https://doi.org/10.1007/s11784-021-00894-w Published online September 6, 2021
c The Author(s) 2021
Journal of Fixed Point Theory and Applications
Periodic solutions for nonresonant parabolic equations on R N with Kato–Rellich type potentials
Aleksander ´ Cwiszewski and Renata Lukasiak
Abstract.A criterion for the existence ofT-periodic solutions of nonau- tonomous parabolic equationut = Δu+V(x)u+f(t, x, u), x ∈ RN, t > 0, where V is Kato–Rellich type potential and f diminishes at infinity, will be provided. It is proved that, under the nonresonance as- sumption, i.e. Ker(Δ +V) = {0}, the equation admits a T-periodic solution. Moreover, in case there is a trivial branch of solutions, i.e.
f(t, x,0) = 0, there exists a nontrivial solution provided the total mul- tiplicities of positive eigenvalues of Δ +V and Δ +V +f0, wheref0 is the partial derivativefu(·,·,0) off, are different mod 2.
Mathematics Subject Classification. 35K55, 35B10, 35A16.
1. Introduction
We shall be concerned with timeT-periodic solutions of the following para- bolic problem
∂u
∂t(x, t) = Δu(x, t) +V(x)u(x, t) +f(t, x, u(x, t)), x∈RN, t >0, (1) where Δ is the Laplace operator (with respect to x), V is a Kato–Rellich type potential andf : [0,+∞)×RN ×R→RisT-periodic in time:
f(t+T, x, u) =f(t, x, u), (2) for allt≥0,u∈Rand a.e.x∈RN.
Periodic problems for parabolic equations were widely studied by many authors by use of various methods. Some early results are due to Brezis and Nirenberg [4], Amman and Zehnder [2], Nkashama and Willem [19], Hirano [15,16], Pr¨uss [25], Hess [14], Shioji [27] and many others; see also [29] and the references therein. These results treat the case where Ω is bounded and are based either on topological degree and coincidence index techniques in the spaces of functions depending both onxand timetor on the translation
along trajectories operator to which fixed point theory is applied. In this paper we shall study the case Ω =RN. In this case arguments based on the compactness of the heat semigroup are no longer valid, sinceH1(RN) is not compactly embedded into L2(RN). Here, we shall prove that the Poincar´e translation along trajectories is eventually compact, which enables us to use fixed point index and averaging techniques (see e.g. [6,7]).
We assume that V is of Kato–Rellich type, i.e. V = V∞+V0 where V∞∈L∞(RN) andV0∈Lp(RN) where
2< p <∞ forN= 1,2 and N≤p <∞ forN ≥3.
The nonlinear perturbationf : [0,+∞)×RN×R→Ris such thatf(t,·, u) is measurable, for allt≥0 andu∈R,
|f(t, x,0)| ≤m0(x), for allt≥0 and a.e.x∈RN, (3) for somem0∈L2(RN), and there areθ∈(0,1),
|f(t, x, u)−f(s, x, u)| ≤(˜k(x) +k(x)|u|)|t−s|θ, (4)
|f(t, x, u)−f(t, x, v)| ≤l(t, x)|u−v|, (5) for all t, s ∈ [0,+∞), u, v ∈ R and a.e. x ∈ RN, where ˜k ∈ L2(RN), k is of Kato–Rellich type (i.e. satisfies the same assumption asV, possibly with differentp),l=l0+l∞withl∞∈L∞([0,+∞)×RN) and supt≥0l0(t,·)Lp<
∞. We shall also assume a sort of relaxed monotonicity:
(f(t, x, u)−f(t, x, v))·(u−v)≤a(x)|u−v|2, (6) for anyu, v ∈ RN, t≥0 and a.e. x∈RN, wherea is of Kato–Rellich type and
r→∞lim esssup
|x|>r (V∞+a∞)(x)<0.
Usually we consider separately two cases looking at the time averaged right hand-side Δ +V +fwhere the time average functionf:RN ×R→R off is given by
f(x, u) := 1 T
T
0
f(t, x, u) dt.
The resonant case is when the linearization of the nonlinear operator Δ+V+f at infinity and/or zero has nontrivial kernel and the non-resonant case when the mentioned linearizations have zero kernels. Both cases can be put in similar settings, however, are different geometrically. The resonant case was studied in [8]. Here we consider the non-resonant case. Our main results are the following criteria for the existence ofT-periodic solutions.
