\\
&\quad\left.\\{2^{p+q-1}3^{2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}5\\}^{-1}\right],\end{array}$
$\begin{array}[]{ll}C_{33}:=\min&\left[\\{2^{p+q-1}3^{2}qAE\|u_{x}^{0}\|_{\infty}(5N)^{p+q-1}N^{-1}\\}^{-1/(p+q-1)},\right.\\\
&\quad\\{2^{p+q-1}3^{2}qAE\|u^{0}\|_{\infty}^{q-1}(5N)^{p+1}N^{-1}\\}^{-1/(p+1)},\\\
&\quad\\{2^{p+q-1}3^{2}qAE\|u^{0}\|_{\infty}^{q-1}\|u_{x}^{0}\|_{\infty}(5N)^{p}N^{-1}\\}^{-1/p},\\\
&\quad\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}(5N)^{q}N^{-1}\\}^{-1/q},\\\
&\quad\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u_{x}^{0}\|_{\infty}(5N)^{q-1}N^{-1}\\}^{-1/(q-1)},\\\
&\quad\left.\\{2^{p+q-1}3^{2}qAE\|u_{t}^{0}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}5\\}^{-1}\right]\end{array}$

and

$\begin{array}[]{ll}C_{34}:=\min&\left[\\{2^{r-1}3rBE\|u_{x}^{0}\|_{\infty}(5N)^{r-1}N^{-1}\\}^{-1/(r-1)},\right.\\\
&\quad\left.\\{2^{r-1}3rBE\|u^{0}\|_{\infty}^{r-1}5\\}^{-1}\right].\end{array}$

Therefore we find that (4.18), (4.19), (4.20) and (4.21) as well as (4.17)
follow from

$\begin{array}[]{ll}T\leq
C_{3}&\min\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)},\right.\\\
&\quad{\varepsilon}^{-[(p+q-2)\min\\{p+q,r\\}+1]/(p+q-1)},{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},\\\
&\quad{\varepsilon}^{-[(p-2)\min\\{p+q,r\\}+q+1]/(p-1)},{\varepsilon}^{-[q\min\\{p+q,r\\}+p-1]/(q+1)},\\\
&\quad{\varepsilon}^{-[(q-1)\min\\{p+q,r\\}+p]/q},{\varepsilon}^{-(p+q-1)},\\\
&\quad{\varepsilon}^{-[p\min\\{p+q,r\\}+q-1]/(p+1)},{\varepsilon}^{-[(q-2)\min\\{p+q,r\\}+p+1]/(q-1)}\\\
&\quad\left.{\varepsilon}^{-[(r-2)\min\\{p+q,r\\}+1]/(r-1)},{\varepsilon}^{-(r-1)}\right\\}\end{array}$
(4.22)

for $0<{\varepsilon}\leq{\varepsilon}_{23}$.

Let us write down this inequality in each case. First we note that

$\begin{array}[]{l}\displaystyle\frac{(\gamma-1)\min\\{p+q,r\\}+p+q-\gamma}{\gamma}\\\
\displaystyle=\min\\{p+q,r\\}-1+\frac{p+q-\min\\{p+q,r\\}}{\gamma}\\\
\displaystyle\geq\min\\{p+q,r\\}-1+\frac{p+q-\min\\{p+q,r\\}}{p+q}\\\
\displaystyle=\frac{(p+q-1)\min\\{p+q,r\\}}{p+q}\end{array}$

holds for any $\gamma$ satisfying $0<\gamma\leq p+q$. Therefore (4.22) can be
diminished as

$\begin{array}[]{ll}T\leq
C_{3}\min&\left\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-\min\\{p+q,r\\}(r-1)/(r+1)},\right.\\\
&\quad\left.{\varepsilon}^{-[(r-2)\min\\{p+q,r\\}+1]/(r-1)},{\varepsilon}^{-(r-1)}\right\\}\end{array}$

  * •

When $(2<)p+q\leq(r+1)/2(<r)$,
$\min\\{p+q,r\\}=p+q$ and the same reason as the argument after (4.13) imply
that (4.22) is equivalent to

$T\leq C_{3}{\varepsilon}^{-(p+q-1)}$

because of

$\begin{array}[]{l}\displaystyle\frac{(r-2)(p+q)+1}{r-1}-\frac{(p+q)(r-1)}{r+1}\\\
\displaystyle=\frac{(r^{2}-r-2)(p+q)+r+1-(p+q)(r^{2}-2r+1)}{(r-1)(r+1)}\\\
\displaystyle=\frac{(r-3)(p+q)+r+1}{(r-1)(r+1)}\geq\frac{(r-2)(p+q)+1}{(r-1)(r+1)}>0.\end{array}$

  * •

When $(r+1)/2\leq p+q\leq r$, similarly to the argument above, we find that
(4.22) is equivalent to

$T\leq C_{3}{\varepsilon}^{-(p+q)(r-1)/(r+1)}.$

  * •

When $p+q\geq r$, $\min\\{p+q,r\\}=r$ and the argument after (4.13) imply that
(4.22) is equivalent to

$T\leq C_{3}{\varepsilon}^{-r(r-1)/(r+1)}$

because of $\min\\{p+q,r\\}=r$ and

$\begin{array}[]{l}\displaystyle\frac{(r-2)r+1}{r-1}-\frac{r(r-1)}{r+1}\\\
\displaystyle=\frac{(r-3)r+r+1}{(r-1)(r+1)}=\frac{r-1}{r+1}>0.\end{array}$

Next, assuming (4.13), (4.16) and (4.22), we shall estimate
$\\{(U_{j+1})_{x}-(U_{j})_{x}\\}$ and $\\{(W_{j+1})_{x}-(W_{j})_{x}\\}$. First
we deal with $\\{(U_{j+1})_{x}-(U_{j})_{x}\\}$. It is easy to see

