数理解析研究所講究録
第 1542 巻 2007 年 58-76

58

Existence and Nonexistence of the Global Solutions
for a Reaction-Diffusion System

雄介 (Yusuke YAMAUCHI)

山内

北海道大学大学院理学研究科

Department of Mathematics,
Hokkaido University

1. INTRODUCTION

We consider the Cauchy problem for the reaction-diffusion system:(1.1)$u_{t}-\Delta u=|x|^{\sigma_{1}}u^{p_{1}}v^{q_{1}}$,$x\in R^{N},$$t>0$ ,

(1.2)

$v_{t}-\Delta v=|x|^{\sigma_{2}}u^{p_{2}}v^{q_{2}}$,$x\in R^{N},$$t>0$ ,

where

$p_{j},$$u(x, 0)=u_{0}(x)\geq 0,$$\not\equiv 0$$v(x, 0)=v_{0}(x)\geq 0,$$\not\equiv 0$$q_{j}\geq 0,$,,$\sigma_{j}\geq 0(j=1,2)$, and$p_{1},$$x\in R^{N}$,$x\in R^{N}$,$q_{2}\neq 1$.

$q_{j}(j=1,2)$
Our aim is to find conditions on the exponents
for the existence and the nonexistence of global solutions to the system$(1.1)-(1.2)$ .
At first we focus on the single equation:. Let $N$ be thespace dimension. In [5], Fujita proved the existence of global solutions
to the equation if $p>1+2/N$ for exponential decaying small initial$\sigma_{j},$$p_{j},$$u_{t}-\triangle u=u^{p}$data. The author also proved the nonexistence of the global solutions
if $p<1+2/N$ . In the critial case, $p=1+2/N$, the nonexistence isproved in Hayakawa [6], Kobayashi, Sirao and Tanaka [7] and Weissler[11]. On the other hand, in. the sublinear case, i.e. $0<p<1$ , it isshown by Aguirre and Escobedo [1] that every solution for the equation
exists globally in time.
There are various extensions of these results. For example, in [10]Pinsky showed the existence and nonexistence for the equation: $u_{t}-$$\Delta u=a(x)u^{p}$ , where $p>1$ and $a(x)$ behaves likefor large .
$|x|^{m}$

$|x|$

59
Next, we introduce the extended results to the system of the equations:$\{\begin{array}{l}u_{t}-\triangle u=F_{1}(u, v)v_{t}-\triangle v=F_{2}(u, v)\end{array}$Escobedo and Herrero [3] studied the system with the nonlinear teams$F_{1}=v^{p}$ and $F_{2}=u^{q}$ for nonnegative, continuous and bounded initial
data, where$q>0$ . The situation is divided into three cases: (i)$pq>1$ and $( \max\{p, q\}+1)/(pq-1)<N/2$ , (ii) $pq>1$ and $( \max\{p, q\}+$$1)/(pq-1)\geq N/2$ , (iii) $pq<1$ . When $pq>1$ and $( \max\{p,q\}+$$1)/(pq-1)<N/2$ , for small initial data there exist global solutions.
For large data, there exist blowing up solutions. When $pq>1$ and$( \max\{p, q\}+1)/(pq-1)\geq N/2$ , there exist no global solutions. When$pq<1$ , every solution exists globally in time.
Now, we introduce two extenteded results of [3]. One is the result
for the system with nonlinear teams
and$(p, q>1,0\leq\sigma_{j}<N(p_{j}+q_{j}-1), j=1,2)$ . In [9], Mochizuki and
Huang showed the existence and nonexistence result and the assymptotic behavior of the solution.
$q_{j}\geq 0,0<$, whereAnother is for the system with
$p_{1}+q_{1}\leq p_{2}+q_{2}$ for each $j=1,2$ .
In [4], the situation is divided
into two cases, $0\leq p_{1}\leq 1$ and $p_{1}>1$ . In the former case, growthof the solutions by the interaction between two equation is stronger
than self-growth of the solutions. In the latter case, self-growth of thesolutions is stronger. These are understood from the following results:
Put $\alpha=(q_{1}-q_{2}+1)/\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\},$ $\beta=(p_{1}-p_{2}+1)/\{p_{2}q_{1}-$$p,$$F_{1}=|x|^{\sigma_{1}}v^{p}$$F_{j}=u^{p_{j}}v^{q_{j}}$$(1-p_{1})(1-q_{2})\}$$F_{2}=|x|^{\sigma_{2}}u^{q}$$p_{j},$.

(i) Let. If $0 \leq\max\{\alpha,\beta\}<N/2$ , then global solution exists forsmall inItial data. If $\max\{\alpha,\beta\}<0$ , then every solution exists globallyin time.
(ii) Let $p_{1}>1$ . If $p_{1}+q_{1}>1+2/N$ , then global solution exists forsmall initial data.
In (i), the condition for blowing up of the solutions consists of the exponents in both two equations. On the other hand, in (ii) the condition
for consists of only the exponents in one equation.
$p_{1}\leq 1$

60
We study $(1.1)-(1.2)$ as an extention of these systems. Since ourproblem includes the sublinear case,or $q_{j}<1$ , the contraction argument does not work to showing the global existence. In this paper,
we show it by iteration argument in weighted
function space.
To show nonexistence theorems, the iteration argument of [4] is often used for reaction-diffusion systems. However, the method does notseem applicable for our problem because the nonlinear terms have the. In this paper, we improve the argument invariable coefficients[9] and apply it to our problem. The argument in [9] is to transform thesystem of PDEs into the ordinary differential inequalities. In our problem, multiplying the equation by negative power of unknown function
makes the transformation possible.
$p_{j}$$L^{\infty}$$|x|^{\sigma_{j}}$REMARK 1.1. In [3], [4], [9], [6], [7], [10] and [11], the authors showthat the solution blows up in critical case. This critical blow-up also
occurs in our system $(1.1)-(1.2)$ .
2. MAIN RESULTS