Theorem 1.1. Suppose that f andV satisfy conditions(2), (3), (4), (5)and (6).If Ker (Δ +V) ={0},
|u|→∞lim
f(t, x, u)
u = 0 for anyt≥0and a.e.x∈RN, (7)
and
r→∞lim esssup
|x|>r V∞(x)<0, (8) then the Eq.(1)admits aT-periodic solution
u∈C([0,+∞), H2(RN))∩C1([0,+∞), L2(RN)).
Our second result applies in the case where there exists a trivial periodic solutionu≡0 and the previous theorem does not imply the existence of a nontrivial periodic one.
Theorem 1.2. Suppose that all the assumptions of Theorem 1.1 are satisfied and, additionally, assume that, for anyt≥0 and a.e.x∈RN,
u→lim0
f(t, x, u)
u =α(x), (9)
whereαis of Kato–Rellich type. IfKer (Δ +V +α) ={0},
r→∞lim esssup
|x|>r (V∞+α∞)(x)<0 (10) and
m+(Δ +V)≡m+(Δ +V +α) mod 2,
wherem+(Δ+V)andm+(Δ+V+α)are the total multiplicities of the positive eigenvalues of Δ + V and Δ + V + α, respectively, then the Eq. (1) admits a nontrivial T-periodic solution u ∈ C([0,+∞), H2(RN))∩ C1([0,+∞), L2(RN)).
Remark 1.3. (a) Due to the spectral theory assumption (8) implies that the essential spectrum σe(Δ +V) is contained in (−∞,0) and all the positive elements of the spectrum are actually eigenvalues. Therefore, 0∈σ(Δ +V) and the numberm+(Δ +V) are well-defined. The same goes for the operator Δ +V +α(see also Remark7.4for more details).
(b) Note that although it is not stated explicitly (9) implies thatf(t, x,0) = 0 for allt≥0 and a.e.x∈RN.
(c) Theorems1.1and1.2are straightforward consequences of more general results from Sect.7, where V may depend on time as well.
The above results are obtained by use of the Poincar´e (translation along trajectories) operatorΦT :H1(RN)→H1(RN) defined as
ΦT(¯u) :=u(T), u¯∈H1(RN),
whereu: [0, T]→H1(RN) is the solution of (1) satisfying the initial value conditionu(0) = ¯u. We obtain T-periodic solutions as fixed points of ΦT. In case of parabolic equations on bounded domains, the Poincar´e operator is compact, which is no longer the case on RN. Actually neither is ΦT a set contraction. In consequence, we can not use standard Leray–Schauder degree/index. However, following the tail estimates techniques from Wang [31] (see also Prizzi [24] and [9]), we show thatΦT is ultimately compact, i.e.
belongs to a wider class of maps for which the fixed point index Ind(ΦT, U), with respect to open subsets of H1(RN), can be considered (see e.g. [1]).
To determine the index Ind(ΦT, U) (for a properly chosen U), we use the averaging method, i.e. we consider the Eq. (1) as one of the family of problems
∂u
∂t = Δu+V(x)u+f(t/λ, x, u), x∈RN, t >0, λ >0. (11) By use of averaging techniques we show that solutions of (11) converge as λ→0+ to a solution of the averaged equation
∂u
∂t = Δu+V(x)u+f(x, u), x∈RN, t >0. (12) Our asymptotic assumptions onf imply a sort of a priori bounds conditions, i.e. that, forλ∈(0,1], the problem (11) has noλT-periodic solution if only initial states are of sufficiently largeH1norm (in case of Theorem1.1) and al- so of sufficiently smallH1norm (in case of Theorem1.2). This will mean that initial states ofλT-periodic solutions are located inside some open bounded setU ⊂H1(RN). Next we prove a sort of “averaged” Krasnosel’skii formula (see also [6]) stating that
Ind(ΦT, U) = lim
t→0+Ind(Φt, U), (13) whereΦtis the translation along trajectories operator for (12). To determine Ind(Φt, U), for smallt >0, we use linearization techniques and strongly rely on spectral properties of Δ +V and Δ +V +α.
Remark 1.4. (a) By the applied approach the (topological) fixed point in- dex of the Poincar´e operatorΦT is determined. The fact that we consid- er operators in the phase space and use topological methods has some stability and continuity implications. Namely, if the nonlinearity f is perturbed by a small term so that the timeT-periodicity and regularity conditions are satisfied, then the conclusions of Theorems1.2 and 1.1 hold. Actually, one can even perturb the nonlinearity, by a term that does not push it out of the class determined by assumptions (2)–(7) as well as (9) and (10). Moreover, it comes from fixed point index theory (see [1,12]) that T-periodic solutions of the problem perturbed with a small term are localized near the T-periodic solutions of (1) with the original f. Another advantage is that the knowledge of the fixed point index of the translation operator carries additional information on the dynamics of the evolution system generated by (1). In fact we can compute the indices of all the iterations ofΦT, which are related to the stability of periodic solutions (see e.g. [20,21]).