$\begin{array}[]{l}|(U_{j+1})_{x}-(U_{j})_{x}|\\\ \leq
L^{\prime}(A||W_{j}+{\varepsilon}u^{0}_{t}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q}-|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p}||U_{j-1}+{\varepsilon}u^{0}|^{q}|)\\\
\quad+L^{\prime}(B||U_{j}+{\varepsilon}u^{0}|^{r}-|U_{j-1}+{\varepsilon}u^{0}|^{r}|)\\\
\leq
2^{q}3^{p-1}pAL^{\prime}\\{(|W_{j-1}|^{p-1}+|W_{j}|^{p-1}+{\varepsilon}^{p-1}|u^{0}_{t}|^{p-1})\times\\\
\qquad\times(|U_{j}|^{q}+{\varepsilon}^{q}|u^{0}|^{q})|W_{j}-W_{j-1}|\\}\\\
\quad+2^{p}3^{q-1}qAL^{\prime}\\{(|W_{j}|^{p}+{\varepsilon}^{p}|u_{t}^{0}|^{p})(|U_{j}|^{q-1}+|U_{j-1}|^{q-1}+{\varepsilon}^{q-1}|u^{0}|^{q-1})\times\\\
\qquad\times|U_{j}-U_{j-1}|\\}\\\
\quad+3^{r-1}rBL^{\prime}\\{(|U_{j-1}|^{r-1}+|U_{j}|^{r-1}+{\varepsilon}^{q-1}|u^{0}|^{r-1})|U_{j}-U_{j-1}|\\}.\end{array}$

Hence it follows from Propositions 4.1 and 4.2 that

$\begin{array}[]{l}\|(U_{j+1})_{x}-(U_{j})_{x}\|_{3}\\\ \leq
2^{q}3^{p-1}pAC(\|W_{j-1}\|_{4}^{p-1}+\|W_{j}\|_{4}^{p-1})\|U_{j}\|_{3}^{q}\|W_{j}-W_{j-1}\|_{4}(T+R)^{p+q}\\\
\quad+2^{q}3^{p-1}pAE{\varepsilon}^{p-1}\|u_{t}^{0}\|_{\infty}^{p-1}\|U_{j}\|_{3}^{q}\|W_{j}-W_{j-1}\|_{4}(T+R)^{q+1}\\\
\quad+2^{q}3^{p-1}pA{\varepsilon}^{q}\|u^{0}\|_{\infty}^{q}(\|W_{j-1}\|_{4}^{p-1}+\|W_{j}\|_{4}^{p-1})\|W_{j}-W_{j-1}\|_{4}(T+R)^{p}\\\
\quad+2^{q}3^{p-1}pA{\varepsilon}^{p+q-1}\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}\|W_{j}-W_{j-1}\|_{4}(T+R)\\\
\quad+2^{p}3^{q-1}qAC\|W_{j}\|_{4}^{p}(\|U_{j}\|_{3}^{q-1}+\|U_{j-1}\|_{3}^{q-1})\|U_{j}-U_{j-1}\|_{3}(T+R)^{p+q}\\\
\quad+2^{p}3^{q-1}qAE{\varepsilon}^{q-1}\|u^{0}\|_{\infty}^{q-1}\|W_{j}\|_{4}^{p}\|U_{j}-U_{j-1}\|_{3}(T+R)^{p+1}\\\
\quad+2^{p}3^{q-1}qAE{\varepsilon}^{p}\|u_{t}^{0}\|_{\infty}^{p}(\|U_{j}\|_{3}^{q-1}+\|U_{j-1}\|_{3}^{q-1})\|U_{j}-U_{j-1}\|_{3}(T+R)^{q}\\\
\quad+2^{p}3^{q-1}qAE{\varepsilon}^{p+q-1}\|u^{0}_{t}\|_{\infty}^{p}\|u^{0}\|_{\infty}^{q-1}\|W_{j}-W_{j-1}\|_{4}(T+R)\\\
\quad+3^{r-1}rBC(\|U_{j-1}\|_{3}^{r-1}+\|U_{j}\|_{3}^{r-1})\|U_{j}-U_{j-1}\|_{3}(T+R)^{r+1}\\\
\quad+3^{r-1}rBE{\varepsilon}^{r-1}\|u^{0}\|_{\infty}^{r-1}\|U_{j}-U_{j-1}\|_{3}(T+R)\end{array}$

Since (4.13) and (4.16) yield (4.14), we have that

$\|U_{j+1}-U_{j}\|_{3}+\|W_{j+1}-W_{j}\|_{3}\leq
O\left(\frac{1}{2^{j}}\right)\quad\mbox{as}\ j\rightarrow\infty.$

This fact implies that

$\|(U_{j+1})_{x}-(U_{j})_{x}\|_{3}=O\left(\frac{1}{2^{j}}\right)\quad\mbox{as}\
j\rightarrow\infty.$ (4.23)

Next, we estimate $\\{(W_{j+1})_{x}-(W_{j})_{x}\\}$. It is also easy to see

$\begin{array}[]{l}|(W_{j+1})_{x}-(W_{j})_{x}|\\\ \leq
pAL^{\prime}\\{||W_{j}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j}+{\varepsilon}u^{0}_{t})((W_{j})_{x}+{\varepsilon}u^{0}_{tx})|U_{j}+{\varepsilon}u^{0}|^{q}\\\
\qquad-|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j-1}+{\varepsilon}u^{0}_{t})((W_{j-1})_{x}+{\varepsilon}u^{0}_{tx})|U_{j-1}+{\varepsilon}u^{0}|^{q}|\\}\\\
\quad+qAL^{\prime}\\{||W_{j}+{\varepsilon}u_{t}^{0}|^{p}|U_{j}+{\varepsilon}u^{0}|^{q-2}(U_{j}+{\varepsilon}u^{0})((U_{j})_{x}+{\varepsilon}u^{0}_{x})\\\
\qquad-|W_{j-1}+{\varepsilon}u_{t}^{0}|^{p}|U_{j-1}+{\varepsilon}u^{0}|^{q-2}(U_{j-1}+{\varepsilon}u^{0})((U_{j-1})_{x}+{\varepsilon}u^{0}_{x})|\\}\\\
\quad+rBL^{\prime}\\{||U_{j}+{\varepsilon}u^{0}|^{r-2}(U_{j}+{\varepsilon}u^{0})((U_{j})_{x}+{\varepsilon}u^{0}_{x})\\\
\qquad-|U_{j-1}+{\varepsilon}u^{0}|^{r-2}(U_{j-1}+{\varepsilon}u^{0})((U_{j-1})_{x}+{\varepsilon}u^{0}_{x})|\\}.\end{array}$