For simplicity, let(2.1)$\{\begin{array}{l}\alpha=\frac{q_{1}(\sigma_{2}+2)+(1-q_{2})(\sigma_{1}+2)}{2\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}}\beta=\frac{p_{2}(\sigma_{1}+2)+(1-p_{1})(\sigma_{2}+2)}{2\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}}\end{array}$(2.2)$\{\begin{array}{l}\delta_{1}=\frac{q_{1}\sigma_{2}+(1-q_{2})\sigma_{1}}{p_{2}q_{1}-(1-p_{1})(1-q_{2})}\delta_{2}=\frac{p_{2}\sigma_{1}+(1-p_{1})\sigma_{2}}{p_{2}q_{1}-(1-p_{1})(1-q_{2})}\end{array}$

For $a\in R$ , we define the function spaces:$I^{a}=${ $w\in C(R^{N});w(x)\geq 0$ , lim $sup|x|^{a}w(x)<\infty$ },$|x|arrow\infty$

and$L_{a}^{\infty}=\{w$is measurable function on$w(x)\geq 0,$$R^{N}$;},$\Vert w||_{\infty,a}\equiv\Vert\langle x\rangle^{a}w(x)\Vert_{\infty}<\infty$

61
where

$\langle x\rangle=(1+|x|^{2})^{1/2}$. We also define$E_{T}=\{(u, v);[0, T]arrow L_{\delta_{1}}^{\infty}\cross L_{\delta_{2}}^{\infty}, ||(u, v)||_{E_{T}}<\infty\}$

,

where
$||(u,v) \Vert_{E_{T}}=\sup_{t\in l0,\eta}(\Vert u(t)\Vert_{\infty,\delta_{1}}+||v(t)\Vert_{\infty,\delta_{2}})$

Now, we state our main results.
$(u_{0}, v_{0})\in I^{\delta_{1}}\cross I^{\delta_{2}}$.

.

We assume that the initial data

THEOREM 2.1. Let $p_{1}<1_{j}q_{2}<1$ .
(i) If $\max(\alpha,\beta)\geq N/2_{f}$ then no nontnmal global solutions of $(1.1)-$$(1.2)$ exzst.
(ii) If $0< \max(\alpha, \beta)<N/2$ , then there exist global solutions of $(1.1)-$$(1.2)$ for small initial data, and there exist no global solutions for large
initial data.
(iii) If $\max(\alpha, \beta)<0$ , then every solution of $(1.1)-(1.2)$ enists globdly
in time.
THEOREM 2.2. Let $p_{1}>1,$ $q_{2}<1$ .
(i) If $\alpha\geq N/2$ or $p_{1}+q_{1}\leq 1+(2+\sigma_{1})/N$ , then no nontri,vial global
solutions of $(1.1)-(1.2)$ exist.
(ii) If $\alpha<N/2$ and $p_{1}+q_{1}>1+(2+\sigma_{1})/N$ , then there enist global
solutions of $(1.1)-(1,2)$ for small initial data, and there erist no global
solutions for large initial data.

THEOREM 2.3. Let $p_{1}>1,$ $q_{2}>1$ .
(i) If $p_{1}+q_{1}\leq 1+(2+\sigma_{1})/N$ or $p_{2}+q_{2}\leq 1+(2+\sigma_{2})/N$ , then no
global solutions of $(1.1)-(1.2)$ exist.
(ii) If $p_{1}+q_{1}>1+(2+\sigma_{1})/N$ and $p_{2}+q_{2}>1+(2+\sigma_{2})/N$ , then there
enist global solutions of $(1.1)-(1.2)$ for small initial data, and there
enist no global solutions for large initial data.
$nont_{7}\cdot ivial$

We can also rewrite the theorems into the way in Escobedo-Levine[4].

COROLLARY 2.4. AsSume that
(2.3)$\frac{p_{1}+q_{1}-1}{\sigma_{1}+2}\leq\frac{p_{2}+q_{2}-1}{\sigma_{2}+2}$,

62
and let $p_{1}<1_{f}q_{2}\neq 1$ .
(i) If $\max(\alpha, \beta)\geq N/2_{f}$ then there evzst no global solutions for large
initial data.
(ii) If $0< \max(\alpha, \beta)<N/2$ , then there enist global solutions for small
initial data, and there exist no global solutions for large initial data.
(iii) If $\max(\alpha, \beta)<0$ , every solutions exists globally in time.

COROLLARY 2.5. Assume (2.3), and let $p_{1}>1_{f}q_{2}\neq 1$ .
(i) If $p_{1}+q_{1}\leq 1+(2+\sigma_{1})/N$, then no nontnvial global solutions exist.
(ii) If $p_{1}+q_{1}>1+(2+\sigma_{1})/N_{f}$ then no global solutions exist for largedata.
3. PROOF OF THEOREMS 2.1-2.3 : GLOBAL EXISTENCEFirst, we show the local existence of classical solutions of $(1.1)-(1.2)$ .

THEOREM 3.1. Let and be defined in (2.2). Assume that
. Then there exist classical solutions $(u(t), v(t))\in E_{T}$ for thesystem $(1.1)-(1.2)$ for some $T>0$ .
$\delta_{1}$$(u_{0},v_{0})\in$$\delta_{2}$$I^{\delta_{1}}\cross I^{\delta_{2}}$PROOF. See Theorem 3.1 in [2].

$\square$Next, we introduce a comparison theorem and the existence of supersolutions.

