数理解析研究所講究録
第 829 巻 1993 年 88-97

88

On the multiplicity of periodic solutions
for semilinear parabolic equations

Norimichi Hirano(横浜国立大学・工)Abstract.
In the present paper, we consider the multiple existence of Tperiodic solutions of semilinear parabolic equations.

1. Introduction.

. Let
be a bounded domain with a smooth boundaryLetbe a second order uniformly strongly elliptic operator of the form$\Omega\subset R^{N}$$\partial\Omega$$L$$Lu=- \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i}}(a_{ij}(x)\frac{\partial u}{\partial x_{\dot{j}}})$

where the cofficient functions $a_{ij}=a_{ji}$ are real valued functions inand satisfies$L^{\infty}(\Omega)$$\sum_{i,j}a_{ij}(x)\xi_{i}\xi\supset\geq C|\xi|^{2}$for all$\xi\in R^{n}$and$x\in\Omega$for some $C>0$ . We impose the Dirichlet boundary condition on
That is$D(L)=$ { $u\in L^{2}(\Omega):Lu\in L^{2}(\Omega)$ ,$u(x)=0$

on$\partial\Omega$

$L$

.

}

Our purpose in note is to report on the multiple existence resultof solutions for the problem of the form

89$\frac{du}{dt}+Lu-g(u)=f(t)$ ,

$(P)$

$t>0$

$u(0)=u(T)$ ,)is a T-periodic function and : $Rarrow$
Here $T>0,$ :is a continuous function with $g(O)=0$ .
The existence of periodic solutions for problems of this kind has
been studied by many authors.(See Amann[l] which also contains
many references.) For the multiple existence of the periodic solutions, Amann[l] established a multiplicity result for the problem (P).
To find a solution of (P), we can make use of two approaches. One
way is to work with Poincare map and find fixed points. Another way
is to find sub-and supersolutions of the problem (P). If one can find a, there exsubsolution and a supersolution tt of (P) satisfying
ists a solution of (P) between and . The method employed in [1] isbased on the super-subsolution method. In [6], the author considered
the multiple exitstence of solutions of (P) by usig the Schauder’s fixed
point theorem and results for multiple solutions of nonlinear elliptic
equations(cf. [2], [3], and [4]). In the present paper, we study themultiplicity of solutions for (P) by using the argument in [6] and thedegree theory for compact mappings.
$f$$[0, \infty$$arrow L^{2}(\Omega)$

$g$

$R$

$\underline{u}<\overline{u}$$\underline{u}$$\overline{u}$$\underline{u}$To state our result, we need some notations. We denote by.
stand for the eigenvalues of thethe norm of
self-adjoint realization inof . The norm ofis given by$|\cdot|$$L^{2}(\Omega)$$0<\lambda_{1}<\lambda_{2}\leq\cdots$$L^{2}(\Omega)$$L$$H_{0}^{1}(\Omega)$

for$||v||^{2}=\langle Lv, v\rangle$$v\in H_{0}^{1}(\Omega)$.

The norm defined above is an equivalent norm with the usual normof
. $W^{1,p}(0, T;X)(1\leq p\leq\infty)$ stands for the space of functions $u\in L^{p}(0, T;X)$ with $du/dt\in L^{p}(0, T;X)$ , where $du/dt$ is thederivative in the sence of distribution.
$H_{0^{1}}(\Omega)$

We can now state our main result.
Theorem. Suppose th at
(g1)

$g$

sa tisfes the following $con$ ditions:$\sup_{t\in R}g’(t)<\lambda_{2}$

,

90
(g2)

$g’(\pm\infty)<\lambda_{1}<g’(0)<\lambda_{2}$,where $g’( \pm\infty)=\lim_{tarrow\pm\infty}g’(t)$ . Then ther exis $tsM>0$ such th atfor each T-periodic fun ction$e$$f\in W^{1,\infty}(0, T;L^{2}(\Omega))$sa tisfyin$\sup\{|f(t)|:t\in[0, T]\}\leq M$ ,$g$

pro blem $(P)$ possesses at le as three $solu$ tions in $W^{1,\infty}(0, T;L^{2}(\Omega))$ .
$t$

Remark. For the existence of a perioidic solution of (P), we do notneed (g2). In fact, the existence of periodic solution of (P) is known
under much more weaker conditions than (g1).
2. Preliminaries.