In order to estimate the quantities in the right hand side of this inequality,
we employ

$\begin{array}[]{l}||W_{j}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j}+{\varepsilon}u^{0}_{t})((W_{j})_{x}+{\varepsilon}u^{0}_{tx})|U_{j}+{\varepsilon}u^{0}|^{q}\\\
-|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j-1}+{\varepsilon}u^{0}_{t})((W_{j-1})_{x}+{\varepsilon}u^{0}_{tx})|U_{j-1}+{\varepsilon}u^{0}|^{q}|\\\
\leq||W_{j}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j}+{\varepsilon}u^{0}_{t})-|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j-1}+{\varepsilon}u^{0}_{t})|\times\\\
\quad\times|(W_{j})_{x}+{\varepsilon}u^{0}_{tx}||U_{j}+{\varepsilon}u^{0}|^{q}\\\
\quad+|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p-1}|(W_{j})_{x}-(W_{j-1})_{x}||U_{j}+{\varepsilon}u^{0}|^{q}\\\
\quad+|W_{j-1}+{\varepsilon}u_{t}^{0}|^{p-1}|(W_{j-1})_{x}+{\varepsilon}u_{tx}|||U_{j}+{\varepsilon}u^{0}|^{q}-|U_{j-1}+{\varepsilon}u^{0}|^{q}|\end{array}$

and

$\begin{array}[]{l}||W_{j}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j}+{\varepsilon}u^{0}_{t})-|W_{j-1}+{\varepsilon}u^{0}_{t}|^{p-2}(W_{j-1}+{\varepsilon}u^{0}_{t})|\\\
\leq\left\\{\begin{array}[]{ll}\begin{array}[]{l}3^{p-2}(p-1)(|W_{j}|^{p-2}+|W_{j-1}|^{p-2}+{\varepsilon}^{p-2}|u^{0}_{t}|^{p-2})\times\\\
\quad\times|W_{j}-W_{j-1}|\end{array}&\mbox{for}\ p\geq 2,\\\
2|W_{j}-W_{j-1}|^{p-1}&\mbox{for}\ 1<p<2.\end{array}\right.\end{array}$

Hence the same manner as estimating $\\{(U_{j+1})_{x}-(U_{j})_{x}\\}$ yields
that

$\begin{array}[]{l}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}\\\ \leq
2^{p+q-1}pA\|L^{\prime}\\{(|W_{j-1}|^{p-1}+{\varepsilon}^{p-1}|u_{t}^{0}|^{p-1})\times\\\
\quad\times(|U_{j}|^{q}+{\varepsilon}^{q}|u^{0}|^{q})|(W_{j})_{x}-(W_{j-1})|\\}\|_{4}+\displaystyle
O\left(\frac{1}{2^{j\min\\{p-1,q-1,r-1,1\\}}}\right)\end{array}$

as $j\rightarrow\infty$. Then it follows from (4.13) as well as (4.10) that

$\begin{array}[]{ll}2^{p+q-1}pA\|L^{\prime}\\{(|W_{j-1}|^{p-1}+{\varepsilon}^{p-1}|u^{0}_{t}|^{p-1})\times\\\
\quad\times(|U_{j}|^{q}+{\varepsilon}^{q}|u^{0}|^{q})|(W_{j})_{x}-(W_{j-1})_{x}|)\|_{4}\\\
\leq
2^{p+q-1}pAC\|W_{j-1}\|_{4}^{p-1}\|U_{j}\|_{3}^{q}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}(T+R)^{p+q}\\\
\quad+2^{p+q-1}pAE{\varepsilon}^{p-1}\|u_{t}^{0}\|_{\infty}^{p-1}\|U_{j}\|_{3}^{q}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}(T+R)^{q+1}\\\
\quad+2^{p+q-1}pAE{\varepsilon}^{q}\|u^{0}\|_{\infty}^{q}\|W_{j-1}\|_{4}^{p-1}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}(T+R)^{p}\\\
\quad+2^{p+q-1}pAE{\varepsilon}^{p+q-1}\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}(T+R).\end{array}$

Therefore we obtain that

$\|(W_{j+1})_{x}-(W_{j})_{x}\|_{4}\leq\frac{1}{2}\|(W_{j})_{x}-(W_{j-1})_{x}\|_{4}+O\left(\frac{1}{2^{j\min\\{p-1,q-1,1\\}}}\right)$
(4.24)

as $j\rightarrow\infty$, which shows the convergence of $\\{(W_{j})_{x}\\}$,
provided

$\left.\begin{array}[]{ll}2^{p+q-1}pAC(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p+q-1}(T+R)^{p+q},\\\
2^{p+q-1}pAE{\varepsilon}^{p-1}\|u_{t}^{0}\|_{\infty}^{p-1}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{q}(T+R)^{q+1},\\\
2^{p+q-1}pAE{\varepsilon}^{q}\|u^{0}\|_{\infty}^{q}(5N{\varepsilon}^{\min\\{p+q,r\\}})^{p-1}(T+R)^{p},\\\
2^{p+q-1}pAE{\varepsilon}^{p+q-1}\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}^{q}(T+R)\end{array}\right\\}\leq\frac{1}{8}$
(4.25)

holds. Since (4.9) yields that

$\begin{array}[]{ll}R\leq
C_{4}\min&\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-[q\min\\{p+q,r\\}+p-1]/(q+1)},\\\
&\quad{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-(p+q-1)}\\}\end{array}$