COMPARISON PRINCIPLEPROPOSITION 3.2. Let $f(u, v.)$ and $g(u, v)$ be stri ctly monotone in$v\geq 0$ . Assume thatcreasing in and forare non$u$

$v$

$u,$

$\overline{u},\overline{v},$$\underline{u},$$\underline{v}$negative and $satisk_{J}$$\{\begin{array}{l}\overline{u}_{t}-\Delta\overline{u}\geq|x|^{\sigma_{1}}f(\overline{u},\overline{v})\overline{v}_{t}-\Delta\overline{v}\geq|x|^{\sigma_{2}}g(\overline{u},\overline{v})R^{N}X(0, T)\Delta\underline{u}\leq|x|^{\sigma_{1}}\underline{v}_{t}-\Delta\underline{v}\leq|x|^{\sigma_{2}}g(\underline{u}, \underline{v})\end{array}$$\{\begin{array}{l}\overline{u}(x, O)-\underline{u}(x, 0)\geq 0,\not\equiv 0\overline{v}(x,O)-\underline{v}(x, 0)\geq 0,\not\equiv 0\end{array}$

Then we have$\overline{u}(x, t)\geq\underline{u}(x, t)$and$x\in R^{N}$$\overline{v}(x, t)\geq\underline{v}(x, t)$.

on $R^{N}x(0, T)$ :63
PROOF. See Proposition 4.1 in [2].

口

EXISTENCE OF SUPER-SOLUTIONSPROPOSITION 3.3. (i) Let $p_{1}>1,$ $q_{2}>1$ or $p_{2}q_{1}-(1-p_{1})(1-q_{2})>0$ ,and let $p_{1}+q_{1}>1,$ $p_{2}+q_{2}>1$ . Assume that one of the following
conditions is satisfied:$(A)p_{i},$ $q_{2}>1,$ $p_{1}+q_{1}>1+(2+\sigma_{1})/N,$ $p_{2}+q_{2}>1+(2+\sigma_{2})/N$ .
$(B)p_{1}>1>q_{2\prime}p_{1}+q_{1}>1+(2+\sigma_{1})/N,$ $\alpha<N/2$$(C)p_{1},$ $q_{2}<1,$$p_{2}q_{1}-(1-p_{1})(1-q_{2})>0,$

Then there exist $C_{1}C_{2},$$\alpha_{1}$,$\beta_{1}>0,$$t_{0}>1$$\alpha,$.

$\beta<N/2$.

such that(3.1)$\overline{u}(x, t)=C_{1}(t+t_{0})^{\alpha_{1}-\frac{N}{2}}$exp$(- \frac{|x|^{2}}{4(t+t_{0})})$(3.2)$\overline{v}(x, t)=C_{2}(t+t_{0})^{\beta_{1}-\frac{N}{2}}$exp$(- \frac{|x|^{2}}{4(t+t_{0})})$,

are super-solutions of $(1.1)-(1.2)$ .
(ii) Let $p_{1}>1,$ $q_{2}>1orp_{2}q_{1}-(1-p_{1})(1-q_{2})>0$ . And let $p_{1}+q_{1}>1$ ,$p_{2}+q_{2}\leq 1$ . Assume that oneof the following conditions is satisfied:$(D)p_{1}>1>q_{2},$ $p_{1}+q_{1}>1+(2+\sigma_{1})/N,$ $\alpha<N/2$ ,$(E)p_{1},$ $q_{2}\leq 1,$ $p_{2}q_{1}-(1-p_{1})(1-q_{2})>0_{f}\alpha,$ $\beta<N/2$

Then there vist

$C_{1}C_{2z}\alpha_{1},$$\beta_{1}>0,$.

$t_{0}>1_{f}a>0$ such that

(3.3)$\overline{u}(x, t)=C_{1}(t+t_{0})^{\alpha_{1}-N}\tau$exp$(- \frac{|x|^{2}}{4(t+t_{0})})$(3.4)$\overline{v}(x,t)=C_{2}(t+t_{0})^{\beta_{1}-\frac{Na}{2}}$exp$(- \frac{a|x|^{2}}{4(t+t_{0})})$

are $suPer$-solutions of $(1.1)-(1.2)$ .

,,(iii) $Letp_{1}<1_{f}q_{2}<1andp_{2}q_{1}-(1-p_{1})(1-q_{2})<0$ . Then there
enist $C_{1}C_{2},$ $k,$ $a>0$ such that

(3.5)$\overline{u}(x,t)=C_{1}\langle x\rangle^{-2\delta_{1}}$exp $(kt)$ ,(3.6)$\overline{v}(x, t)=C_{2}\langle x\rangle^{-2\delta_{2}}$exp $(akt)$ ,are super-solutions of $(1.1)-(1.2)$ .

64
Proof of Proposition 3.3 (i) Put(3.7)$\overline{u}(x, t)=C_{1}(t+t_{0})^{\alpha_{1}}$$(- \frac{|x|^{2}}{4(t+t_{0})})$

,

(3.8)

$\overline{v}(x, t)=C_{2}(t+t_{0})^{\beta_{1}-\frac{N}{2}}\exp(-\frac{|x|^{2}}{4(t+t_{0})})$,“

where $C_{1}C_{2},$$\alpha_{1},$$\beta_{1}>0,$

super-solutions for small$t_{0}>1$$C_{1},$$\frac{N}{2}$exp. We can see that

$C_{2}>0$

are global$(\overline{u},\overline{v})$and large $t_{0}>1$ , provided that

(3.9)$\{\begin{array}{ll}\alpha_{1}-N/2-1>p_{1}(\alpha_{1}-N/2)+q_{1}(\beta_{1}-N/2)-\sigma_{1}/2, and\beta_{1}-N/2-1>p_{2}(\alpha_{1}-N/2)+q_{2}(\beta_{1}-N/2)-\sigma_{2}/2, \end{array}$which (3.9) is equivalent to(3.10)$(p_{1}-1)\alpha_{1}+q_{1}\beta_{1}<(p_{1}+q_{1}-1)N/2-(\sigma_{1}+2)/2$, and(3.11)$p_{2}\alpha_{1}+(q_{2}-1)\beta_{1}<(p_{2}+q_{2}-1)N/2-(\sigma_{2}+2)/2$.