(b) There are several other approaches to forced oscillations for nonlinear evolution problems where the solutions are found in the space of time- periodic functions, see e.g. [30] or [17] for problems on bounded domains.
Another method uses a separation of the so-called steady state part and the oscillatory one like in recent paper [11,18], in which maximal regularity for semi-linear problems onRN were obtained (it is not clear yet if this setting adapts to nonlinear state dependingf).
(c) We may assume that the timeT-periodicf is defined onR×RN and con- siderT-periodic solutions onR. Then we are able to consider the prob- lem where∂u/∂tis replaced by−∂u/∂tprovided−V and−f(−t, x, u) satisfy the above assumptions forV andf.
The paper is organized as follows. In Sect.2 we recall the concept of ultimately compact maps and fixed point index theory. In Sect.3we strengthen in a general setting of sectorial operators the initial condition continuity prop- erty and Henry’s averaging principle, that we apply to the parabolic equation in Sect.4. Section5 is devoted to the ultimate compactness property of the translation operator. In Sect.6we adapt the ideas of [6] to the case Ω =RN, proving the averaging index formula (13) as well as verify a priori bounds conditions forλT-periodic solutions of (11) withλ∈(0,1]. Finally, in Sect.7 the main results are proved.
2. Preliminaries
Notation. IfX is a normed space with the norm · , then, forx0 ∈X and r >0, we putBX(x0, r) :={x∈X | x−x0< r}. By∂U andU we denote the boundary and the closure ofU ⊂X. convV and convXV stand for the convex hull and the closed (in X) convex hull of V ⊂ X, respectively. By (·,·)0 is denoted the inner product inX.
Measure of noncompactness. IfX is a Banach space andV ⊂X is bounded, then by βX(V) we denote the infimum over all r > 0 such that V can be covered with a finite number of open balls of radiusr. ClearlyβX(V) is finite and it is called the Hausdorff measure of noncompactness of the setV in the spaceX. It is not hard to show thatβX(V) = 0 implies thatV is relatively compact in X. More properties of the measure of noncompactness can be found in [10] or [1].
Fixed point index. Below we recall basic definitions and facts from the fixed point index theory for ultimately compact maps. For details we refer to [1].
We say that a map Φ :D→X, defined on a subsetDof a Banach space X is ultimately compact ifW ⊂X is such that conv Φ(W ∩D) =W, then W is compact. We shall say that an ultimately compact map Φ :U →X, defined on the closure of an open bounded setU ⊂X, is calledadmissibleif Φ(u)=ufor allu∈∂U. By anadmissible homotopybetween two admissible maps Φ0,Φ1 :U →X we mean a continuous map Ψ: U×[0,1]→X such that Ψ(·,0) = Φ0, Ψ(·,1) = Φ1, Ψ(u, μ)=ufor all u∈∂U and μ∈[0,1], and, for anyW ⊂X, if convΨ((W ∩U)×[0,1]) =W, thenW is relatively compact. Φ0,Φ1are called homotopic then. A fixed point index for ultimately compact maps was constructed in [1, 1.6.3 and 3.5.6]. Basic properties of the fixed point index are collected in the following.
Proposition 2.1. (i) (Existence) If Ind(Φ, U)= 0, then there exists u∈U such thatΦ(u) =u.
(ii) (Additivity)IfU1, U2⊂Uare open andΦ(u)=ufor allu∈U\(U1∪U2), then
Ind(Φ, U) = Ind(Φ, U1) + Ind(Φ, U2).
(iii) (Homotopy invariance) If Φ0,Φ1:U →X are homotopic,then Ind(Φ0, U) = Ind(Φ1, U).
(iv) (Normalization) Let u0 ∈X andΦu0 :U →X be defined byΦu0(u) = u0for allu∈U. ThenInd(Φu0, U)is equal0ifu0∈U and1ifu0∈U.
Remark 2.2. If Φ :U →X is a compact map then Ind(Φ, U) is equal to the Leray–Schauder index IndLS(Φ, U) (see e.g. [12]).
