Existence of multiple periodic solutions for semilinear parabolic equations with sublinear growth nonlinearities

DOI 10.4134/JKMS.2009.46.4.691

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH SUBLINEAR GROWTH NONLINEARITIES

Wan Se Kim

Abstract. In this paper, we establish a multiple existence result of T - periodic solutions for the semilinear parabolic boundary value problem with sublinear growth nonlinearities. We adapt sub-supersolution scheme and topological argument based on variational structure of functionals.

1. Introduction

Let Ω ⊂ R ⁿ , n ≥ 2, be a bounded domain with smooth boundary ∂u. In this paper, we are concerned with the multiple existence result of T -periodic solutions for the semilinear parabolic boundary value problem

(P )

 



 

u t − 4 x u + u = g(u) + h(t, x) in (0, T ) × Ω,

u = 0 on (0, T ) × ∂Ω,

u(0) = u(T ) in ¯ Ω.

We assume u = u(t, x), g : R → R is continuous, and h : R × ¯ Ω → R is a continuous function which is T -periodic with respect to the first variable and h > 0 on R×Ω. There are many results for the multiple existence of T -periodic solutions for seminear parabolic equations with this type of nonlinearity in [6, 7, 8, 9], and for elliptic equations also in [2, 4, 10].

Here, we denote Q T the open set (0, T ) × Ω. For q ≥ 1, we denote by | · | q

and || · || q the norms of L ^q (Ω) and W ^1,q (Ω), respectively. || · || stands for the norm of H ₀ ¹ (Ω). We put V = H ₀ ¹ (Ω), H = L ² (Ω). The norm of the dual space V ^∗ of V is denoted by || · || ∗ . h·, ·i stands for the paring of V and V ^∗ . A function u ∈ C([0, T ]; H ₀ ¹ (Ω)) ∩ C ¹ ([0, T ]; L ² (Ω)) is said to be a solution of (P ) if u satisfies (P ). Here, we assume

(H 1 ) g is Lipschitz continuous, nondecreasing, odd function and g(0) = 0,

Received August 20, 2007.

2000 Mathematics Subject Classification. 35K20, 35K55.

Key words and phrases. multiplicity, periodic solutions, semilinear parabolic equations, sublinear nonlinearity.

This work was supported by the grant R01-2003-000-11731-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

°2009 The Korean Mathematical Societyc

691

(H 2 ) there exist C 1 > 0 and 0 < α < 1 such that |g(u)| ≤ C 1 |u| ^α on R, (H 3 ) there exists C 2 > 0 such that

lim inf

|u|→0

G(u)

|u| ≥ C ₂ , where G(u) = R _u

0 g(s)ds, (H 4 )

|u|→∞ lim g(u)

u < λ 1 , (H 5 )

λ 1 < lim

|u|→0

g(u) u ,

where we denote by λ 1 < λ 2 ≤ · · · the eigenvalues of the problem

−4u = λu, u ∈ H ₀ ¹ (Ω)

and by φ 1 the normalized eigenfunction corresponding to λ 1 .

Such a function exists; for example, we first fix a smooth function φ : (−∞, ∞)

→ [0, 1] such that φ ⁰ (t) ≤ a, and φ(t) =

( 0 for t ∈ (−∞, −1] ∪ [1, ∞) 1 for t ∈ [− ¹ ₂ , ¹ ₂ ].

Let n ≥ 1 and t ^± _n be the numbers such that t ⁻ _n < 0 < t ⁺ _n and g(2t ^± _n ) = 2nt ^± _n . We put

g n (t) =

( nφ ⁺ _n (t)t + (1 − φ ⁺ _n (t))h(t) for t ≥ 0 nφ ⁻ _n (t)t + (1 − φ ⁻ _n (t))h(t) for t ≤ 0, where h(t) = |t| ^α−1 t, φ ⁺ _n (t) = φ( _2t ^t

n

) and φ ⁻ _n (t) = φ( _2t ^−t

n

). Then we have that g n (t) = nt on [t ⁻ _n , t ⁺ _n ] and g n (t) = h(t) on (−∞, 2t ⁻ _n ) ∪ (2t ⁺ _n , ∞). For Lipschitz continuity of g n , let consider the case that t > 0. From the definition, we have g n (t) = nt on [0, t ⁺ _n ]. On the other hand, we have that for t ∈ [t ⁺ _n , 2t ⁺ _n ],

g _n ⁰ (t) = n((φ ⁺ _n (t)) ⁰ t + φ ⁺ _n (t)) + (1 − φ ⁺ _n (t))h ⁰ (t) − (φ ⁺ _n (t)) ⁰ h(t)