-plane inon theNow, we shall show the existence ofeach case of Proposition 3.3.
$q_{2}>1,$ $p_{1}+q_{1}>1+(2+\sigma_{1})/N,$ $p_{2}+q_{2}>1+(2+\sigma_{2})/N$ .
Case (A):Since the right hand sides of (3.10) and (3.11) are positive, we can takesatisfying (3.10) and (3.11).
small$\alpha_{1},$$\beta_{1}>0$$(\alpha_{1}, \beta_{1})$$p_{1},$$\alpha_{1},$$\beta_{1}>0$

Case (B): $p_{1}>1>q_{2},$ $p_{1}+q_{1}>1+(2+\sigma_{1})/N,$ $\alpha<N/2$ .
We remark that the intersection of (3.10) and (3.11) is$(N/2-\alpha, N/2-\beta)$ . From the assumption, we can see that the in-$(\alpha_{1}, \beta_{1})=$

tersection lies above the -axis and that the boundary of (3.10) lies$\{(p_{1}+q_{1}-1)N/2-$, putabove the origin. For$(\sigma_{1}+2)/2\}/q_{1}+\epsilon_{2})$ . Then there exist small constantssuchsatisfy (3.10) and (3.11).
that$q_{2}<1,$ $p_{2}q_{1}-(1-p_{1})(1-q_{2})>0,$ $\alpha,\beta<N/2$ .
Case (C):From the assumption, we can see that the intersection lies in the first$q_{2}<1$ and $p_{2}q_{1}-(1-p_{1})(1-q_{1})>0$ , we havequadrant. Since$(1-p_{1})/q_{1}<p_{2}/(1-q_{2})$ , that is, the angular coefficient of (3.11) islarger than that of (3.10). Hence, there exist small constants
such that $(\alpha_{1}, \beta_{1})=(N/2-\alpha-\epsilon_{1}, N/2-\beta-\epsilon_{2})$ satisfy (3.10) and$\alpha_{1}$$\epsilon_{1},$$\epsilon_{2}>0$$(\alpha_{1}, \beta_{1})=(\epsilon_{1},$$\epsilon_{1},$$\epsilon_{2}>0$$(\alpha_{1}, \beta_{1})$$p_{1},$$p_{1},$$\epsilon_{1},$$\epsilon_{2}>0$65(3.11).

$\square$

Proof of Proposition 3.3 (ii) Case (D): $p_{1}>1>q_{2},$ $p_{1}+q_{1}>$$1+(2+\sigma_{1})/N,$ $\alpha<N/2$.

Put $a>0$ such that
(3.12)$\max\{0,$ $\frac{(1-p_{1})N+(\sigma_{1}+2)}{q_{1}N}\}<a<\frac{p_{2}}{1-q_{2}}$In fact, since $q_{2}<1,$ $p_{2}q_{1}-(1-p_{1})(1-q_{2})>0$ and $\alpha<N/2$ , we have$\frac{p_{2}}{1-q_{2}}-\frac{(1-p_{1})N+(\sigma_{1}+2)}{q_{1}N}$$= \frac{1}{Nq_{1}(1-q_{2})}\{Nq_{1}p_{2}-N(1-q_{2})(1-p_{1})-(1-q_{2})(\sigma_{1}+2)\}$$= \frac{2\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}}{Nq_{1}(1-q_{2})}\{\frac{N}{2}$一$\frac{(1-q_{2})(\sigma_{1}+2)}{2(p_{2}q_{1}-(1-p_{1})(1-q_{2}))}\}$$\geq\frac{2\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}}{Nq_{1}(1-q_{2})}(\frac{N}{2}-\alpha)$

$>0$

.

Therefore we can take $a>0$ satysfying (3.12). Let(3.13)$\overline{u}(x, t)=C_{1}(t+t_{0})^{\alpha_{1}-\frac{N}{2}}$exp$(- \frac{|x|^{2}}{4(t+t_{0})})$(3.14)$\overline{v}(x, t)=C_{2}(t+t_{0})^{\beta_{1}-\frac{Na}{2}}$exp$(- \frac{a|x|^{2}}{4(t+t_{0})})$$\beta_{1}>0,$ $t_{0}>1$ .
We can see that
where $C_{1}C_{2},$super-solutions provided that$\alpha_{1},$,,$(\overline{u},\overline{v})$are globai(3.15)$\{\begin{array}{l}\alpha_{1}-N/2-1>p_{1}(\alpha_{1}-N/2)+q_{1}(\beta_{1}-Na/2)-\sigma_{1}/2\beta_{1}-Na/2-1>p_{2}(\alpha_{1}-N/2)+q_{2}(\beta_{1}-Na/2)-\sigma_{2}/2\end{array}$and large $t_{0}>1$ . And (3.15) is equivalent tofor small$C_{1},$(3.16)$(p_{1}-1)\alpha_{1}+q_{1}\beta_{1}<(p_{1}+aq_{1}-1)N/2-(\sigma_{1}+2)/2$, and(3.17)$p_{2}\alpha_{1}+(q_{2}-1)\beta_{1}<(p_{2}+aq_{2}-a)N/2-(\sigma_{2}+2)/2$.

$C_{2}>0$

66
We remark that the intersection of$(p_{1}-1)\alpha_{1}+q_{1}\beta_{1}=(p_{1}+aq_{1}-1)N/2-(\sigma_{1}+2)/2$, and$p_{2}\alpha_{1}+(q_{2}-1)\beta_{1}=(p_{2}+aq_{2}-a)N/2-(\sigma_{2}+2)/2$.

is $(\alpha_{1}, \beta_{1})=(N/2-\alpha, Na/2-\beta)$ . From the \"assumption $\alpha<N/2$ ,we see that the intersection lies above the -axis. From $a>\{(1-$$p_{1})N+(\sigma_{1}+2)\}/q_{1}N$ , we can easily see that the boundary of (3.16)
lies above the origin. Hence, we can prove the existence ofsatisfying (3.16) and (3.17) in the same way as in Case (B).
$\alpha_{1}$$(\alpha_{1}, \beta_{1})$Case (E):$p_{1},$$q_{2}\leq 1,$$p_{2}q_{1}-(1-p_{1})(1-q_{2})>0,$ $\alpha,\beta<N/2$

