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4 $BH>@~7AJ}Dx<0$N2r$NGzH/(B

3 $B@a$G$O!"(B(11) $B$@$1$G$J$/!"1&JU$,(B $u$ $B$N(B 1 $B

$$u_t+\alpha(t,x)u_x=\beta(t,x,u)$$ (13)

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$$u_t+\alpha(t,x)u_x$$

$B$H$J$C$F$$$F!"@~7A$NJ}Dx<0$N$=$l$HF1$8$G$"$j!"Hs@~7AItJ,$O$3$N@~7A$H8F$P$l$k!#(B

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$BJ}Dx<0(B

$$\left\{\begin{array}{l} \displaystyle \frac{d y}{d t}=y^2,\\ y(0)=1\end{array}\right.$$

$B$r2r$/$H!"2r$O(B

$$y=\frac{1}{1-t}$$

$B$H$J$j!"(B$t=0$ $B$N$H$-$K(B $y=1$ $B$H$J$k2r$O(B $t\rightarrow 1-0$ $B$N$H$-$KL58B(B $BBg$KH/;6$7$F$7$^$$!"$3$N2r$O(B $0\leq t<1$ $B$NHO0O$G$7$+B8:_$7$J$$!#$3$N$h(B $B$&$J>uBV$r!"2r$,(B $t=1$ $B$G(B $BGzH/(B (blowup) $B$9$k!"$H$$$&!#(B

$B?^(B 8: $y=1/(1-t)$

$BJ}Dx<0(B

$$\frac{d y}{d t}=y^2$$

$B$O!"2r$N%0%i%U$N79$-$,!"(B$(t,y)$ $B$H$$$&E@$G$O(B $y^2$ $B$K$J$k!"$H$$$&$3$H$r(B $B<($7$F$$$k!#(B$t=1$ $B$H$$$&D>@~$N>e$G$O!"FC$K$=$N79$-$NCM$,Bg$-$/$J$C$F$$$k(B $B$o$1$G$O$J$$$,!"2r$O$3$3$GL58BBg$KH/;6$7$F$$$k!#$3$l$O@~7A$NJ}Dx<0$G$O(B $B8+$i$l$J$$8=>]$G$"$k!#(B

$B$3$NJ}Dx<0$N>l9g!"(B$y$ $B$,Bg$-$/$J$k$H(B $y^2$ $B$O$5$i$KBg$-$/$J$j!"$=$l$K$7(B $B$?$,$C$F(B $y$ $B$NA}2CN((B $dy/dt$ $B$,Bg$-$/$J$j!"$^$9$^$9(B $y$ $B$,Bg$-$/$J$k!"(B $B$H$$$&Aj>h8z2L$,Hs>o$K6/$/F/$/$?$a$K$3$N$h$&$J$3$H$,5/$3$k$N$G$"$j!"(B $B@~7A$NJ}Dx<0$N>l9g$NAj>h8z2L$O!"$"$kM-8B$N(B $t$ $B$G(B $y$ $B$NCM$rL58BBg$K$7(B $B$F$7$^$&$[$I$N$b$N$G$O$J$$!#(B

$BJPHyJ,J}Dx<0$KOC$rLa$9!#(B $B:#!"Nc$($PH>@~7A0\N.J}Dx<0(B

$$u_t+au_x=u^2 \hspace{1zw}(a \mbox{ $B$ODj?t(B})$$ (14)

$B$r9M$($F$_$k!#(B $B=i4|CM$r(B $u(0,x)=f(x)$ $B$H$9$k!#FC@-6J@~$O(B $x=at+x_0$ $B$@$+$i(B $v(t)=u(t,at+x_0)$ $B$H$9$k$H(B

$$\frac{d }{d t}v(t)=\frac{\partial u}{\partial t}(t,at+x_0)+a\frac{\partial u}{\partial x}(t,at+x_0) =u(t,at+x_0)^2 =v(t)^2$$

$B$h$j!"(B$v(t)$ $B$O(B

$$\frac{d v}{d t}=v^2$$

$B$rK~$?$7!"$3$l$r2r$/$H(B

$$v(t)=\frac{v(0)}{1-v(0)t}$$ (15)

$B$H$J$j!"$3$l$r(B $u$ $B$KLa$7$F!"(B$x_0=x-at$ $B$rBeF~$9$l$P!"7k6I(B

$$u(t,x)=\frac{f(x-at)}{1-f(x-at)t}$$ (16)

$B$,F@$i$l$k!#(B $B$3$N2r$r3FFC@-6J@~>e$G8+$F$_$k!#$9$k$H!"(B (15) $B$K$h$l$P!"(B$v(0)>0$ $B$N$H$-$O(B $v(t)$, $B$9$J$o$A$3$NFC@-6J@~>e$G$N(B $u$ $B$NCM$O(B $t=1/v(0)$ $B$N$H$-$KL58B(B $BBg$KH/;6$7!"GzH/$,5/$3$k$3$H$,$o$+$k!#(B $B0lJ}(B $v(0)=0$ $B$N$H$-$O(B $v(t)=0$, $v(0)<0$ $B$N$H$-$O$9$Y$F$N@5$N(B $t$ $B$N(B $BCM$KBP$7$FM-8B$JCM$G(B $v(t)\rightarrow 0$ $B$H$J$k!#(B