≤ n µ at

2t ⁺ n

+ 1

¶

+ α

t ^1−α + at 2t ⁺ n

t ^α

≤ n ³ a 2 + 1 ´

+ α

(t ⁺ n ) ^1−α + a 2 (2t ⁺ _n ) ^α .

Then we find g _n ⁰ (t) ≤ C max {n, h ⁰ (t)} for some C > 0. Moreover recalling that

n(2t ⁺ _n ) ^1−α ∼ = 1, we find that h ⁰ (t) ≤ Cn on [t ⁺ _n , 2t ⁺ _n ] for some C > 0, and hence

each g n is Lipschitz continuous on R. Therefore (H 1 )-(H 5 ) follows from the

definition.

2. Preliminary results

Let us consider a initial boundary value problem associated with (P )

(I)

 



 

u _t − 4 _x u + u = g(u) + h in (0.∞) × Ω u(t) = 0 on (0, ∞) × Ω u(0) = u 0 in ∂Ω,

where u 0 ∈ L ² (Ω) and h ∈ C ¹ ( ¯ Q T ). We denote by t(u 0 ) the number such that [0, t(u 0 )) is the maximal interval for u(t) to exist. If u is a solution of problem (I) on [0, t(u 0 )), u can be represented by the integral form

(2.1) u(t) = S(t)u 0 + Z _t

0

S(t − s)(g(u(s)) − u(s) + h(s, x))ds

for 0 < t < t(u 0 ). Here, {S(t)} is the semigroup of linear operators generated by −4 _x . It is known that for each q ≥ 2, there exists c(q) > 0 satisfying (2.2) ||S(t)f || q ≤ c(q)t ^−1/2 |f | q for all f ∈ L ^q (Ω) and t > 0

(cf. Amann [1], Tanabe [12]). If we set X + = {u ∈ C ₀ ¹ ( ¯ Ω); u ≥ 0 on Ω}, then X + is a closed cone in C ₀ ¹ ( ¯ Ω). We employ the standard order in C ₀ ¹ ( ¯ Ω) as

u ≥ v ⇔ u − v ∈ X + , u > v ⇔ u ≥ v, u 6= v, u À v ⇔ u − v ∈ intX + . For each u, v ∈ C ₀ ¹ ( ¯ Ω), we put

[v, u] = {w ∈ C ₀ ¹ ( ¯ Ω); v ≤ w ≤ u}.

A mapping S : [u, v] → C ₀ ¹ ( ¯ Ω) is said to be order preserving if Sx À Sy for x, y ∈ [u, v] with x > y. Here, we denote by S the Poincare mapping associated with problem (I). That is Su 0 = u(T ), u 0 ∈ H. It is obvious that the Poincare mapping S is well defined only when t(u 0 ) > T . It follows from the parabolic maximal principle that S is strictly monotone with respect to the order defined above. That is, if u > v in C ₀ ¹ ( ¯ Ω) and Su, Sv exist, then Su À Sv. A function u ∈ C ^1,2 ((0.T ) × Ω) ∩ C ^0,1 ((0, T ) × ¯ Ω) is called subsolution (cf. Hess [5]) for the T -periodic problem (I) if

 



 

u t − 4 x u + u ≤ g(u) + h in (0, ∞) × Ω

u = 0 on (0, ∞) × ∂Ω

u(0) = u 0 in Ω.

A subsolution is said to be a strict subsolution if it is not a solution of (I).

Similarly, a supersolution and strict supersolution are defined by the inequality sign, correspondingly.