Putting $a>0$ satisfying(3.18)$\max\{\frac{1-p_{1}}{q_{1}},$ $\frac{2\beta}{N}\}<a<\frac{p_{2}}{1-q_{2}}$,we can prove in the same way as in Case (C). In fact, since $q_{2}<1$ ,$p_{2}q_{1}-(1-p_{1})(1-q_{2})>0$ and $\alpha<N/2$ , we have$\frac{p_{2}}{1-q_{2}}-\frac{2\beta}{N}$$=^{p_{2}N\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}-(1}\ovalbox{\tt\small REJECT}^{-p_{1})p_{2}(\sigma_{1}+2)-(1-p_{1})(1-q_{2})(\sigma_{2}+2)}(1-q_{2})\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}$$=^{p_{2}N\{p_{2}q_{1}-(1-p_{1})(1}\ovalbox{\tt\small REJECT}^{-q_{2})\}-(1-p_{1})p_{2}(\sigma_{1}+2)-p_{2}q_{1}(\sigma_{2}+2)}(1-q_{2})\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}$$+^{p_{2}q_{1}(\sigma_{2}+2)}\ovalbox{\tt\small REJECT}^{-(1-p_{1})(1-q_{2})(\sigma_{2}+2)}(1-q_{2})\{p_{2}q_{1}-(1-p_{1})(1-q_{2})\}$$= \frac{2p_{2}N}{1-q_{2}}(\frac{N}{2}-\alpha)+\frac{\sigma_{2}+2}{1-q_{2}}$$>0$ ,and since$p_{1},$$p_{2}/(1-q_{2})$$q_{2}\leq 1,$$p_{2}q_{1}\cdot-(1-p_{1})(1-q_{2})>0$, we have $(1-p_{1})/q_{1}<$. Therefore, we can take $a>0$ satisfying (3.18).

Proof of Proposition 3.3 (iii) Let$a=\overline{1}-p_{\frac{2}{q_{2}}}$. Put(3.19)$\overline{u}(x, t)=C_{1}\langle x\rangle^{-2\delta_{1}}$exp $(kt)$ ,(3.20)$\overline{v}(x, t)=C_{2}\langle x\rangle^{-2\delta_{2}}$exp $(akt)$ ,$\square$67$k>0$ . We can see that
wherefor large $k>0$ .
$C_{1}C_{2},$$(\overline{u},\overline{v})$are global super-solutions

We are now in a position to prove the global existence theorems.

Proof of Theorems 2.1(i), 2.2 and 2.3. Let
be the maximal
existenoe time of the classical solutions for $(1.1)-(1.2)$ . Erom the local
existence theorem in Section 3, it is clear that $\tau*\neq 0$ . Assume $\tau*<\infty$ .
If the initial data
are sufficiently small, then the solutions $(u, v)$
are estimated above by the super-solutions in Proposition 3.3. Using
Theorem 3.1, we can extend the solutions $(u, v)$ with new initial data$(u(T^{*}), v(T^{*}))$ to time $\tau**>\tau*$ . This contradicts the maximality of. Hence $\tau*=\infty$ .
$\tau*$$(u_{0}, v_{0})$$\tau*$$\square$

Proof of Theorem 2.1 (ii). The constants
and $C_{2}>0$ in
Proposition 3.3 (iii) have no restriction. Hence, the argument as above
works for arbitrary initial data in.
$C_{1}$$I^{\delta_{1}}\cross I^{\delta_{2}}$$\square$

4. PRELIMINARIES TO NONEXISTENCE THEOREMSIn this section, we prepare several estimates for the solutions. To
show them, we introduce the system of integral equations associatedto (1.1) and (1.2):(4.1)$u(t)=S(t)u_{0}+ \int_{0}^{t}S(t-s)|\cdot|^{\sigma_{1}}u(s)^{p_{1}}v(s)^{q_{1}}ds$(4.2)$v(t)=S(t)v_{0}+ \int_{0}^{t}S(t-s)|$.

$|^{\sigma_{2}}u(s)^{p_{2}}v(s)^{q_{2}}ds$

,
,

where
$S(t)f(x)=(4 \pi t)^{-\frac{N}{2}}\int_{R^{N}}$exp $(- \frac{|x-y|^{2}}{4t})f(y)dy$.

The following lemma is a well-known estimate for the heat equations.

68
LEMMA 4.1. Let and be solutions of the system (1.1) and (1.2).
There exists $C>0$ such that
$u$

$v$

exp

,

$(t>0)$ ,$v(x, t)\geq C(1+t)^{--r}N$ exp $(- \frac{|x|^{2}}{2t}I,$$(t>0)$ .

$u(x, t)\geq C(1+t)^{-\frac{N}{2}}$$(- \frac{|x|^{2}}{2t})$

Moreover, we can add logarithmic growth to the bounds in the critical case.

LEMMA 4.2. ([4]) Let(1.2). Assume that

$u$

and

$u(x, t)\geq C_{1}(1+t)^{-\frac{N}{2}}$$v(x, t)\geq C_{2}(1+t)^{m}$where$C_{1},$exp

exp$C_{2_{f}}C_{3}>0_{f}t_{0}\geq 0$

be solutions of the system (1.1) and$v$$(- \frac{|x|^{2}}{t})$$(- \frac{C_{3}|x|^{2}}{t})$$C_{4},$$(t>0)$ ,,$(t>t_{0})$and $m\in R$ . If $m$ and$- \frac{Np_{1}}{2}+mq_{1}+\frac{\sigma_{1}+2}{2}=-\frac{N}{2},$then there exist constants,$C_{5}>0$$u(x, t)\geq C_{4}(1+t)^{-\tau}\log(1N+t)$$\sigma_{1}$,satish$\sigma_{1}>\max(-2, -N)$,

and $t_{1}>t_{0}$ such thatexp$(- \frac{C_{5}|x|^{2}}{t})$,$(t>t_{1})$PROOF. See Proposition 1 in [4].

.
$\square$

The following two lemmas are for the sublinear case.