3. Remarks on abstract continuity and averaging principle
LetA:D(A)→X be a sectorial operator such that for somea >0,A+aI has its spectrum in the half-plane{z∈C|Rez >0}. LetXα, 0≤α <1, be the fractional power space determined byA+aI. It is well-known that there existC0, Cα>0 such that for allt >0
e−tAuα≤C0eatuα for allu∈Xα, e−tAuα≤Cαt−αeatu0 for allu∈X,
where{e−tA}t≥0 is the semigroup generated by−A. Consider the equation u(t) =˙ −Au(t) +F(t, u(t)), t >0,
u(0) = ¯u, (14)
where ¯u∈Xα and F : [0,+∞)×Xα →X is such that there exists C ≥0 with
F(t, u) ≤C(1 +uα) for allu∈X, t >0, (15) and, for any boundedZ ⊂Xα×[0,+∞) there existD, L≥0 andθ∈(0,1) with the property
F(t, u)−F(s, v) ≤D|t−s|θ+Lu−vα for all (u, t),(v, s)∈Z.(16) We shall say that u : [0,+∞) → Xα is a solution of above initial value problem if
u∈C([0,+∞), Xα)∩C((0,+∞), D(A))∩C1((0,+∞), X)
and satisfies (14). By classical results (see [5] or [13]), the problem (14) ad- mits a unique global solution u ∈ C([0,+∞), Xα)∩C((0,+∞), D(A))∩ C1((0,+∞), X). Moreover, it is known thatubeing solution of (14) satisfies the following Duhamel formula:
u(t) =e−tAu(0) + t
0
e−(t−s)AF(s, u(s)) ds, t >0. (17)
Remark 3.1. Assume that u: [0, T]→Xα is a solution of (14) with T >0.
Then clearly, by (15) and (17), there is a constant ˜C= ˜C(C, C0, Cα, a, T)>0 such that for allt∈(0, T]
u(t)α≤C0eat¯uα+ t
0
Cα(t−s)−αea(t−s)F(s, u(s))ds,
≤C(1 +˜ u¯α) + ˜C t
0
(t−s)−αu(s)αds.
This in view of [5, Lemma 1.2.9] implies that there exists ¯C= ¯C(C, C0, Cα, a, T, α)>0 such that
u(t)α≤C(1 +¯ u¯ α) for allt∈[0, T]. (18) Theorem 3.2. Assume that (αn) is a sequence of positive numbers such that αn →α0 as n→+∞ for some α0 >0 and thatAn :=αnA forn≥0. Let Fn : [0, T]×Xα → X, T > 0, n ≥ 0, satisfy (15) and (16) with common constantsC, L(independent of n)and let, for eachu∈Xα,
t
0
Fn(s, u) ds→ t
0
F0(s, u) ds inX asn→+∞
uniformly with respect tot∈[0, T]. If un: [0, T]→Xα, n≥0,are solutions of
˙
u(t) =−Anu(t) +Fn(t, u(t)), t∈[0, T],
andun(0)→ u0(0) in X, then un(t)→u0(t) in Xα uniformly with respect totfrom compact subsets of(0, T].
Remark 3.3. Recall that Henry’s result from [13] states that, under the above assumptions withαn≡1, ifun(0)→u0(0) inXα, asn→+∞, thenun(t)→ u0(t) in Xα uniformly on compact subsets of [0, T). Here, inspired by the proof of Proposition 2.3 of [24], we modify Henry’s proof.
In the proof we shall use the following lemma.
Lemma 3.4. Under the assumptions of Theorem 3.2, for any continuous u: [0, T]→Xα,
t
0 e−(t−s)AnFn(s, u(s)) ds→ t
0 e−(t−s)A0F0(s, u(s)) ds inXα as n→+∞, uniformly with respect tot∈[0, T].
Proof. We shall adjust arguments from the proof of [13, Lemma 3.4.7]. First observe that due to the assumptions concerning the constantLforFn’s it is sufficient to show the assertion for u ≡u¯ where ¯u∈ Xα. Take any ε >0.
There existδ >0, ˜C >0 and ˜a >0 such that, for anyn≥0 andt∈[0, δ]
t
0
e−(t−s)AnFn(s,u) ds¯ α
≤ t
0
Cαα−αn (t−s)−αeαna(t−s)C(1 +u¯ α) ds
≤C˜ t
0
τ−αe˜aτdτ ≤Ce˜ ˜aT(1−α)−1δ1−α≤Ce˜ ˜aTδ1−α< ε/4 (19)
and, for anyn≥0 andt∈[δ, T], t
t−δe−(t−s)AnFn(s,u¯) ds α
≤C˜ δ
0 τ−αe˜aτdτ≤Ce˜ ˜aT(1−α)−1δ1−α< ε/4. (20) Observe that, for anyn≥0 andt∈[δ, T],
t−δ
0
e−(t−s)AnFn(s,u) ds¯ =e−tAn t
0
Fn(τ,u) dτ¯ −e−δAn t
t−δFn(τ,u) dτ¯ +
t−δ
0
Ane−(t−s)An t
s
Fn(τ,¯u) dτ ds.