for $0<{\varepsilon}\leq{\varepsilon}_{24}$, where

$\begin{array}[]{ll}C_{4}:=\displaystyle\frac{1}{2}\min&\left[\\{2^{p+q+2}pAC(5N)^{p+q-1}\\}^{-1/(p+q)},\right.\\\
&\quad\\{2^{p+q+2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}(5N)^{q}\\}^{-1/(q+1)},\\\
&\quad\\{2^{p+q+2}pAE\|u\|_{\infty}^{q}(5N)^{p-1}\\}^{-1/p},\\\
&\quad\left.\\{2^{p+q+2}pAE\|u_{t}^{0}\|_{\infty}^{p-1}\|u^{0}\|_{\infty}\\}^{-1}\right]\end{array}$

we find that (4.25) as well as (4.23) and (4.24) follows from

$\begin{array}[]{ll}T\leq
C_{4}\min&\\{{\varepsilon}^{-\min\\{p+q,r\\}(p+q-1)/(p+q)},{\varepsilon}^{-[q\min\\{p+q,r\\}+p-1]/(q+1)},\\\
&\quad{\varepsilon}^{-[(p-1)\min\\{p+q,r\\}+q]/p},{\varepsilon}^{-(p+q-1)}\\}\end{array}$
(4.26)

for $0<{\varepsilon}\leq{\varepsilon}_{24}$. We note that (4.26) is a part of
(4.16) for which $C_{4}$ is replace with $C_{2}$.

Continuation of the proof.

The convergence of the sequence $\\{(U_{j},W_{j})\\}$ to $(U,W)$ in the closed
subspace of $Y$ satisfying

$\|U\|_{3},\|(U_{x})\|_{3},\|W\|_{4},\|(W)_{x}\|_{4}\leq
5N{\varepsilon}^{\min\\{p+q,r\\}}$

is established by (4.6), (4.7), (4.8), (4.9), (4.13), (4.16), (4.22) and
(4.26). Therefore the statement of Theorem 2.2 is established with

$\left\\{\begin{array}[]{l}\displaystyle
c=\min\left\\{C_{1},C_{2},C_{3},C_{4}\right\\},\\\
{\varepsilon}_{2}=\min\\{1,{\varepsilon}_{21},{\varepsilon}_{22},{\varepsilon}_{23},{\varepsilon}_{24}\\}.\end{array}\right.$

$\Box$

##  5 Proof of Proposition 4.2

In this section, we prove a priori estimate (4.5). Note that three estimates
with $|U|^{r}$ are already obtained by Proposition 5.2 in Morisawa, Sasaki and
Takamura [17], so that we shall prove other three estimates with
$|W|^{p}|U|^{q}$. Here a positive constant $C$ independent of $T$ and
${\varepsilon}$ may change from line to line.

It follows from the assumption on the supports and the definition of $L$ that

$|L(|W|^{p}|U|^{q})(x,t)|\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}J(x,t)\quad\mbox{for}\ |x|\leq t+R,$

where we set

$\begin{array}[]{ll}J(x,t):=&\displaystyle\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}\\{\chi_{D}(s,y)+(1-\chi_{D}(s,y))(s+|y|+R)^{p}\\}\times\\\
&\quad\times(s+|y|+R)^{q}\chi_{\mbox{\footnotesize
supp}(U,W)}(s,y)dy.\end{array}$

First, we consider the case of $x\geq 0$. From now on, we employ the change of
variables

$\alpha=s+y,\ \beta=s-y.$ (5.1)

For $(x,t)\in D$, extending the domain of the integral, we have that

$\begin{array}[]{ll}J(x,t)&\displaystyle\leq
C\int_{-R}^{R}d\beta\int_{-R}^{x+t}(\alpha+R)^{p+q}d\alpha\\\
&\displaystyle\quad+C\int_{R}^{t-x}(\beta+R)^{p+q}d\beta\int_{-R}^{R}d\alpha\\\
&\displaystyle\quad+C\int_{R}^{t-x}d\beta\int_{R}^{t+x}(\alpha+R)^{q}d\alpha\\\
&\leq C(t+x+R)^{p+q+1}\\\ &\quad+C(t-x+R)^{p+q+1}\\\
&\quad+C(t+x+R)^{q+1}(t-x+R)\\\ &\leq C(T+R)^{p+q}(t+x+R).\end{array}$

For $t+x\geq R$ and $|t-x|\leq R$, we also have that

$\begin{array}[]{ll}J(x,t)&\displaystyle\leq
C+C\int_{-R}^{t-x}d\beta\int_{R}^{t+x}(\alpha+R)^{p+q}d\alpha\\\
&\displaystyle\leq C(T+R)^{p+q}(t+x+R).\end{array}$

For $t+x\leq R$, it is trivial that

$J(x,t)\leq C.$

Summing up, we obtain that

$\begin{array}[]{c}|L(|W|^{p}|U|^{q})(x,t)|\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}(T+R)^{p+q}(t+x+R)\\\ \mbox{for}\ 0\leq x\leq
t+R.\end{array}$

The case of $x\leq 0$ is similar to the one above, so we omit the details.
Therefore we obtain the first inequality in (4.5).

Next, we shall show the third inequality in (4.5). It follows from the
assumption on the supports and the definition of $L^{\prime}$ that

$\begin{array}[]{l}|L^{\prime}(|W|^{p}|U|^{q})(x,t)|\\\ \quad\leq
C\|W\|_{4}^{p}\|U\|_{3}^{q}\\{J_{+}(x,t)+J_{-}(x,t)\\}\end{array}\mbox{for}\
|x|\leq t+R,$

where the integrals $J_{+}$ and $J_{-}$ are defined by

$\begin{array}[]{ll}J_{\pm}(x,t):=&\displaystyle\int_{0}^{t}\\{\chi_{I\pm}(x,t;s)+(1-\chi_{I\pm}(x,t;s))(s+|t-s\pm
x|+R)^{p}\\}\times\\\ &\quad\times\chi_{E\pm}(x,t;s)(s+|t-s\pm
x|+R)^{q}ds\end{array}$

and the characteristic functions $\chi_{I+},\chi_{I-},\chi_{E+}$ and
$\chi_{E-}$ are defined by