LEMMA 4.3. Let $0\leq q_{2}<1,$ $\sigma_{2}>\max(-2, -N)$ and define$\overline{v}(x, t)=\overline{c}t^{21-\neg}q_{2}(S(t)u_{0}(x)^{\epsilon})^{\epsilon 1q_{2}}\sigma+2$

for

. If

and
lution for the problem:$\overline{C},$$\epsilon>0$$\tilde{C}$$\epsilon$

are sufficiently small, then$v_{t}-\Delta v=|x|^{\sigma_{2}}u^{p_{2}}v^{q_{2}}$$v(x, 0)=v_{0}(x)$ ,,$x\in R^{N},$$x\in R^{N}$.

$\overline{v}(x, t)$$t>0$ ,

is a subso-

69
PROOF. Let $k> \max\{(\sigma_{2}+N)/N, 1\}$ and $0< \epsilon<\min(1,p_{2}/\{(1-$. It suffices to prove that$q_{2})k\})$$\overline{v}(x, t)\leq\int_{0}^{t}S(t-s)|x|^{\sigma_{2}}(S(s)u_{0}(x))^{p_{2}}\overline{v}(x, s)^{q_{2}}ds$.

By Jensen’s inequality, we have$\int_{0}^{t}S(t-s)|x|^{\sigma_{2}}(S(s)u_{0}(x))^{p_{2}}\overline{v}(x, s)^{q_{2}}ds$(4.3)$\geq\overline{C}^{q_{2}}\int_{0}^{t_{q(}}s^{1-q_{2}}\mapsto^{\sigma+2)}S(t-s)|x|^{\sigma_{2}}(S(s)u_{0}(x)^{\epsilon})$社$\iota_{-q}^{p}\urcorner_{ds}$.

Using the inverse H\"older inequality and Jensen’s inequality again, we
have for $k>1$ ,$S(t-s)|x|^{\sigma_{2}}(S(s)u_{0}(x)^{\epsilon})^{\frac{p2}{\epsilon(1-q_{2})}}$$\geq\{S(t-s)|x|^{B}\overline{1}-\overline{k}\}^{1-k}\{S(t-s)\sigma(S(s)u_{0}(x)^{\epsilon})^{\frac{p2}{ke(1-q_{2})}}\}^{k}$$\geq\{C_{1}(t-s)\}^{1-k}\{S(t-s)(S(s)u_{0}(x)^{\epsilon})\}^{e1-q_{2}\neg}$(4.4)$=C_{1}1-k^{\underline{\sigma}_{2}}(t-s)2(S(t)u_{0}(x)^{\epsilon})^{\epsilon 1-q_{2}}p\neq 7$.

Substituting (4.4) into (4.3), we obtain$\int_{0}^{t}S(t-s)|x|^{\sigma_{2}}(S(s)u_{0}(x))^{p_{2}}\overline{v}(x, s)^{q_{2}}ds$$\geq\tilde{C}^{q_{2}}C_{1}^{1-k}(S(t)u_{0}(x)^{\epsilon})^{\epsilon 1-q_{2}}\neq p\urcorner\int_{0}^{t_{q(\sigma+2)}}s^{2(1-q_{2})}(t-s)^{\text{血}}ds\infty$$\geq\overline{C}^{q_{2}}C_{1}^{1-k}C_{2}t^{21-\neg}q_{2}(S(t)u_{0}(x)^{\epsilon})^{\epsilon 1-q_{2}T}\sigma+2p$$=\tilde{C}^{q_{2}-1}C_{1}^{1-k}C_{2}\overline{v}(x, t)$$\geq\overline{v}(x, t)$

for sufficiently small$\tilde{C}>0$. This completes the proof.

$\square$

LEMMA 4.4. Let $0\leq q_{2}<1$ , and. $\sigma_{2}>(-2, -N)$ and let and be
solutions of the system (1.1) and (1.2). Then there exist constants ,$C_{2}>0$ such that
$u$

$v$

$C_{1}$$v(x, t)\geq C_{1}t^{21}\urcorner(1\sigma_{B_{\frac{+2}{-q_{2})}}}+t)^{-\neq}21-q_{2}\urcorner pN$

exp$(- \frac{C_{2}|x|^{2}}{t})$,$(t>0)$ .

70
PROOF. Fix arbitrary $s>0$ , and apply Lemma 4.3 to $U(t)=u(t+s)$
and $V(t)=v(t+s)$ . Then, we have$V(x, t)\geq Ct^{21-\neg}\neq\sigma+2q_{2}(S(t)U(x, 0)^{\epsilon})^{\epsilon 1-q_{2}}\neq p\neg$.

Putting $s=t$ and using Lemma 4.1, we obtain$v(x, 2t)\geq Ct^{21-\neg}\neq\sigma+2q_{2}(S(t)u(x, t)^{\epsilon})^{\epsilon 1-q_{2}\urcorner}\neq L$$\geq c_{t^{21-q\neg(1+t)^{-\neq}q_{2}}}^{\sigma+2}\neq 221-\urcorner pN\{(4\pi t)^{-\tau}N_{-}/\exp(-\frac{|x-y|^{2}}{4t}-\frac{\epsilon|y|^{2}}{2t})dy\}^{\epsilon}\urcorner p+$$\geq q_{2}\sigma+2T^{p}N\mapsto 2$

exp$(- \frac{C|x|^{2}}{t})$.

This completes the proof.

$\square$

5. PROOF OF THEOREM 2.1 : NONEXISTENCE
In this section we prove Theorem 2.1 (i). For Theorem 2.1 (ii) and(iii), see [12].

NECESSARY CONDITION FOR THE GLOBAL EXISTENCE Assume
that $(u, v)$ are global solutions for (1.1) and (1.2). Sinoe $p_{1}<1,$ $q_{2}<1$
and $p_{2}q_{1}-(1-p_{1})(1-q_{2})>0$ , we can take a positive constant $k>0$
such that $(1-q_{2})/p_{2}<k<q_{1}/(1-p_{1})$ . For this , fix positive constants$k$$r_{1},$$r_{2}>0$

satisfying$r_{2}=kr_{1}$$r_{1}<$,

min $\{1-p_{1}, p_{2}\}$ ,$r_{2}< \min\{1-q_{2}, q_{1}\}$,$r_{1} \sigma_{1}<\frac{N(q_{1}-k(1-p_{1}))}{k}$,$r_{2} \sigma_{2}<\frac{N(kp_{2}-(1-q_{2}))}{k}$

.