$B$D$^$j(B $v(t)$ $B$,GzH/$9$k$+!"GzH/$7$J$$$+$,(B $v(0)$ $B$NCM$K$h$C$FJQ$o$C$F(B $B$$$k!#$^$?!"GzH/$9$k>l9g$G$b$=$NGzH/;~9o$O(B $v(0)$ $B$NCM$K$h$C$FJQ$o$C$F(B $B$$$k$3$H$,$o$+$k!#(B $B$3$l$O!"3FFC@-6J@~$4$H$KGzH/;~9o$,0[$J$k$3$H$r0UL#$7$F$$$k!#(B

$B?^(B 9: $BGzH/6-3&$HFC@-6J@~(B

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$$t=\frac{1}{v(0)}=\frac{1}{f(x_0)}$$

$B$@$+$i!"(B$(t,x)$ $BJ?LL>e$N(B

$$(t,x)=\left(\frac{1}{f(x_0)},\frac{a}{f(x_0)}+x_0\right)$$

$B$H$$$&E@$GGzH/$7$F$$$k$3$H$K$J$k!#(B $B$3$N!"3FFC@-6J@~>e$G$NGzH/$9$kE@(B $(t,x)$ $B$r$9$Y$F7k$s$G$$$/$H(B $B0l$D$N6J@~$,=PMh>e$,$k!#(B $B$3$l$r(B $BGzH/6-3&(B (blowup boundary), $B$"$k$$$O(B $BGzH/6J@~(B (blowup curve) $B$H8F$V!#(B

$B:#$NJ}Dx<0$N>l9g$K$O!"(B

$$\left\{\begin{array}{l} x=at+x_0,\\ \displaystyle t=\frac{1}{f(x_0)}\end{array}\right.$$

$B$+$i(B $x_0$ $B$r>C5n$9$k$H(B

$$tf(x-at)=1$$

$B$H$J$k$,!"$3$l$,J}Dx<0(B (14) $B$NGzH/6-3&$NJ}Dx<0$G$"$k!#(B

$BJ}Dx<0(B (14) $B$N1&JU$r$h$j0lHL2=$7$F(B

$$u_t+au_x=\vert u\vert^p \hspace{1zw}(p>0)$$

$B$H$9$k$H!"F1MM$N9M;!$K$h$j!"(B$p>1$ $B$N$H$-$K$O(B $f(x_0)>0$ $B$G$"$k$h$&$J(B $x_0$ $B$KBP$9$kFC@-6J@~(B $x=at+x_0$ $B$N>e$GGzH/$,5/$3$j!"GzH/6-3&$NJ}Dx<0$O(B

$$\{(p-1)t\}^{1/(p-1)}f(x-at)=1$$

$B$H$J$k$3$H$,$o$+$k!#(B

1 $B@~7AGHF0J}Dx<0(B

$$u_{tt}-a^2u_{xx}=\vert u\vert^p$$

$B$N>l9g$b!"2CF#IRIW$i$K$h$j(B ([3]) $p>1$ $B$J$i$P2r$,GzH/$9$k$3$H$,<($5$l$F$$$F!"$3$N>l9g$K$b(B $B3j$i$+$JGzH/6-3&$,B8:_$9$k$3$H$,$o$+$C$F$$$k(B ([4])$B!#(B $B$7$+$7$3$N>l9g$O!"0\N.J}Dx<0$N$h$&$KFC@-6J@~$K$h$C$F2r$rF@$k$3(B $B$H$,$G$-$k$o$1$G$O$J$/!"GzH/6-3&$N<0$,6qBNE*$K=q$-2<$9$3$H$,$G(B $B$-$k$o$1$G$O$J$$!#(B

$x$ $B$Nl9g!"$9$J$o$A(B $n$ $B@~7AGHF0J}Dx<0(B

$$u_{tt}-a^2\Delta u = \vert u\vert^p \hspace{1zw}\left(\Delta... ...tial x_2^2}+\ldots+\frac{\partial^2 u}{\partial x_n^2} \right)$$

$B$K$D$$$F$b!"$3$3?tG/@9$s$K8&5f$5$l!"(B$p$ $B$,$I$N$h$&$JCM$N>l9g$K2r(B $B$,GzH/$9$k!"$H$$$&$3$H$,?'!9D4$Y$i$l$F$$$k!#(B $B$*$*$^$+$K$=$N7k2L$r=R$Y$k$H!"$=$l$>$l$N $B$KBP$7$F(B $p(n)$ $B$H$$$&

$$\begin{eqnarray*} 1<p\leq p(n) & \Rightarrow & \mbox{$B=i4|CM$,>.$5$/$F$b2r$OM-8B(B... ... p> p(n) & \Rightarrow & \mbox{$B=i4|CM$,>.$5$1$l$P2r$OGzH/$7$J$$(B} \end{eqnarray*}$$

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$B$3$N(B $p(n)$ $B$O!"$"$k(B 2 $B, $p(2)=(3+\sqrt{17})/2$, $p(3)=1+\sqrt{2}$ $B$N$h$&$JCM$r $B$H$$$&$N$O!"(B$n=1$ $B$N>l9g$K$O!"(B$p$ $B$r$I$l$[$IBg$-$/ $B$J$i$P$I$s$J$K=i4|CM$,>.$5$/$F$b2r$OM-8B;~4V$GGzH/$9$k!"(B $B$H$$$&0UL#$G$"$k!#(B

$B$J$*!"$3$l$i$NGzH/6-3&(B ($n>1$ $B$N>l9g$OGzH/6J@~$G$O$J$/GzH/6JLL$K$J$k(B) $B$N@-l9g$K$O(B $B3j$i$+$JGzH/6-3&$,B8:_$9$k$+$I$&$+$b$o$+$C$F$$$J$$!#(B