In the following we assume that (g1) and (g2) hold. we set $H=L^{2}(\Omega)$ ,$V=H_{0}^{1}(\Omega)$ , and $V^{*}=H^{-1}(\Omega)$ . We denote by
the pairing of. For each subset $A\subset V$ ,.
andstands for the norm of
int $(A)$ denotes the set of interior point of . For each $i\geq 1,$ denotes
spanned by the eigenfunctions corresponding
the subspace of, andis a normalized eigenfunction
to the eigenvalues
and $V_{1}=\{k\varphi_{1} : k\in R\}$ .
corresponding to . Thenis the projection from $H$ onto for each $i\geq 1$ .
$V$$\langle\cdot, \cdot\rangle$$V^{*}$$H^{-1}(\Omega)$$||\cdot||_{*}$$A$$V_{i}$$H_{0^{1}}(\Omega)$$\{\lambda_{1}, \cdots, \lambda_{i}\}$$\varphi_{i}$$\varphi_{1}\in L^{\infty}(\Omega)$$\lambda_{i}$$P_{i}$$V_{i}$We define a functional $F:Varrow R$ by$F(v)= \frac{1}{2}\{Lv,$$v\rangle$$- \int_{\Omega}\int_{0}^{v(x)}g(\tau)d\tau dx$

for each $v\in V$.

We setfor each $c\in R$ .

$A_{c}=\{v\in H_{0}^{1}(\Omega):F(v)\leq c\}$Then the problem (P) can be rewritten as$u_{t}+F’(u)=f(t)$ ,$u(O)=u(T)$ .

(2.1)

Lemma 1.
(1)
The set $\{s\in R:F(s\varphi_{1})<0\}consists$ of at le $ast$ two intervals :91
The$re$ exis $ts\omega>0$ such that for each $w\in V_{1}$ ,

(2)

(2.2)

$<F’(v_{1}+w)-F’(v_{2}+w),$ $v_{1}-v_{2}>\geq\omega||v_{1}-v_{2}||^{2}$for all.

$v_{2}\in V_{1}^{\perp}$$v_{1},$

Proof. Since $\lambda_{1}<g’(O)$ , we can see from the definition of $F$ that
if I $s|$ is sufficiently small, $F(s\varphi_{1})<0(=F(O))$ . This implies thatthe set $A_{0}=\{s\in R : F(s\varphi_{1})<0\}$ is nonempty. It is easy to seefrom the continuity of $F$ that $D$ consists of open intervals. Then since$F(O)=0$ , the assertion (1) follows.
,for
We put $\omega=1-g’(0)/\lambda_{2}$ . Then since I
we have that$v\Vert\geq\lambda_{2}|v|$$v\in V_{1}^{\perp}$$<F’(v_{1}+w)-F’(v_{2}+w),$ $v_{1}-v_{2}>\geq||v_{1}-v_{2}||^{2}-g’(O)|v_{1}-v_{2}|^{2}$$\geq\omega||v_{1}-v_{2}||^{2}$for all$v_{1},$$v_{2}\in V_{1}^{\perp}$

1.

Remark.
The inequality (2.2) implies that for each $w\in V_{1}$ , thefunctional $F(\cdot+w)$ :is strictly convex.
$V_{1}^{\perp}arrow R$

Let

$u$

-and

$u_{+}$

be elements of$H_{0}^{1}(\Omega)$such thatv\in V, <P_{i}v, \varphi_{1}><0\}$,$F(u_{+})= \min\{F(v):v\in V, <P_{i}v, \varphi_{1}>>0\}$.

$F(u_{-})= \min\{F(v) :andFrom Lemma 1, -andintervalsand$u$$(a_{-}, b_{-})$$P_{1}u_{-}\in\{c\varphi_{1} :$u_{+}$$(a_{+}, b_{+})$

are well defined and there exist open
such thata_{-}<c<b_{-}\}$,$P_{1}u_{+}\in\{c\varphi_{1} :a_{+}<c<b_{+}\}$

and$F(c)<0$

for $c\in\{c\varphi_{1} : a_{-}<c<b_{-}\}\cup\{c\varphi_{1} : a_{+}<c<b_{+}\}$ .

92
Here we define subsets$A^{\pm}$

of

$V$

by

$A^{\pm}=\{v\in V : F(v)<0, <P_{1}v, \varphi_{1}>\in(a\pm, b\pm)\}$

(2.3)

,

respectively. We put$c \pm=\min${$F(s\varphi_{1})$For each $i\geq 1$ , we denote by$A(i)_{c}$ the intersection of level set$F_{i}(v)$: sgns $=\pm 1$ }.

the restriction of $F$ towith . That is$A_{c}$$A(i)_{c}=\{v\in V_{i} :We put$A_{c}^{\pm}=\overline{A\pm}\cap A_{c}$$V_{i}$, and by$V_{i}$

F(v)\leq c\}$.

for each $c>0$ .