$\begin{array}[]{l}\chi_{I\pm}(x,t;s):=\chi_{\\{s:s-|t-s\pm x|\geq R\\}},\\\
\chi_{E\pm}(x,t;s):=\chi_{\\{s:|t-s\pm x|\leq s+R\\}}\end{array}$

respectively. First we note that it is sufficient to estimate $J_{\pm}$ for
$x\geq 0$ due to its symmetry,

$J_{+}(-x,t)=J_{-}(x,t).$

For $(x,t)\in D\cap\\{x\geq 0\\}$, we have

$\begin{array}[]{ll}J_{+}(x,t)&\displaystyle\leq
C\int_{(t+x-R)/2}^{(t+x+R)/2}(t+x+R)^{p+q}ds\\\
&\displaystyle\quad+C\int_{(t+x+R)/2}^{t}(t+x+R)^{q}ds\\\ &\leq
C(t+x+R)^{p+q}+C(t+x+R)^{q+1}\\\ &\leq C(T+R)^{p+q}\end{array}$

and

$\begin{array}[]{ll}J_{-}(x,t)&\displaystyle\leq
C\int_{(t-x-R)/2}^{(t-x+R)/2}(s+|t-s-x|+R)^{p+q}ds\\\
&\displaystyle\quad+C\int_{(t-x+R)/2}^{t}(s+|t-s-x|+R)^{q}ds\\\ &\leq
C(t+x+R)^{p+q}+C(t+x+R)^{q+1}\\\ &\leq C(T+R)^{p+q}.\end{array}$

For $t+x\geq R$ and $|t-x|\leq R$, we have

$J_{+}(x,t)\leq C\int_{(t+x-R)/2}^{t}(t+x+R)^{p+q}ds\leq C(T+R)^{p+q}(t+x+R)$

and

$\begin{array}[]{ll}J_{-}(x,t)&\displaystyle\leq
C\int_{0}^{t}(s+|t-s-x|+R)^{p+q}ds\\\ &\displaystyle\leq
C\int_{0}^{t-x}(t-x+R)^{p+q}ds+C\int_{t-x}^{t}(2s-t+x+R)^{p+q}ds\\\ &\leq
C(t-x+R)^{p+q+1}+C(t+x+R)^{p+q+1}\\\ &\leq C(T+R)^{p+q}(t+x+R).\end{array}$

It is trivial that $J_{\pm}(x,t)\leq C$ for $t+x\leq R$. Therefore we obtain
the third inequality in (4.5).

The fifth inequality in (4.5) readily follows from the computations above. The
proof of Proposition 4.2 is now completed. $\Box$

##  6 Proofs of Theorem 2.3 and Theorem 2.4

The essential argument to obtain the upper bound of the lifespan is the
following. Let $u$ be a classical solution of (1.1) in a time interval
$[0,T]$. If $T$ is bigger than some quantity depending on ${\varepsilon}$, we
will meet a contradiction to the fact that $u$ is a classical solution. This
situation gives us that the lifespan should be less than the quantity due to
its definition.

Proof of Theorem 2.3.

Neglecting the second term of our equation as

$A|u_{t}|^{p}|u|^{q}+B|u|^{r}\geq A|u_{t}|^{p}|u|^{q},$

in (1.1), we have that there exists a constant
${\varepsilon}_{31}={\varepsilon}_{31}(f,g,p,q,A,R)$ such that the
contradiction appears provided

$T>C_{31}{\varepsilon}^{-(p+q-1)}\quad\mbox{if}\ \int_{{\bf R}}g(x)dx>0$ (6.1)

holds for $0<{\varepsilon}\leq{\varepsilon}_{31}$ and some positive constant
$C_{31}$ independent of ${\varepsilon}$. Because it is already obtained by
Zhou [23] for the equation

$u_{tt}-u_{xx}=|u_{t}|^{p}|u|^{q}.$

in which it is trivial that “$=$” can be replaced with “$\geq A\times$”. As
stated in Introduction, we shall repeat its proof in Appendix below.

By virtue of the same reason and Zhou [21], making use of

$A|u|_{t}^{p}|u|^{q}+B|u|^{r}\geq B|u|^{r},$

we have that there exists a constant
${\varepsilon}_{32}={\varepsilon}_{32}(f,g,r,B,R)$ such that the contradiction
appears provided

$T>C_{32}{\varepsilon}^{-(r-1)/2}\quad\mbox{if}\ \int_{{\bf R}}g(x)dx>0$ (6.2)

holds for $0<{\varepsilon}\leq{\varepsilon}_{32}$ and some positive constant
$C_{32}$ independent of ${\varepsilon}$. Therefore, taking

${\varepsilon}_{3}=\min\\{{\varepsilon}_{31},{\varepsilon}_{32},1\\}\quad\mbox{and}\quad
C=\min\\{C_{31},C_{32}\\},$

we have the desired lifespan estimate by (6.1) and (6.2). $\Box$

Proof of Theorem 2.4.

First we shall prove that there exists a constant
${\varepsilon}_{41}={\varepsilon}_{41}(f,p,q,A,R)$ such that the contradiction
appears provided

$T>C_{41}{\varepsilon}^{-(p+q-1)}\quad\mbox{if (\ref{positive_zero}) is
fulfilled}$ (6.3)

holds for $0<{\varepsilon}\leq{\varepsilon}_{41}$ and some positive constant
$C_{41}$ independent of ${\varepsilon}$. To this end, define the blow-up set

$\Sigma:=\left\\{(x,t)\in{\bf R}\times[0,T]\ :\ x\geq 0,\ t+x\geq R,\
0<t-x<\frac{R}{2}\right\\}.$ (6.4)

Then, it follows from the positiveness of the nonlinear term and
representation of $u,u_{t}$ in (2.3), (2.6) that

$u(x,t)\geq\frac{{\varepsilon}}{2}\\{f(x+t)+f(x-t)\\}$

and

$u_{t}(x,t)\geq\frac{{\varepsilon}}{2}\\{f^{\prime}(x+t)-f^{\prime}(x-t)\\}.$

Hence the assumption on the initial data, (2.21), implies that

$u(x,t),u_{t}(x,t)\geq\frac{1}{2}f_{0}{\varepsilon}\quad\mbox{in}\ \Sigma.$
(6.5)