3. Multiplicity result We set

C([0, T ]; u 0 , H) = {u ∈ C([0, T ], H); u(0, x) = u 0 (x) on Ω}

for each u 0 ∈ H. For each u 0 ∈ H, we define a mapping K u

: C([0, T ]; u 0 , H)

→ C([0, T ]; u 0 , H) by

(K _u

u)(t) = S(t)u ₀ + Z _t

0

S(t − s)(g(u(s)) − u(s) + h(s, x))ds for each u ∈ C([0, T ]; u 0 , H). Then we have:

Lemma 3.1. For each u 0 ∈ H, K u

is compact and has a unique fixed point in v u

∈ C([0, T ]; u 0 , H).

Proof. See the proofs of Theorems 1.7 and 2.1 in Chapter 6 of Pazy [11]. ¤ Remark. Since Su 0 = v u

(T ) and v u

is a solution of (I), v u

is a periodic solution of (I).

By (H 5 ), there exists µ 1 > 0 such that ^g(u) _u > λ 1 for all |u| ≤ µ 1 .

Let 0 < ² < 1 be such that h − ²φ 1 > 0 and |²φ 1 | ∞ ≤ µ 1 on Ω. Then we have

−∆(²φ 1 ) + ²φ 1 = ²λ 1 φ 1 + ²φ 1 < g(²φ 1 ) + h on Ω.

Hence ²φ 1 is a strict subsolution of (I). Let 0 < λ < λ 1 . By (H 4 ), there exists µ 2 > 0 such that g(u) < λu for all |u| ≥ µ 2 . Put c = max{g(u) : 0 ≤ u ≤ µ 2 }.

Since λ < λ 1 . Dirichlet boundary value problem

−∆ x u = λu + c + h

has a solution v ∈ H ₀ ¹ (Ω). Note that c + h > 0, we have that v ∈ C ¹ ( ¯ Ω) and v > 0 on Ω. Let b > 0 and put ˜ u = bφ 1 + v. Then

λv(x) + λ 1 bφ 1 (x) > λ(v(x) + bφ 1 (x))

> g(v(x) + bφ 1 (x)) for x ∈ Ω with ˜ u(x) ≥ µ 2

and c > g(˜ u(x)) for x ∈ Ω with ˜ u(x) < µ 2 . Hence, we have

−∆ x u + ˜ ˜ u ≥ λv + λ 1 bφ 1 + c + h > g(˜ u) + h.

Therefore, ˜ u is a strict supersolution of (I). Recall that ∂φ 1 /∂n < 0 and

∂v/∂n < 0 on ∂Ω by the maximal principle. Then we can choose b > 0 sufficiently large so that ²φ 1 ¿ ˜ u on Ω. We know that S is strongly order preserving on [²φ 1 , ˜ u] and

S[²φ 1 , ˜ u] ⊂ [²φ 1 , ˜ u].

We know that S[²φ 1 , ˜ u] is relatively compact in C ₀ ¹ ( ¯ Ω) (cf. Proposition 21.2 of

[5]). Hence, by Theorem 4.2 of [5], we have two sequences u ⁽¹⁾ n ≡ S ⁿ (²φ) and

u ⁽²⁾ n ≡ S ⁿ (˜ u) which converges to a fixed point u ⁽¹⁾ and u ⁽²⁾ of S as n → ∞,

respectively and ²φ 1 < u ⁽¹⁾ ≤ u ⁽²⁾ < ˜ u. From Remark 21.3 of [5], the problem

(P ) has a solution u 1 ∈ C ^1,2 ([0, T ] × ˜ Ω) with u 1 (0) = u 1 (T ) = u ⁽ⁱ⁾ for i = 1, 2

(cf. Lemma 20.1 of [5]). Therefore we have:

Lemma 3.2. For each h ∈ C ¹ ( ¯ Q T ) and h > 0, there exist a solution u 1 ∈ C ^1,2 ([0, T ] × ¯ Ω) of (P ) such that ²φ 1 < u 1 (t) < ¯ u on [0, T ].

Next, we prove the existence of the second solution.