Lemma 2. Let $c<0$ such that $c\pm<c$ . Then
$A_{c}^{\pm}$

are non $empty$ bounded an closed.
$d$, we have that $F(v)arrow\infty$ , as Il $v||arrow\infty$ .
Proof. Since
is bounded. It is obvious from the definition ofThis implies that1
are closed.
that$g’(\pm\infty)<\lambda_{1}$$A_{c}$$A_{c}^{\pm}$$A_{c}^{\pm}$

For each $i\geq 1$ , we denote bysubspace . We set$A(i)_{c}^{\pm}$the restriction of$A_{c}^{\pm}$to the$V_{i}$$K(i)_{\pm}=\overline{co}A(i)_{c}^{\pm}$

Since$A(i)_{c}^{\pm}\subset A^{\pm}$

and$K\pm=\overline{co}A_{c}^{\pm}$.

, we have by (2.3) that
$K(i)_{+}\cap K(i)_{-}=\phi$.

Then we have that

Lemma 3. There exis $tc\pm,\overline{c}\pm<0wi$ th $c\pm<\overline{c}\pm andd>0$ such that

Il Lv–g(v)$||_{*}\geq d$for all$v\in A_{\overline{c}}^{+_{+}}\backslash A_{c}^{+_{+}}\cup A_{\overline{c}}^{+_{-}}\backslash A_{c-}^{-}$

.

(2.4)

93
are disjoint from
Proof. We choose $c\pm and\overline{c}\pm such$ thatthe set of critical points of $F$ . It is well known that the functional$F$ satisfies Palais-Smale condition, i.e., any sequencesatisfying$\{F(x_{n})\}$ is bounded and $F’(x_{n})arrow 0$ contains a convergent subsequence. If (2.4) does not hold for any $d>0$ , there exists a sequencesuch that$cl(A_{\overline{c}\pm}^{\pm}\backslash A_{c}^{\pm_{\pm}})$$\{x_{n}\}$$\{x_{n}\}$$x_{n} \in D=A\frac{+}{c}\backslash A_{c}^{+}\cup A_{\overline{c}}^{-}\backslash A_{c}^{-}$

are bounded, by Palais-Smale
and $F’(x_{n})arrow 0$ , as $narrow\infty$ . Since
condition, we have that there exists a convergence subsequence. Let $v\in V$ such that $x_{m}arrow v$ . Then we have that $v\in D$ andof$\nabla F(v)=0$ . This contradicts the definition of
1$A \frac{\pm}{c}$$\{x_{m}\}$$\{x_{n}\}$$c\pm and\overline{c}\pm\cdot$

For simplicity of notations, we put$c=c\pm and\overline{c}=\overline{c}\pm\cdot$

Lemma 4. For each $i\geq 1$ , there exist mappings
such thatare continuous an$A(i)_{c}^{\pm}$$Q(i)_{\pm}$: $K(i)_{\pm}arrow$$d$$Q(i)_{\pm}$$Q(i)_{\pm}x=x$for each

Proof. Fix $i\geq 1$ . Let $x\in K(i)_{+}$ . Thenas $x=x_{1}+x_{2}$ , whereand$x_{1}\in V_{1}$$x\in A(i)_{c}^{\pm}$

$x$

.

(2.5)

is uniquely decomposed. Then since$x_{2}\in V_{1}^{\perp}\cap V_{i}$$C_{x_{1}}=\{v\in V_{1}^{\perp}\cap V_{i} :F(x_{1}+v)\leq c\}$is nonempty and strictly convex by Lemma 2, we have that there
exists an unique element
such that$\overline{x}\in C_{x_{1}}$$||x_{2}- \overline{x}||=\min\{||x_{2}-y||:y\in C_{x_{1}}\}$.

We put $Q(i)_{+}x=x_{1}+\overline{x}$ . Then from the definition, it is obviousthat $Q(i)_{+}x\in A(i)_{c}^{+}$ and that (2.5) holds. The mapping $Q(i)_{-}$ isdefined by the same way. It is easy to see thatare continuous
on $K(i)_{\pm}$ .
1$Q(i)_{\pm}$

3. Proof of Theorem.

94
We consider initial value problems of the form$\frac{du}{dt}-\triangle u-g(u)=f(t)$(I)$u(0)=u_{0}$ ,$(u_{0}\in V)$

,

$t>0$

,

and
$\frac{dv}{dt}-\triangle v-P_{i}g(v)=P_{i}f(t)$

,

$t>0$

$(I_{i})$$v(0)=v_{0}$where $i\geq 1$ and$v_{0}\in V_{i}$,.