71
For

$\epsilon>0$, define the cut off function$\rho_{\epsilon}(x)=\{\begin{array}{ll}\epsilon^{\frac{N}{2}} exp (-\frac{1}{1-\epsilon|x|^{2}}) (|x|<\epsilon^{-\frac{1}{2}})0 (|x|\geq\epsilon^{-\frac{1}{2}}),\end{array}$and set(5.1)$F_{\epsilon}(t)= \int_{R^{N}}u(x, t)^{r_{1}}\rho_{\epsilon}(x)dx$

,

(5.2)

$G_{\epsilon}(t)= \int_{R^{N}}v(x, t)^{r_{2}}\rho_{\epsilon}(x)dx$.

Then the following inequalities hold.
LEMMA 5.1. Let $p_{1}<1,$ $q_{2}<1$ and $\sigma_{j}>-N(j=1,2)$ . Then there$C_{4}>0$ such thatenist constants$C_{1},$

(5.3)

(5.4)

$C_{2},$$C_{3},$$F_{\epsilon}’(t)\geq-C_{1}\epsilon F_{\epsilon}(t)+C_{2}\epsilon-\sigma_{2}\lrcorner^{(1-p)-r}F_{\epsilon}(t)^{r_{1}}G_{\epsilon}(t)^{r}q_{2}$$G_{\epsilon}’(t)\geq-C_{3}\epsilon G_{\epsilon}(t)+C_{4}^{\underline{\sigma}_{2}}\epsilon^{-a\frac{p}{r}2^{(1)-r}}F_{\epsilon}(t)\iota G_{\epsilon}(t)^{-j}2$.

with, and integrating over
PROOF. Multiplying (1.1) byrespect to , we obtain the desired inequality (5.3). Indeed, integration
by parts implies that
$R^{N}$$u^{r_{1}-1}\rho_{\epsilon}$$x$$\int_{R^{N}}\rho_{\epsilon}u^{r_{1}-1}u_{t}dx=\frac{1}{r_{1}}\frac{d}{dt}F_{\epsilon}(t)$,$\int_{R^{N}}\rho_{\epsilon}u^{r_{1}-1}\triangle udx\geq-\int_{R^{N}}\nabla\rho_{\epsilon}\cdot u^{r_{1}-1}\nabla udx$$=- \frac{1}{r_{1}}\int_{R^{N}}\nabla\rho_{\epsilon}\cdot\nabla(u^{r_{1}})dx$$= \frac{1}{r_{1}}\int_{R^{N}}u^{r_{1}}\Delta\rho_{\epsilon}dx$$\geq-\frac{C\epsilon}{r_{1}}F_{\epsilon}(t)$Here, we have used the property of$C>0$ depending only on $N$ such that$\rho_{\epsilon}$.

that tbere exists a constant. The normal and$\Delta\rho_{\epsilon}\geq-C\epsilon\rho_{\epsilon}$

72
inverse H\"older inequalities also imply that$\int_{R^{N}}\rho_{\epsilon}|x|^{\sigma_{1}}u^{r_{1}-(1-p_{1})}v^{q_{1}}dx$$\geq(\int_{|x|<\epsilon z}-1\rho_{\epsilon}v^{r_{2}}dx)^{\frac{q}{r}\perp}2(\int_{|x|<\epsilon^{-:^{\rho_{\epsilon}|x|^{r_{2}-q1}u^{r_{2}-q_{1}}dx}}}A^{r(r})^{r}B\frac{-q}{r_{2}}$$\geq G_{\epsilon}^{r_{2}}q\lrcorner(\int_{|x|<\epsilon^{-t^{\rho_{\epsilon}u^{r_{1}}dx}}})^{\frac{r-(1-p)}{r_{1}}}(\int_{|x|<\epsilon^{-t^{\rho_{\epsilon}|x|^{r_{1}q_{1}-r}dx}}}-rrR_{2}^{\sigma_{1-p_{1}}})^{r_{1}r_{2}}-\ovalbox{\tt\small REJECT}$$=c^{\sigma}\neq 1-p)-r$

Multiplying (1.2) by
to , we can also get (5.4).

$v^{r_{2}-1}’\rho_{\epsilon}$, and integrating over$R^{N}$

with respect$\square$$x$

Setting$\overline{F_{\epsilon}}(t)=F_{\epsilon^{1}}^{\frac{1-}{r}p}(t)\lrcorner$$\overline{G_{\epsilon}}(t)=G_{\epsilon}^{r_{2}}(t)\underline{1}-\simeq q$

we simplify the inequalities (5.3) and (5.4).
LEMMA 5.2. Let $p_{1}<1,$ $q_{2}<1$ and $\sigma_{j}>-N(j=1,2)$ . Then there$C_{8}>0$ such thatenist constants$C_{5},$$C_{6},$$C_{7},$$\overline{F_{\epsilon}}’(t)\geq-C_{5}\epsilon\overline{F_{\epsilon}}(t)+c_{6^{\mathcal{E}^{-\lrcorner^{\frac{q}{1-q}-}}}}^{\sigma_{2\overline{G_{\epsilon}}(t)2}}$$\overline{G_{\epsilon}}(t)\geq-C_{7}\epsilon\overline{G_{\epsilon}}(t)+C_{8}^{\underline{\sigma_{2}}}\epsilon^{-a^{p_{B-}}}\overline{F_{\epsilon}}(t)^{\overline{1}-p_{1}}’$,.