Clearly,
e−tAn t
0
Fn(τ,u) dτ¯ →e−tA0 t
0
F0(τ,u) dτ,¯ inXα,
uniformly with respect tot∈[δ, T]. Note also that, for all t∈[δ, T] and all n≥1,
e−δAn t
t−δFn(τ,u) dτ¯ α
≤Ce˜ ˜aδδ1−α≤ε/4.
Finally, for largenand allt∈[δ, T] ands∈[0, t−δ], one has Ane−(t−s)An
t
s Fn(τ,u¯) dτ−A0e−(t−s)A0 t
s F0(τ,u¯) dτ α
≤ |αn−α0|
Ae−(t−s)An t
s Fn(τ,u) dτ¯ α
+α0
Ae−(t−s)An t
s Fn(τ,¯u) dτ−Ae−(t−s)A0 t
s F0(τ,u¯) dτ α
≤C|α¯ n−α0|
e−(t−s)An t
s Fn(τ,u¯) dτ 1+α
+ ¯Cα0
e−(t−s)An t
s Fn(τ,¯u) dτ− t
s F0(τ,¯u) dτ 1+α
+ ¯Cα0
e−(t−s)An−e−(t−s)A0 t
s F0(τ,u) dτ¯ 1+α
≤ |αn−α0|CC¯ 1+αe˜aT δ1+α
t
s Fn(τ,u¯) dτ +α0
CC¯ 1+αe˜aT δ1+α
t
s Fn(τ,u¯) dτ− t
s F0(τ,u¯) dτ +α0
CC¯ 1+αe˜aT (α0δ/2)1+α
e−((t−s)αn/α0−δ/2)A0−e−(t−s−δ/2)A0 t
s F0(τ,u) dτ¯ ,
where ¯C > 0 is such that Awα ≤Cw¯ 1+α for all w∈X1+α. Therefore, for largenand allt∈[δ, T]
t−δ
0 e−(t−s)AnFn(s,u¯) ds−t−δ
0 e−(t−s)A0F0(s,¯u) ds
α≤ε/4 +ε/4 +ε/4 = 3ε/4,
which together with (19) and (20) ends the proof.
Proof of Theorem3.2. By the Duhamel formula, fort∈(0, T] and n≥1, un(t)−u0(t) =e−tAnun(0)−e−tA0u0(0) +
+ t
0
e−(t−s)AnFn(s, u0(s))−e−(t−s)A0F0(s, u0(s))) ds +
t
0
e−(t−s)An(Fn(s, un(s))−Fn(s, u0(s))) ds.
This gives, for allt∈(0, T] and n≥1, un(t)−u0(t)α≤γn(t) +CαL
t
0
eaαn(t−s)(αn(t−s))−αun(s)−u0(s)αds with
γn(t) :=Cαeaαnt
(αnt)α un(0)−u0(0)0+(e−tAn−e−tA0)u0(0)
α
+ t
0
e−(t−s)AnFn(s, u0(s))−e−(t−s)A0F0(s, u0(s)) ds α
. This means that there are ˜a >0 and ˜C >0 such that, for allt∈(0, T] and n≥1,
un(t)−u0(t)α≤γn(t) + ˜C t
0
e˜a(t−s)(t−s)−αun(s)−u0(s)αds.
By use of Lemma 7.1.1 of [13], we get un(t)−u0(t)α≤γn(t) +K
t
0
(t−s)−αγn(s) ds
for some constant K > 0. Now let us fix t ∈ [0, T] and take an arbitrary δ∈(0, t). Observe also that
t
0
(t−s)−αγn(s) ds≤ 2α δα
t−δ/2 0
γn(s) ds+ t
t−δ/2(t−s)−αγn(s) ds
≤ 2α δα
T
0
γn(s) ds+(δ/2)1−α
1−α · sup
s∈[δ/2,T]
γn(s).
Since, in view of Lemma3.4,γn(t)→0 uniformly with respect totfrom com- pact subsets of (0, T] and the functions γn,n≥1, are estimated from above by an integrable function we infer, by the dominated convergence theorem, thatun(t)−u0(t)α→0 asn→+∞uniformly with respect tot∈[δ, T].
The above theorem allows us to strengthen Henry’s averaging principle.