By Lemma 3.1, we have:

Lemma 3.3. If lim n→∞ |u ⁽¹⁾ n − u ⁽²⁾ n | _c

0

( ¯ Ω) > 0, then we have two solutions u 1 , u 2 of (P ) such that ²φ 1 < u 1 (0) = u ⁽¹⁾ < u 2 (0) = u ⁽²⁾ < ¯ u.

Proof. cf. Lemma 3.1 and Remark 21.3 in [5]. ¤

To complete our assertion, we assume that

(3.1) lim

n→∞ |u ⁽¹⁾ _n − u ⁽²⁾ _n | _c

0

( ¯ Ω) = 0.

Now, we let I : V → R be a functional defined by I(v) = 1

2 ||v|| ² ₂ − Z

Ω

G(v)dx for v ∈ V.

By I ^c , we denote the level set I ^c = {v ∈ V : I(v) ≤ c}. From the definition of I and (H 2 ), we can see that lim _||v||

_→∞ I(v) = ∞. Thus we have that

−∞ < m 1 = min{I(v) : v ∈ V }.

(H 3 ) implies that for any nonzero v ∈ V , there is sufficiently small t > 0 that I(tv) < 0. That is m 1 < 0.

Lemma 3.4. For any δ ∈ [m 1 , 0], there exist m ≥ 1 and a continuous function h : S ^m → I ^δ such that h(S ^m ) is not contractible in I ^δ , where S ^m denotes the unit sphere in R ^m .

Proof. We put V k = span{φ 1 , φ 2 , . . . , φ k }. Fix δ ∈ [m 1 , 0]. Let v ∈ V with

||v|| 2 = 1. From (H 1 ), we have that the mapping s → I(sv) is decreases on interval [0, t], where t > 0 satisfies

I(tv) = min{I(sv) : s ≥ 0}

and increases on [t, ∞). From the definition of I, we have, by (H 2 ), t ² ||v|| ² ₂ =

Z

Ω

g(tv)tvdx ≤ C 1

Z

Ω

t ^α+1 |v| ^α+1 dx.

Suppose that v ∈ V _k−1 ^⊥ for some k ≥ 2. Since |v| ^α+1 _α+1 ≤ C 3 |v| ^α+1 ₂ for some C 3 > 0 and λ k |v| ² ₂ ≤ |∇v| ² ₂ , we have that

t ^1−α ||v|| ² ₂ ≤ C 1 |v| ^α+1 _α+1

≤ C 1 C 3 |v| ^α+1 ₂

≤ C 1 C 3

µ 1 λ k

¶ α+1

||v|| ^2(α+1) ₂

and hence 0 < t ≤ (C 1 C 3 )

( _λ ¹

k

)

. This implies that t → 0 when k → ∞.

By (H 1 ) and (H 2 ), I(tv) = t ²

2 ||v|| ² ₂ − Z

Ω

Z _tv

0

g(s)ds ≥ t ²

2 − C 1 t ^α+1 Z

Ω

|v| ^α+1 dx

≥ t ²

2 − C 1 C 3 t ^α+1 |v| ^α+1 ₂

≥ t ²

2 − C 1 C 3 t ^α+1

µ |∇v| 2

√ λ _k

¶ _α+1

≥ t ²

2 − C 1 C 3

µ t

√ λ k

¶ _α+1 .

Thus I(tv) → 0 as k → ∞. Therefore there exists k 0 ≥ 0 such that I ^δ ∩V _k ^⊥

= φ.

Let v 0 ∈ I ^δ , then since I is an even function, −v 0 ∈ I ^δ . If {v 0 , −v 0 } is contractible in I ^δ , by Krasonalski’s result (cf. Lemma 3.2 of Bahri [3]), we can define an odd continuous function h 1 : S ¹ → I ^δ such that h 1 (S ¹ ) ⊂ I ^δ . By induction, if h k

−1 (S ^k

⁻¹ ) is contractible, we can construct an odd and continuous function h k

: S ^k ₀ → I ^δ . but since h k

(S ^k

) ∩ V _k ^⊥

6= φ, this is

impossible. Hence, this proves our theorem. ¤

By (H 4 ), there exists c 3 > 0 such that λu − c 3 < g(u) for all u ≤ 0, where 0 < λ < λ 1 . Then there exists a negative solution v ∈ c ¹ ( ¯ Ω) of the Dirichlet problem

−∆ x u = λu − c 3 . Let a ≥ 1. If we put u = av, then

−∆ x u + u = λav − c 3 + av < λav − c 3 < g(u) + h.