From now on, we employ the routine iteration procedure. Assume an estimate

$\left\\{\begin{array}[]{ll}u(x,t)&\geq M_{n}(t+x-R)^{a_{n}}(t-x)^{b_{n}},\\\
u_{t}(x,t)&\geq M_{n}(t+x-R)^{a_{n}}(t-x)^{c_{n}}\end{array}\right.\mbox{for}\
(x,t)\in\Sigma$ (6.6)

holds, where $a_{n},b_{n},c_{n}\geq 0$ and $M_{n}>0$. All the sequences
$\\{a_{n}\\},\\{b_{n}\\},\ \\{c_{n}\\}$ and $\\{M_{n}\\}$ are defined later.
Then it follows from (2.3), (2.6) and (5.1) that

$\begin{array}[]{ll}u(x,t)&\displaystyle\geq\frac{AM_{n}^{p+q}}{4}\int_{0}^{t-x}\beta^{qb_{n}+pc_{n}}d\beta\int_{R}^{t+x}(\alpha-R)^{(p+q)a_{n}}d\alpha\\\
&\displaystyle\geq\frac{AM_{n}^{p+q}}{4\\{(p+q)a_{n}+1\\}\\{qb_{n}+pc_{n}+1\\}}\times\\\
&\quad\times(t+x-R)^{(p+q)a_{n}+1}(t-x)^{qb_{n}+pc_{n}+1}\end{array}$

and

$\begin{array}[]{ll}u_{t}(x,t)&\displaystyle\geq\frac{A}{2}\int_{(t-x+R)/2}^{t}|u_{t}(x-t+s,s)|^{p}|u(x-t+s,s)|^{q}ds\\\
&\displaystyle\geq\frac{AM_{n}^{p+q}}{2}(t-x)^{qb_{n}+pc_{n}}\int_{(t-x+R)/2}^{t}(2s-t+x-R)^{(p+q)a_{n}}ds\\\
&\displaystyle\geq\frac{AM_{n}^{p+q}}{4\\{(p+q)a_{n}+1\\}}(t+x-R)^{(p+q)a_{n}+1}(t-x)^{qb_{n}+pc_{n}}\end{array}$

for $(x,t)\in\Sigma$. Hence (6.6) holds for all $n\in{\bf N}$ provided

$\left\\{\begin{array}[]{ll}a_{n+1}=(p+q)a_{n}+1,&a_{1}=0,\\\
b_{n+1}=qb_{n}+pc_{n}+1,&b_{1}=0,\\\
c_{n+1}=qb_{n}+pc_{n},&c_{1}=0\end{array}\right.$

and

$M_{n+1}\leq\frac{AM_{n}^{p+q}}{4\\{(p+q)a_{n}+1\\}\\{(p+q)(b_{n}+c_{n})+1\\}},\
M_{1}=\frac{1}{2}f_{0}{\varepsilon}.$

It is easy to see that

$a_{n}=b_{n}+c_{n}=\frac{(p+q)^{n-1}-1}{p+q-1}\quad(n\in{\bf N}),$

which implies

$\begin{array}[]{l}\\{(p+q)a_{n}+1\\}\\{(p+q)(b_{n}+c_{n})+1\\}\\\
\displaystyle\leq\\{(p+q)a_{n}+1\\}^{2}=a_{n+1}^{2}\leq\frac{(p+q)^{2n}}{(p+q-1)^{2}}.\end{array}$

Therefore $M_{n}$ should be defined by

$M_{n+1}=AC_{5}(p+q)^{-2n}M_{n}^{p+q},\ M_{1}=\frac{1}{2}f_{0}{\varepsilon},$

where we set

$C_{5}:=\frac{(p+q-1)^{2}}{4}>0,$

so that (6.6) implies that

$\begin{array}[]{l}u(x,t)u_{t}(x,t)\\\ \geq
M_{n}^{2}(t+x-R)^{2a_{n}}(t-x)^{a_{n}}\\\ \geq
C_{6}\\{(t+x-R)^{2}(t-x)\\}^{-1/(p+q-1)}\exp\left\\{Z(x,t)(p+q)^{n-1}\right\\}\end{array}$
(6.7)

for $(x,t)\in\Sigma$, where

$\begin{array}[]{ll}Z(x,t)&\displaystyle:=\frac{1}{p+q-1}\log\\{(t+x-R)^{2}(t-x)\\}\\\
&\displaystyle\quad+\frac{2}{p+q-1}\log(AC_{5})-4S_{p+q}\log(p+q)+2\log\left(\frac{1}{2}f_{0}{\varepsilon}\right),\\\
C_{6}&\displaystyle:=\exp\left(-\frac{2}{p+q-1}\log(AC_{5})\right)>0.\end{array}$

Indeed, $M_{n}$ satisfies

$\log M_{n+1}=\log(AC_{5})-2n\log(p+q)+(p+q)\log M_{n},$

which implies

$\begin{array}[]{l}\log M_{n+1}\\\
=\\{1+(p+q)+\cdots+(p+q)^{n-1}\\}\log(AC_{5})\\\
\quad-2\\{n+(p+q)(n-1)+\cdots+(p+q)^{n-1}(n-(n-1))\\}\log(p+q)\\\
\quad+(p+q)^{n}\log M_{1}\\\
\displaystyle=\frac{(p+q)^{n}-1}{p+q-1}\log(AC_{5})-2(p+q)^{n-1}\log(p+q)\sum_{j=0}^{n-1}\frac{j+1}{(p+q)^{j}}\\\
\quad+(p+q)^{n}\log M_{1}\\\ \displaystyle\geq-\frac{1}{p+q-1}\log(AC_{5})\\\
\displaystyle\quad+(p+q)^{n}\left\\{\frac{1}{p+q-1}\log(AC_{5})-2S_{p+q}\log(p+q)+\log
M_{1}\right\\},\end{array}$

where we set

$S_{r}:=\sum_{j=0}^{\infty}\frac{j+1}{r^{j+1}}<\infty.$

In view of (6.7), if there exists a point $(x_{0},t_{0})\in\Sigma$ such that
$Z(x_{0},t_{0})>0$, we have a contradiction
$u(x_{0},t_{0})u_{t}(x_{0},t_{0})=\infty$ to the fact that $u$ is a classical
solution on the time interval $[0,T]$ with $T\geq t_{0}$ of (1.1) by letting
$n\rightarrow\infty$. Let us set