We assume that mappingsFn : [0,+∞)×Xα→X,n≥1, satisfy (15) and (16) with common constants C, L(independent of n) and that there exists F:Xα→X such that, for all ¯u∈Xα,
τ→+∞, n→lim +∞
1 τ
τ
0
Fn(t,u) dt¯ =F(¯u) inX. (21) Theorem 3.5. Suppose Fn and F are as above, u¯n → u¯ in X, λn → 0+ as n→+∞, andun : [0,+∞)→Xα, n≥1,are solutions of
u(t) =˙ −Au(t) +Fn(t/λn, u(t)), t >0, u(0) = ¯un.
Thenun(t)→u(t) in Xα uniformly with respect tot from compact subsets of(0,+∞)whereu: [0,+∞)→Xα is the solution of
u(t) =˙ −Au(t) +F(u(t)), t >0, u(0) = ¯u.
Proof. Let ˜Fn :=Fn(·/λn,·) and ˜F0:=F. Observe that, using (21), we get, for any ¯u∈Xα andt >0,
t
0
F˜n(s,u¯) ds=λn
t/λn
0 Fn(ρ,¯u) dρ→tF(¯u) = t
0
F˜0(¯u) ds inX, asn→+∞.
Clearly, ˜Fn,n≥1, and ˜F0satisfy (15) and (16) with the common constants C, L. It can be easily verified that the convergence above is uniform with respect to t from bounded subintervals of [0,+∞). Now, an application of
Theorem3.2yields the assertion.
Remark 3.6. (a) The above result is an improvement of the continuation the- orem and the Henry averaging principle [13, Th. 3.4.9] to the case when initial values fromXα converge in the topology of X (not Xα). This will appear crucial when establishing the ultimate compactness property of the Poincar´e operator and verifying a priori estimates in the proofs of main results. We shall need to consider solutions in the phase space X1/2 = H1(RN) while the compactness of sequences of initial values is possible with respect to the L2(RN) topology only.
(b) An averaging principle for parabolic equations on RN was also proved in [3] in a specific case, where the elliptic operator with time de- pendent coefficients was considered.
4. Continuity and averaging for parabolic equations
We transform (1) into an abstract evolution equation. To this end define an operatorA:D(A)→X in the space X:=L2(RN) by
Au:=− N i,j=1
aij ∂2u
∂xj∂xi, foru∈D(A) :=H2(RN),
whereaij∈R,i, j= 1, . . . , N, are such that N
i,j=1
aijξiξj >0, for anyξ∈RN,
and aij = aji for i, j = 1, . . . , N. It is well-known thatA is a self-adjoint, positive and sectorial operator inL2(RN).
Suppose thatf satisfies (3)–(5). DefineF: [0,+∞)×H1(RN)→L2(R2) by [F(t, u)](x) :=f(t, x, u(x)) for a.e.x∈RN.
Lemma 4.1. Under the above assumptions there are constants D > 0, de- pending only on k, ˜k, N and p, L > 0, depending only l, N and p, and C >0, depending only on m0, l, N and p, such that, for all t1, t2 ≥0 and u1, u2∈H1(RN),
F(t1, u1)−F(t2, u2)L2 ≤D(1 +u1H1)|t1−t2|θ+Lu1−u2H1 and F(t, u)L2 ≤C(1 +uH1) for any t≥0 andu∈H1(RN).
In the proof of Lemma4.1 we shall use the following technical lemma (see, e.g. [8, Lemma 4.2]).
Lemma 4.2. There exist constantsC1=C1(N, p)>0andC2=C2(N, p)>0 such that for anyu∈H1(RN)
uL2p/(p−1) ≤C1uH1, (22)
and
uL2p/(p−2) ≤C2uH1. (23)
Proof of Lemma4.1. By use of (4) and (5), the H¨older inequality and Lemma4.2, one finds constantsD=D(k,k, N, p)˜ >0 andL=L(l, N, p)>0 such that, for anyt1, t2≥0 andu1, u2∈H1(RN),
F(t1, u1)−F(t2, u2)L2 ≤(˜kL2+C2k0Lpu1H1+k∞L∞u1L2)|t1−t2|θ +C2l0(t2,·)Lpu1−u2H1+l∞(t2,·)L∞u1−u2L2
≤D(1 +u1H1)|t1−t2|θ+Lu1−u2H1.