That is u is a strict subsolution of (P ).

Lemma 3.5. For any δ < 0, there exists δ 1 , δ 2 < 0 such that δ < δ 1 < δ 2 < 0 and the interval [δ 1 , δ 2 ] contains no critical point of I.

Proof. Let δ 0 < 0 and suppose contrary that there exists no interval in (δ 0 , 0) satisfying the condition. Then, for any δ 0 < δ < 0, there exists a sequence {u n } ⊂ V such that ∇I(u n ) = 0; i.e., −∆u n +u n = g(u n ) and lim n→∞ I(u n ) = δ.

Then, by (H 2 ), we have δ = lim

n→∞ I(u _n ) = lim

n→∞

à 1

2 ||u _n || ² ₂ − Z

Ω

Z _u

_(x)

0

g(t)dtdx

!

≥ lim

n→∞

µ 1

2 ||u n || ² ₂ − C 1

1 + α |u| ^1+α _1+α

¶ .

Hence {u n } is bounded in W ^1,2 (Ω) and hence bounded in V . Therefore there

exists a subsequence, say again {u n }, such that {u n } converges to u ∈ V

strongly in H and weakly in V.

Since g is Lipschitz continuous and

||u m − u n || ² ₂ ≤ |g(u m ) − g(u n )| 2 |u m − u n | 2 ≤ L|u m − u n | 2

for some constant L > 0, {u n } converges to u strongly in V . Therefore, we have

∇I(u) = 0 and I(u) = δ. This is impossible and completes our assertion. ¤ Lemma 3.6. Let δ 0 < 0 and δ 1 , δ 2 be constants, δ 0 < δ 1 < δ 2 < 0, satis- fying the assertion of Lemma 3.5. Then there exists m 0 such that, for each h ∈ C ¹ ( ¯ Q T ) with |h| _C

( ¯ Q

) < m 0 , if v is the solution of (I) with v(0) ∈ I ^δ for some δ ∈ [δ 1 , δ 2 ], then v(t) ∈ I ^δ for t ≥ 0.

Proof. Let δ 0 such that I(²φ 1 ) < δ 0 . Let δ 1 , δ 2 be constants such that δ 0 <

δ 1 < δ 2 < 0 and satisfying the assertion of Lemma 3.5. Then we define ˜ m 0 = inf{||∇I(v)|| ∗ : v ∈ I ^δ

\ I ^δ

}, then we have ˜ m 0 > 0. We put m 0 = ˜ m 0 /|Ω| ^1/2 . Now let h ∈ C ¹ ( ¯ Q T ) with |h| _C

_{( ¯ Q}

₎ < m 0 . Suppose δ ∈ [δ 1 , δ 2 ], v(0) ∈ I ^δ and v(t) ∈ I ^δ

on an interval [0, t v(0) ]. From the definition of m 0 , we have that for t ∈ [0, t v(0) ], using the Holder inequality,

I(v(t)) − I(v(0)) = Z _t

0

∇I(v(s)) · dv ds

≤ Z _t

0

(−||∇I(v)|| ² _∗ + ||h(s)||||∇I(v)|| ∗ )

≤ Z _t

0

||∇I|| ∗ (−||∇I|| ∗ + ||h(s)||) < 0.

Then we have I(v(t)) < I(v(0)). Hence, we have that v(t) ∈ I ^δ for all t ≥ 0.

This completes our assertion. ¤

Theorem. There exists m 0 > 0 such that for each h ∈ C ¹ ( ¯ Q T ) with |h| _C

( ¯ Q

)

< m 0 , there exists a periodic solution u 2 in V \ [²φ 1 , ¯ u].