$t_{0}=x_{0}+\frac{R}{4}\quad\mbox{and}\quad t_{0}\geq\frac{5R}{4}(>1).$

Then, since we have

$(t_{0}+x_{0}-R)^{2}(t_{0}-x_{0})\geq\frac{R}{4}t_{0}^{2},$

$Z(x_{0},t_{0})>0$ follows from

$t_{0}^{2}>\frac{4}{R}\cdot\frac{(p+q)^{4(p+q-1)S_{p+q}}}{(AC_{5})^{2}(2^{-1}f_{0})^{2(p+q-1)}}{\varepsilon}^{-2(p+q-1)}.$

Therefore this inequality completes the proof of (6.3) with

$C_{41}=\frac{2}{\sqrt{R}}\cdot\frac{(p+q)^{2(p+q-1)S_{p+q}}}{AC_{5}(2^{-1}f_{0})^{p+q-1}}$

and ${\varepsilon}_{41}>0$ satisfying

$C_{41}{\varepsilon}_{41}^{-(p+q-1)}=\frac{5R}{4}.$

Now we note that there exists a constant
${\varepsilon}_{42}={\varepsilon}_{42}(f,r,B,R)$ such that a contradiction
appears provided

$T>C_{42}{\varepsilon}^{-r(r-1)/(r+1)}\quad\mbox{if (\ref{positive_zero}) is
fulfilled}$ (6.8)

holds for $0<{\varepsilon}\leq{\varepsilon}_{42}$ and some positive constant
$C_{42}$ independent of ${\varepsilon}$. The assumption (2.21) is stronger
than the one of Theorem 5.1 in Takamura [18] for the equation
$u_{tt}-u_{xx}=|u|^{r}$. Its proof is available also for $u_{tt}-u_{xx}\geq
B|u|^{r}$ with a trivial modification.

Finally, we shall deal with the case of the generalized combined effect,
namely

$\frac{r+1}{2}\leq p+q\leq r.$

Set

$F(t):=\int_{{\bf R}}u(x,t)dx.$

Then the equation in (1.1) and the support of the solution (2.2) yield that

$F^{\prime\prime}(t)=\int_{{\bf
R}}\\{A|u_{t}(x,t)|^{p}|u(x,t)|^{q}+B|u(x,t)|^{r}\\}dx$

Neglecting the first term in the integrand and making use of Hölder’s
inequality, we have that

$F^{\prime\prime}(t)\geq B\int_{{\bf R}}|u(x,t)|^{r}dx\geq
2^{1-r}B(t+R)^{-(r-1)}|F(t)|^{r}\quad\mbox{for}\ t\geq 0.$ (6.9)

On the other hand, neglecting the second term of the integrand, we have that

$F^{\prime\prime}(t)\geq A\int_{{\bf R}}|u_{t}(x,t)|^{p}|u(x,t)|^{q}dx\geq
A\int_{t-R/2}^{t}|u_{t}(x,t)|^{p}|u(x,t)|^{q}dx$

for $t\geq R/2$. Then it follows from (6.5) that

$F^{\prime\prime}(t)\geq\frac{ARf_{0}^{p+q}}{2^{p+q+1}}{\varepsilon}^{p+q}\quad\mbox{for}\
t\geq\frac{R}{2}.$

Integrating this inequality and employing the fact that
$F^{\prime\prime}(t)\geq 0$ for $t\geq 0$ and

$\left\\{\begin{array}[]{ll}F^{\prime}(t)&\displaystyle\geq
F^{\prime}(0)={\varepsilon}\int_{{\bf R}}g(x)dx=0,\\\ F(t)&\displaystyle\geq
F(0)={\varepsilon}\int_{{\bf R}}f(x)dx>0\end{array}\right.\mbox{for}\ t\geq
0,$

we have that

$F^{\prime}(t)\geq\frac{ARf_{0}^{p+q}}{2^{p+q+2}}{\varepsilon}^{p+q}t\quad\mbox{for}\
t\geq R$

which yields that

$F(t)\geq\frac{ARf_{0}^{p+q}}{2^{p+q+4}}{\varepsilon}^{p+q}t^{2}\quad\mbox{for}\
t\geq 2R.$ (6.10)

We are now in a position to employ Lemma 2.2 in Takamura [18]. The assumption
in (2.9) in [18] is satisfied by (6.10) and setting

$\frac{ARf_{0}^{p+q}}{2^{p+q+4}}{\varepsilon}^{p+q}t_{0}^{2}=2F(0)=2{\varepsilon}\int_{{\bf
R}}f(x)dx.$

Also with

$p=r,\ q=r-1,\ a=2,$

the blow-up condition (2.1) in [18] implies the trivial inequality

$M=\frac{r-1}{2}\cdot 2-\frac{r-1}{2}+1=\frac{r+1}{2}>0.$

If we put

$T_{0}=C_{0}^{\prime}\left(\frac{ARf_{0}^{p+q}}{2^{p+q+4}}{\varepsilon}^{p+q}\right)^{-(r-1)/(r+1)}$

where $C_{0}^{\prime}=C_{0}^{\prime}(r,B)>0$ is the $C_{0}$ in Lemma 2.2 in
[18], we have

$T_{0}\geq\max\\{t_{0},2R\\}$

provided

$C_{0}^{\prime}\left(\frac{ARf_{0}^{p+q}}{2^{p+q+4}}{\varepsilon}^{p+q}\right)^{-(r-1)/(r+1)}\geq\left(\frac{2^{p+q+5}}{ARf_{0}^{p+q}}\int_{{\bf
R}}f(x)dx{\varepsilon}^{-(p+q-1)}\right)^{1/2}\geq 2R$

holds. This inequality can be possible to be established for small
${\varepsilon}$ by the fact that