Furthermore, by (5), one also has|f(t, x, u)| ≤ |f(t, x,0)|+l(t, x)|u|fort≥ 0, x ∈ RN, u ∈ R. This, together with (3), gives the existence of C = C(m0, l, N, p)>0 such that
F(t, u)L2≤ m0L2+C2l0(t,·)LpuH1+l∞(t,·)L∞uL2≤C(1 +uH1)
for anyt≥0 andu∈H1(RN).
Consider now the evolutionary problem
˙
u(t) =−Au(t) +F(t, u(t)), t≥0, u(0) = ¯u∈H1(RN). (24) Due to Lemma 4.1 and standard results in theory of abstract evolution equations (see [13] or [5]), the problem (24) admits a unique global solution u ∈ C([0,+∞), H1(RN)) ∩ C((0,+∞), H2(RN))∩C1((0,+∞), L2(RN)).
We shall say thatu: [0, T0)→H1(RN),T0>0, is asolution(H1-solution)
of ∂u
∂t(x, t) =Au(x, t) +f(t, x, u(x, t)), x∈RN, t∈(0, T0),
u(x,0) = ¯u(x), x∈RN,
for some ¯u∈H1(RN), whereA=N
i,j=1aij∂x∂2
j∂xi, if
u∈C([0,+∞), H1(RN))∩C((0,+∞), H2(RN))∩C1((0,+∞), L2(RN)) and (24) holds. In this sense we have global in time existence and uniqueness of solutions for the parabolic partial differential equation.
The continuity of solutions properties are collected below.
Proposition 4.3 (Compare [24, Prop. 2.3]). Assume that functions fn: [0,+∞)×RN×R→R, n≥0,satisfy the assumptions(3)with common m0 and (4) with common l and that fn(t, x, u) →f0(t, x, u), for all t ≥0, u ∈ R, a.e. x ∈ RN, and fn(t,·,0) → f0(t,·,0) in L2(RN) for all t ≥ 0.
Suppose that(αn) is a sequence of positive numbers such that αn →α0, as n→+∞, for some α0 >0.Let un : [0, T]→H1(RN), n≥0, be a solution of
∂u
∂t(x, t) =αnAu(x, t) +fn(t, x, u(x, t)), x∈RN, t∈(0, T],
such that, for someR >0,un(t)H1 ≤R, for allt∈[0, T]andn≥0. Then fn(t,·, u(·))→f0(t,·, u(·))in L2(RN) for anyu∈H1(RN)andt≥0and
(i) ifun(0)→u0(0) inL2(RN) asn→ ∞, then un(t)→u(t)in H1(RN) fort from compact subsets of(0, T].
(ii) ifun(0)→u0(0)inH1(RN)asn→ ∞,thenun(t)→u0(t)inH1(RN) uniformly fort∈[0, T].
Proof. Define Fn: [0,+∞)×H1(RN)→L2(RN),n≥0, by [Fn(t, u)](x) :=
fn(t, x, u(x)). Note that, in view of (4), for anyt≥0,u∈H1(RN) and a.e.
x∈RN,
|fn(t, x, u(x))−f0(t, x, u(x))|2≤2|fn(t, x,0)−f0(t, x,0)|2+4|l(x, t)u|2. Since, for any t ≥ 0, fn(t,·,0) → f0(t,·,0) in L2(RN) as n → +∞, the right hand side can be estimated by an integrated function, which due to the Lebesgue dominated convergence theorem impliesFn(t, u)→F0(t, u) in L2(RN) as n → +∞. Moreover, by use of Lemma 4.1, we may pass to the limit under the integral to get t
0Fn(s, u) ds → t
0F0(s, u) ds in L2(RN) for any u ∈ H1(RN) and t ≥ 0. This in view of Theorem 3.2 implies the assertion (i). The assertion (ii) comes from the standard continuity theorem
from [13].
Let us also state an averaging principle.
Proposition 4.4. Assume that functions fn : [0,+∞)×RN ×R→R, n≥0, satisfy the assumptions of Proposition4.3and additionally(2). Suppose that
¯
un→¯u0inL2(RN), λn→0+asn→+∞and thatun : [0,+∞)→H1(RN), n≥1,are solutions of
∂u
∂t =Au+fn(t/λn, x, u), x∈RN, t >0, u(x,0) = ¯un(x), x∈RN.
Then un(t) → u(t) in H1(RN) uniformly on compact subsets of (0,+∞), whereu: [0,+∞)→H1(RN)is the solution of
∂u
∂t =Au+f0(x, u), x∈RN, t >0, u(x,0) = ¯u0(x), x∈RN, with f0 :RN ×R→Rgiven by f0(x, u) := T1T
0 f0(t, x, u) dt for allu∈R and a.e.x∈RN.