We define mappings$T_{f}$$T_{f}(u_{0})=u(T)$: $Varrow V$ and

,

and

,

$T_{f^{i}}$:$V_{i}arrow V_{i}$by$T_{f^{i}},(v_{0})=v(T)$and , and continuous on
andThen it is easy to verify thatis a periodic
V. From the definition of , each fixed point ofsolution of (P). To prove Theorem, we need a few lemmas.
$T_{f}$$V$$T_{f^{i}}$$u$$T_{f}$$T_{f}$

Lemma 5. There exists a posi tive numb er $M$ and such th at if $\sup\{|$$f(t)|:t\in[0, T]\}<M$ , then$F_{i}(v_{i}(t))<F_{i}(v_{i})$for all $i\geq 1,$ $v_{i}\in D$ and $t>0$ satisfying
for all $s\in[0, t]$ ,$v_{i}(s)\in D$

where$v_{i}(\cdot)$$A_{\overline{c}}^{-}\backslash A_{c}^{-}$

is the solution of$(I_{i})$with $v_{0}=v_{i}$ . an

$dD=A \frac{+}{c}\backslash A_{c}^{+}\cup$.

Proof. We choose $M>0$ such that $M<d/2$ . Let $i\geq 1$ anbethe solu tion ofwith $v_{i}(0)=v;\in D$ an suppose that ther exis
$dv_{i}$$(I_{i})$

$d$

$e$

$ts$

95
$t>0$ an $dv_{i}(s)\in D$ for all $s\in[0, t]$ . Then by Lemma 4, we have$F_{i}(v_{i}(s))-F_{i}(v_{i})$$= \int_{0}^{s}<F’(v_{i}(\tau)),$ $u_{t}(\tau)>d\tau$$= \int_{0}^{s}<Lv_{i}(\tau)-g(v_{i}(\tau)),$$-Lv_{i}(\tau)+g(v_{i}(\tau))+f(\tau)>d\tau$$\leq\int_{0}^{s}(-|Lv_{i}(\tau)-g(v_{i}(\tau))|^{2}+|Lv_{i}(\tau)-g(v_{i}(\tau))||f(\tau)|)d\tau$$\leq\int_{0}^{s}|Lv_{i}(\tau)-g(v_{i}(\tau))|(-|Lv_{i}(\tau)-g(v_{i}(\tau))|+|f(\tau)|)d\tau$$\leq\int_{0}^{s}||Lv_{i}(\tau)-g(v_{i}(\tau))||_{*}(-||Lv_{i}(\tau)-g(v_{i}(\tau))||_{*}+|f(\tau)|)$$\leq-(d/2)^{2}s+(d/2)\cdot\sup\{|f(t)|:t\in[0, T]\}s<0$1
From Lemma 5, we have the following lemma.

Lemma 6.
$T_{f^{i}},(A(i)_{c}^{\pm})\subset int(A(i)_{c}^{\pm})$,for each $i\geq 1$ .

(3.1)

be the solution of theProof. Let $i\geq 1$ and $v\in A(i)_{c}^{+}$ . Let
problemwith $v_{0}=v$ . If there exists an interval $[0, t]$ such that$v_{i}$$(I_{i})$$v_{i}(s)\in D\cap V_{i}$for all $s\in[0, t]$ ,then by Lemma 5,$F_{i}(v_{i}(s))<F_{i}(v)\leq c$

for all $s\in[0, t]$ .

(3.2), this implies that $v_{i}(s)\in A(i)_{c}^{+}$ for all
From the definition of$s\in[0, t]$ . Recalling that the boundary $\{v\in V_{i} : F_{i}(v)=c\}\cap A(i)_{c}$ of$A(i)_{c}$ is contained in $D$ , we obtain from the observation above that
$A(i)_{c}^{+}$$F_{i}(v_{i}(s))<F_{i}(v)\leq c$

for all $s>0$ .

96Thus we find that $v_{i}(s)\in int(A(i)_{c}^{+})$ for all $s>0$ . Then from the, this implies that $T_{f^{i}},v\in int(A(i)_{c}^{+})$ . By the samedefinition of.
1
argument, we have that
$T_{-f^{i}},$$T_{f^{i}},(A(i)_{c}^{-})\subset int(A(i)_{c}^{-})$

Lemma 7. For each $i\geq 1$ ,$deg(I-T_{f^{i}},, K(i)_{\pm}, 0)=1$.