$\frac{p+q-1}{2}<(p+q)\frac{r-1}{r+1}$

is equivalent to

$(p+q)\frac{3-r}{r+1}<1.$

This inequality is trivial when $r\geq 3$ and also follows from a trivial
inequality

$r\frac{3-r}{r+1}<1$

for $1<r<3$ in this case, $(r+1)/2\leq p+q\leq r$. Therefore it is possible to
take $T_{2}=T_{0}$ in Lemma 2.2 in [18], so that there exists a constant
${\varepsilon}_{43}={\varepsilon}_{42}(f,p,q,r,A,B,R)$ such that the
contradiction appears provided

$T>C_{43}{\varepsilon}^{-r(r-1)/(r+1)}\quad\mbox{if (\ref{positive_zero}) is
fulfilled}$ (6.11)

holds for $0<{\varepsilon}\leq{\varepsilon}_{43}$ and some positive constant
$C_{43}$ independent of ${\varepsilon}$. According to the computations above,
we can choose

${\varepsilon}_{43}=\min\\{{\varepsilon}_{431},{\varepsilon}_{432}\\},$

where ${\varepsilon}_{431}$ and ${\varepsilon}_{432}$ are defined by

$C_{0}^{\prime}\left(\frac{ARf_{0}^{p+q}}{2^{p+q+4}}{\varepsilon}_{431}^{p+q}\right)^{-(r-1)/(r+1)}=\left(\frac{2^{p+q+5}}{ARf_{0}^{p+q}}\int_{{\bf
R}}f(x)dx{\varepsilon}_{431}^{-(p+q-1)}\right)^{1/2}$

and

$\left(\frac{2^{p+q+5}}{ARf_{0}^{p+q}}\int_{{\bf
R}}f(x)dx{\varepsilon}_{432}^{-(p+q-1)}\right)^{1/2}=2R.$

Also it is possible to set

$C_{43}=2^{4/(r+1)}C_{0}^{\prime}\left(\frac{ARf_{0}^{p+q}}{2^{p+q+4}}\right)^{-(r-1)/(r+1)}>0.$

Recall Remark 2.1. Summing up (6.3), (6.8) and (6.11), we obtain the statement
of Theorem 2.4 by taking

${\varepsilon}_{4}=\min\\{1,{\varepsilon}_{41},{\varepsilon}_{42},{\varepsilon}_{43}\\}\quad\mbox{and}\quad
C=\min\\{C_{41},C_{42},C_{43}\\}.$

The proof of Theorem 2.4 is completed now. $\Box$

## Appendix

As stated in Introduction, we repeat here the result and its proof, restricted
in one space dimension although, of the unpublished paper by Zhou [23], for
the sake of the completeness of this paper.

######  Theorem 6.1 (Zhou [23])

Let $A>0$ and $B=0$. Assume (2.1) and (2.19). Then, there exists a positive
constant ${\varepsilon}_{5}={\varepsilon}_{5}(f,g,p,q,A,R)>0$ such that the
lifespan $T({\varepsilon})$ of a classical solution of (1.1) satisfies

$T({\varepsilon})\leq C{\varepsilon}^{-(p+q-1)}$ (6.12)

where $0<{\varepsilon}\leq{\varepsilon}_{5}$, and $C$ is a positive constant
independent of ${\varepsilon}$.

Proof. The proof is almost same as Zhou [22] in which only the nonlinear term
$|u_{t}|^{p}$ is considered.

Let $u$ be a classical solution of (1.1) in the time interval $[0,T]$. Set

$G:=\frac{1}{2}\int_{{\bf R}}g(x)dx>0.$

From now on, we restrict ourselves in the domain

$D^{\prime}:=\\{(x,t)\in{\bf R}\times[0,T]\ :\ x\geq R\quad\mbox{and}\quad
t-x\geq R\\}.$

Then, it follows from (2.1) and (2.3) that

$u=G{\varepsilon}+AL(|u_{t}|^{p}|u|^{q})\quad\mbox{in}\ D^{\prime}.$

Inverting the order of variables and diminishing the domain in $L$, we have

$L(v)(x,t)\geq\frac{1}{2}\int_{R}^{x}dy\int_{y-R}^{y+R}v(y,s)ds\quad\mbox{for}\
(x,t)\in D^{\prime}$

for any non-negative function $v=v(x,t)$ Applying $v=|u_{t}|^{p}|u|^{q}$ to
the equation above and making use of (1.4), we obtain that

$P(x)\geq G{\varepsilon}+C_{7}\int_{R}^{x}|P(y)|^{p+q}dy\quad\mbox{for}\ x\geq
R,$ (6.13)

where we set

$P(x):=u(x,x+R)\quad\mbox{and}\quad
C_{7}:=\frac{A}{2}\left(\frac{p}{p+q}\right)^{p}(2R)^{1-p}>0.$

Because we have employed Hölder’s inequality to have

$\int_{y-R}^{y+R}\left(|u(y,s)|^{(p+q)/p}\right)_{s}ds\leq\left\\{\int_{y-R}^{y+R}\left|\left(|u(y,s)|^{(p+q)/p}\right)_{s}\right|^{p}ds\right\\}^{1/p}(2R)^{1-1/p}$

and (2.2) implies that

$\int_{y-R}^{y+R}\left(|u(y,s)|^{(p+q)/p}\right)_{s}ds=|u(y,y+R)|^{(p+q)/p}.$

Once (6.13) is obtained, it is easy to reach to the desired conclusion by
completely the same argument in [22] in which $p$ is replaced with $p+q$. The
proof is now completed. $\Box$.

## Acknowledgement

The second author is partially supported by the Grant-in-Aid for Young
Scientists (No.18K13447), Japan Society for the Promotion of Science. The
fourth author is partially supported by the Grant-in-Aid for Scientific
Research (A) (No.22H00097), Japan Society for the Promotion of Science.

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