Proof. Define Fn: [0,+∞)×H1(RN)→L2(RN),n≥0, by [Fn(t, u)](x) :=
fn(t, x, u(x)) and F0 : H1(RN) → L2(RN) by F0(u) := T1T
0 F0(t, u) dt.
Clearly, for allu∈H1(RN), F(u)(x) = f(x, u(x)) for a.e.x∈RN. Fix any
¯
u∈ H1(RN) and (τn) in (0,+∞) such that τn → +∞. Clearly, Fn,n ≥1, areT-periodic in time. Consequently, one has
In:= 1 τn
τn
0 Fn(t,u) dt¯ =[τn/T] τn/T · 1
T T
0 Fn(t,¯u) dt+ 1 τn
τn−[τn/T]T
0 Fn(t,u) dt.¯ Hence, to see thatIn→F0(¯u) it is sufficient to prove that
In(T):= 1 T
T
0
Fn(t,u) dt¯ →F0(¯u) in L2(RN), asn→+∞.
To this end observe that, for a.e.x∈RN, In(T)(x) = 1
T T
0 fn(t, x,u¯(x)) dt→ 1 T
T
0 f0(t, x,u¯(x)) dt=f0(x, u(x)) = [F0(¯u)](x). Moreover, by use of the assumptions onfn’s, one has
|In(T)(x)|= 1 T
T
0
fn(t, x,u(x)) dt¯
≤m0(x) +g(x), whereg(x) :=T1T
0 |l(t, x)||u(x)¯ |dt. and, by use of Jensen’s inequality,
RN|g(x)|2dx≤ 1 T
RN
T
0
|l(t, x)|2|¯u(x)|2dtdx <+∞
(see the proof of Lemma4.1). Hence, by the dominated convergence theorem we infer that In(T) → F0(¯u) in L2(RN). Since (τn) was arbitrary it follows that
τ→+∞, n→lim +∞
1 τ
τ
0
Fn(t,u) dt¯ →F0(¯u).
Finally, we get the assertion by use of Theorem3.5.
5. Ultimate compactness of Poincar´ e operator
We start the section with the following version of standard tail estimates (see [24]).
Lemma 5.1. Assume that f : [0,+∞)×RN×R→R satisfies(3)–(5)and (f(t, x, u)−f(t, x, v))·(u−v)≤a(x)|u−v|2. (25) whereais of Rellich–Kato type and
¯
a:= lim
r→+∞essup
|x|>r a∞(x)<0. (26) Suppose thatu: [0, T]→H1(RN)is a solution of
∂u
∂t = Δu+f(t, x, u), x∈RN, t≥0, (27) such that u(t)H1 ≤ R for all t ∈ [0, T]. Then, for any γ ∈ (0,|¯a|) there exists a sequence(αn)withαn→0+ asn→ ∞ such that
RN\B(0,n)|u(t)|2dx≤R2e−2γt+αn for allt∈[0, T], andn≥1, whereαns depend only onN, p, R, m0, a, γ andaijs.
Proof. It goes along the lines of [24, Prop. 2.2]. The only difference is that here we have the modified dissipativity condition (25) that implies
f(t, x, u)u≤a(x)|u|2+f(t, x,0)u
for allt ≥0, u∈R and a.e. x∈ RN. Hence, there exists r =rγ >0 such that
f(t, x, u)u≤ −γ|u|2+a0(x)|u|2+m0(x)u
for all t ≥0,u∈R and a.e. x∈RN\B(0, r). Therefore, one only needs to modify the proof of [24, Prop. 2.2] (compare also the proof of Lemma5.3).
Remark 5.2. Lemma5.1is sufficient to show the asymptotic compactness or Conley index admissibility of the parabolic semi-flow (whenf is independent of time). However, to prove the ultimate compactness of the Poincare´e oper- ator we shall need tail estimates involving any two solutions of the parabolic equations as well as parameter dependence.
Suppose thataij ∈C([0,1],R),i, j= 1, . . . , N, are such that N
i,j=1
aij(μ)ξiξj>0 for anyξ∈RN andμ∈[0,1]
and consider h : [0,+∞)×RN×R×[0,1] → R such that, for all t, s ≥ 0, u, v∈R,μ, ν∈[0,1] and a.e.x∈RN,
h(t,·, u, μ) is measurable and|h(t, x,0, μ)| ≤m0(x), (28)
|h(t, x, u, μ)−h(s, x, v, μ)| ≤(˜k(x) +k(x)|u|)|t−s|θ+l(s, x)|u−v|, (29)