From the phase field argument in [9], we get upper bounds ofas follows:
and$G_{\epsilon}(t)$PROPOSITION 5.3. Let $p_{1}<1,$ $q_{2}<1$ and $\sigma_{j}>-N(j=1,2)$ .
(i) There exist constants $A>0$ and $B>0$ such that
(5.5)

(5.6)

$\overline{F_{\epsilon}}(t)\leq A\epsilon^{\alpha(1-p_{1})}$,$\overline{G_{\epsilon}}(t)\leq B\epsilon^{\beta(1-q_{2})}$,$F_{\epsilon}(t)$73
and are defined in (2.1).
(ii) (upperbounds) There exist constants $A>0$ and $B>0$ such thatfor all $t>0$ and $6>0$ , where$\beta$$\alpha$(5.7)$F_{\epsilon}(t)\leq A\epsilon^{\alpha r_{1}}$

,

(5.8)

$G_{\epsilon}(t)\leq B\epsilon^{\beta r_{2}}$,for all $t>0$ and$\epsilon>0$.

We COnSider the case $\alpha\geq N/2$ .
PROOF OF THEOREM 2.1(i).
Lemmas 4.1, 4.2, 4.4, and the definition of in (5.1) give lower bounds:of$F_{\epsilon}$$F_{\epsilon}(\epsilon^{-1})$(5.9)$F_{\epsilon}(\epsilon^{-1})\geq\{\begin{array}{ll}C_{5}\epsilon^{\frac{Nr}{2}}, (\alpha>\frac{N}{2}),C_{6}\epsilon^{\frac{Nr}{2}}\log(1+\epsilon^{-1}), (\alpha=\frac{N}{2}).\end{array}$Indeed, in the critical case $\alpha=N/2$ , we have$u(x, t)\leq C(1+t)^{-\frac{N}{2}}$exp$v(x, t)\leq C(1+t)\mapsto^{\sigma_{2^{+}\iota_{-q_{2}}^{2-pN}}}$$(- \frac{|x|^{2}}{t})$exp$(t>0)$ ,,$(- \frac{C|x|^{2}}{t})$,$(t>1)$

from Lemmas 4.1 and 4.4. Applying Lemma 4.2, we have
(5.10)$u(x, t)\leq C(1+t)^{-\frac{N}{2}}\log(1+t)$

exp$(- \frac{|x|^{2}}{\backslash t})$,$(t>t_{0})$for some $t_{0}>1$ . Substituting (5.10) into (5.1), we obtain (5.9). This
contradicts (5.7) for small. This completes the proof.
$\epsilon>0$$\square$

6. PROOFS OF THEOREMS 2.2 AND 2.3 : NONEXISTENCE
In this section we prove Theorems 2.2 (i) and 2.3 (i). In order toprove the theorems, it suffices to show the following propositions.

PROPOSITION 6.1. Let $p_{1}>1,$ $q_{2}<1$ . If $\alpha\geq N/2$ , then no nontrivial global solutions exist.

PROPOSITION 6.2. Let $p_{1}>1$ . If $p_{1}+q_{1}\leq 1+\cdot(2+\sigma_{1})/N$, then no
nontrivial global solutions exist.

74
NECESSARY CONDITION FOR THE GLOBAL EXISTENCE Assume
that $(u, v)$ are global solutions for (1.1) and (1.2). For, define$\epsilon>0$(6.1)$F_{\epsilon}(t)= \int_{R^{N}}u(x, t)^{r}\rho_{\epsilon}(x)dx$,.
where $r>0$ satisfying
Multiplying (1.1) by $p_{e}(x)u^{r-1}$ and integrating by parts, we have$r \sigma_{1}<\frac{N}{p_{1}-1}$$(t\geq\epsilon^{-1})$

,

where
and $C_{2}>0$ . Indeed, from the inverse H\"older inequality andLemma 4.4,
$C_{1}$$\int_{R^{N}}\rho_{\epsilon}|x|^{\sigma_{1}}u^{r+p_{1}-1}v^{q_{1}}dx$$\geq(\int_{R^{N}}\rho_{\epsilon}u^{r}dx)^{\underline{r+p}_{\frac{-1}{r}}}(\int_{R^{N}}\rho_{\epsilon}|x|arrow^{r\sigma}1\frac{rq}{1-}\llcorner\underline{1}p$Puttingyields the following inequality:$(s\geq 1)$

A comparison argument and the global existence of$\overline{F_{\epsilon}}(1)\leq K$,

where $K>0$ is independent of $0<\epsilon\leq 1$ . Hence,(6.2)

for $0<\epsilon\leq 1$ .

$\overline{F_{\epsilon}}(s)$.

imply that75
PROOF OF PROPOSITION 6.1. Lemmas 4.1 and 4.2, and the defi:
nition ofin (6.1) give lower bounds of$F_{\epsilon}(\epsilon^{-1})$$F_{\epsilon}$$F_{\epsilon}(\epsilon^{-1})\geq\{\begin{array}{ll}C_{3}\epsilon^{\frac{Nr}{2}}, (\alpha>\frac{N}{2}),C_{4}\epsilon^{\frac{Nr}{2}}\log(1+\epsilon^{-1}), (\alpha=\frac{N}{2}),\end{array}$which contradicts (6.2) for small
is equivalent to$\epsilon>0$. Indeed, one can see that$\alpha\geq\frac{N}{2}$$- \frac{q_{1}(\sigma_{2}+2)+(1-q_{2})(\sigma_{1}+2)-p_{2}q_{1}N}{2(1-p_{1})(1-q_{2})}\geq\frac{N}{2}$This completes the proof.

$\square$

PROOF OF PROPOSITION 6.2. Using Lemma 4.1 instead of Lemma
4.4 for the estimate of $v(x, t)$ , we can prove Proposition 6.2 in the same
way as the proof of Proposition 6.1.
$\square$

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76[10] Ross G.Pinsky, Existence and nonexistence of global solutions for $u_{t}=\Delta u+$$a(x)u^{P}$ in, J. Diff. Eqns. 133 (1997), 152-177.
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$R^{d}$

DEPARTMENT OF MATHEMATICSHOKKAIDO UNIVERSITYSAPPORO 060-0810
JAPAN
E-mail address: yamauchiQmath. sci. hokudai. ac. jp