Proof. Fix $i\geq 1$ . We set
for $v\in K(i)_{\pm}$ .

$G_{\pm}(v)=T_{f},;Q(i)_{\pm}v$

Then by Lemma 6, we have thatfor all $v\in K(i)_{\pm}$$G_{\pm}(v)\in int(A(i)_{c}^{\pm})$are continuous mappings on bounded closed convex sets in a
Since
finite dimensional space and $G\pm have$ no fixed point on the boundary$G_{\pm}$

of $K(i)_{\pm}$ ,$deg(I-G_{\pm}, K(i)_{\pm},$ $0$) $=1$ .

From the definition ofand Lemma 6, we have that the sets of, respectively. Then it
fixed points of $G\pm are$ contained in int
follows that$c_{\pm}$$(A(i)_{c}^{\pm})$$deg(I-G\pm, A(i)_{c}^{\pm},$ $O$

Since $c_{\pm}=T_{f,i}$ on$A(i)_{c}^{\pm}$) $=deg(I-G_{\pm}, K(i)_{\pm},$ ) $=1$ .
$0$, we find that$deg(I-T_{f};, A(i)_{c}^{\pm},$ $O$) $=deg(I-G_{\pm}, A(i)_{c}^{\pm},$ ) $=1$ .
$0$

1This completes the proof.

Lemma 8. There exis $tse>0$ such th at$deg(I-T_{f,;}, A(i)_{e},$ $0$) $=1$$A_{c}^{+}\cup A_{c}^{-}\subset A_{e}$

an

$d$

for all $i\geq 1$ .

Proof. Let $e>0$ such that the set of critical points of $F$ is containedin the interior of . Fix $i\geq 1$ . Then since, we have$A_{e}$$A_{c}^{+}\cup A_{c}^{-}\subset A_{e}$97by Lemma 5 that $T_{f^{i}},(A(i)_{e})\subset int(A(i)_{e})$ . On the other hand, by thesame argument as in Lemma 4, we can define a continuous mapping
such that $Q_{e}v=v$ for all $v\in A(i)_{e}$ . Then from:, we canthe same argument as in Lemma 7 with $Q\pm replaced$ by
1
see that the assertion follows.
$Q_{e}$$\overline{co}A(i)_{e}arrow A(i)_{e}$$Q_{e}$

Proof of Theorem. Let $i\geq 1$ . Then by Lemma 7, there exist fixed. On the other hand, by Lemma 7
andpointsand Lemma 8, we have that$v_{i}^{+}\in A(i)_{c}^{+}$$v_{i}^{+}\in A(i)_{c}^{+}$$deg(I-T_{f^{i}}, A(i)_{e}\backslash (A_{c}^{+}\cup A_{c}^{-}),$ $0$) $=-1$ .

.
This implies that there exists a fixed point
andbe sequences obtained by the argument above.
Now letare bounded in , we may assume that
andThen sinceconverge weakly to $v\pm andv_{0}\in V$ , respectively. Then it isandeasy to verify that $v\pm\in K\pm andv_{0}\in V\backslash (K_{+}\cup K_{-})$ are fixed pointsof . This completes the proof.
1$v_{i}^{0}\in A(i)_{e}\backslash (A_{c}^{+}\cup A_{c}^{-})$$\{v_{i}^{\pm}\}$$\{v_{i}^{\pm}\}$$\{v_{i}^{0}\}$$V$$\{v_{i}^{0}\}$$v_{i}^{\pm}$$v_{i}^{0}$$T_{f}$

Refere nces
[1] Amann. , Periodic solutions for semi-linear parabolic equations, in“Nonlinear Analysis:A collection of Papers in Honor of Erich Rothe,“
Academic Press, New York, 1-29, 1978.
[2] H. Amann&E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations,Ann. Scuola Norm. Sup. Pisa 7(1980), 539-603.
[3] A. Ambrosetti&G. Mancini, Sharp nonuniqueness results for somenonlinear problems, Nonlinear Analysis, 3(1979), 635-645.
[4] A. Castro &A. C. Lazer, Critical point theory and the number ofsolutions of a nonlinear Dirichlet problem, Ann. Math., 18(1977),$H$

113-137.
[5] N. Hirano, Existence of nontrivial solutions of semilinear elliptic
equations, Nonlinear Analysis, Nonlinear Analysis, 13(1989), 695-

705.

[6] N. Hirano, Existence of multiple periodic solutions for a semilinear
evolution equation, Proc. Amer. Math. Soc., 106(1989), 107-114.