Proof. Let δ ₀ , m ₀ be as in Lemma 3.5. Let u ₁ be the solution of (P ) obtained in Lemma 3.2.

Suppose there in no fixed point of S in V \ [²φ 1 , ¯ u]. Let δ 0 , δ 2 be constants such that δ 0 < δ 1 < δ 2 < 0 satisfying the assertion of Lemma 3.5. We recall Lemma 3.6. Since ²φ 1 ∈ I ^δ

and u ⁽¹⁾ = lim n→∞ u ⁽¹⁾ n = lim n→∞ S ⁿ (²φ 1 ), we find that u ⁽¹⁾ ∈ I ^δ

. Let ² > 0 be such that δ 1 + 2² < δ 2 . Then, by (3.1), there exists n 0 such that for all n ≥ n 0 , such that

u ⁽¹⁾ _n , u ⁽²⁾ _n ∈ I ^δ

^+²/2 and

(3.2)

¯ ¯

¯ ¯ Z

Ω

G(v)dx − Z

Ω

G(z)dx

¯ ¯

¯ ¯ < ² for all v, z ∈ [u ^{(1) n} , u ⁽²⁾ _n ].

Since || · || ² ₂ is a convex function, by (3.2), I(αv + (1 − α)u ⁽¹⁾ n ) < δ 1 + 2² for all

v ∈ [u ⁽¹⁾ n , u ⁽²⁾ n ] ∩ I ^δ

^+² and α ∈ [0, 1], and hence [u ⁽¹⁾ n , u ⁽²⁾ n ] ∩ I ^δ

^+² is star convex

with respect to u ⁽¹⁾ n in I ^δ

^+2² . Therefore, for any sufficiently small r > 0 , (3.3) [v, z] ∩ I ^δ

^+² is cotractible in I ^δ

,

where v, z ∈ C ₀ ¹ ( ¯ Ω) such that ||v − u ⁽¹⁾ n || 2 < r and ||z − u ⁽²⁾ n || 2 < r.

By Lemma 3.4, there exist m > 0 and a continuous function h : S ^m → I ^δ

such that h(S ^m ) is not contractible in I ^δ

. Since C ₀ ¹ ( ¯ Ω) ∩ I ^δ

is dense in I ^δ

, we may have h(S ^m ) ⊂ C ₀ ¹ ( ¯ Ω) ∩ I ^δ

. Let u z be the solution of (I) with u z (0) = h(z), z ∈ S ^m . By choosing b > 0 sufficiently large in the definition of ˜ u, we have another strict supersolution ¯ u > ˜ u such that v < ¯ u for all v ∈ h(S ^m ). Similarly, by choosing a > 0 in the definition of u, we have that u < ²φ 1 and u < v for all v ∈ h(S ^m ). We recall that u and ¯ u are strict sub and supersolution of (I), respectively and that u < ²φ 1 , ˜ u < ¯ u. Since S has no fixed point in V \ [²φ 1 , ˜ u], we have that S ⁿ (u) → u ⁽¹⁾ and S ⁿ (¯ u) → u ⁽²⁾ as n → ∞. Therefore, there exists n ≥ n 0 such that ||S ⁿ (u) − u ⁽¹⁾ n || 2 < r and

||S ⁿ (¯ u) − u ⁽²⁾ n || 2 < r. Since S ⁿ (u) ≤ u z (nT ) ≤ S ⁿ (¯ u) for all z ∈ S ^m , by (3, 3), {u z (nT ) : z ∈ S ^m } is contractible in I ^δ

.

By Lemma 3.6, we can define a homotopy

ρ : [0, nT ] × h(S ^m ) → C ₀ ¹ (Ω) ∩ I ^δ

by

ρ(s, h(z)) = u z (s) for 0 ≤ s ≤ nT and z ∈ S ^m .

Then h(S ^m ) is contractible in I ^δ

. This is a contraction. Hence, S has a fixed point u ₂ in V \ [²φ ₁ , ˜ u]. This proves assertion. ¤

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Department of Mathematics Hanyang University

Seoul 133-791, Korea

E-mail address: [email protected]