NLS-LIKE EQUATIONS IN BOUNDED DOMAINS: PARABOLIC APPROXIMATION PROCEDURE

. The article is devoted to semilinear Schr¨odinger equations in bounded domains. A uniﬁed semigroup approach is applied following a concept of Trotter-Kato approximations. Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in H 10 (Ω).

1. Introduction. We study a family of initial-boundary value problems of the form (± 1 − η 2 i − η)u t + ∆u + f (x, u) = 0, t > 0, x ∈ Ω, u | ∂Ω = 0, u(0, x) = u 0 (x), x ∈ Ω, where η ∈ [0, 1] plays a role of a parameter, Ω is a smooth bounded domain in R N and f : Ω × C → C is a continuous map satisfying a suitable growth condition. The equations in (1) contain as the limiting problems some well known models. Namely, η → 1 leads to a parabolic equation in R + × Ω and when η → 0 one gets a semilinear Schrödinder equation iu t + ∆u + f (x, u) = 0 (2) in R + × Ω or in R − × Ω respectively. Here note that changing u(t; x) into u(−t; x) leads from the equation (2) in R − × Ω to −iu t + ∆u + f (x, u) = 0 in R + × Ω and that the latter equation can be viewed as the limit of (1) with the minus sign as η → 0.
Nonlinear Schrödinger like problems have brought a lot of attention in recent years and much progress has been achieved. See, for example, [6,8,13,16,17,19] and references therein, which nonetheless are merely samples of the rich literature devoted to this subject.
It is known that if Ω is a bounded domain, which situation we consider here, then Strichartz's estimates are not applicable (see [6,Remark 2.7.3]). In the profound studies of [6] existential results where obtained in H 1 0 (Ω) via regularization of the nonlinear term f in the case when f behaved subcritically; namely for N ≥ 3 the image f (H 1 0 (Ω)) was contained in L q (Ω) for some q ∈ [2, 2N N −2 ). Consequently, f was then well defined from H 1 0 (Ω) into an intermediate space between (H 1 0 (Ω)) and H 1 0 (Ω).
In this article we describe complementary regularization procedure, relying on a regularization of the linear main part operator, which in a natural way reveals the critical exponents. Such parabolic approximation also seems advantageous in the consideration of (2) as on the one hand one can obtain global solutions of (2) under some mild assumptions on the nonlinear term and, on the other, one can even consider some situation when f takes H 1 0 (Ω) into (H 1 0 (Ω)) but the image f (H 1 0 (Ω)) is not contained in any intermediate space between (H 1 0 (Ω)) and H 1 0 (Ω). Within this approach one can thus handle nonlinearities, which behave in a critical manner (see Examples 3 and 4 v) below, for which Theorem 2.5 can be applied).
A brief description of this work is as follows. In Section 2 below we tersely describe the main results. Moreover, we exhibit critical exponents and give examples of some typical nonlinearities, involving even a critically growing one, to which the results are applicable. The results are then proved in the following two sections. Section 3 deals with approximate problems (1) η∈(0,1] and Section 4 is devoted to the limit equation (2). Additional comments concerning a critical regime appear in Section 5 and some auxiliary results are included in the Appendix. 2. Notation and main results. We will use spaces H s p (Ω), s ≥ 0, p ∈ [1, ∞) as in [20] and write H s p (Ω) with s < 0 for (H −s p (Ω)) . If p = 2 it is typical to write H s (Ω) instead of H s 2 (Ω). Actually, to keep the notation short, given a smooth bounded domain Ω ⊂ R N , we will omit from now on the dependence on Ω denoting H s p (Ω) =: H s p . Some of the spaces will involve zero trace boundary condition. In particular, as in [20], we now use the space H 2 p,{Id} (Ω) := {ϕ ∈ H 2 p : ϕ | ∂Ω = 0} for which we use a short notationḢ 2 p . To express our results better let us consider the negative Laplacian operator with the domain D(A p ) =Ḣ 2 p and let θ η = Arg(η + 1 − η 2 i). Then η + 1 − η 2 i = e iθη , η ∈ [0, 1] and the first equation in (1) rewrites as u t + e ±iθη A p u = e ±iθη f (·, u), t > 0, θ η ∈ [0, Concerning the linear main part operator in (3), Stone's theorem (see [16,Theorem 1.10.8]) implies the following result in the limit case η = 0.
Note that we sometimes use below the notation of the "dotted" scale as in Remark 1 and that in this conventionḢ 0 p = L p (Ω) andḢ 1 p = {ϕ ∈ H 1 p : ϕ | ∂Ω = 0}. Concerning properties of a nonlinear right hand side in (3) we associate with f the operator f e , where f e (u)(x) = f (x, u(x)) a.e. in Ω for any measurable u : Ω → C, and consider the following hypothesis H k p relative to the phase space of initial dataḢ k p with k = 1 or k = 0 respectively. Remark 2. Note that the case k = 1, p = 2 is of the primary interest below because in this case we will construct global solutions of (2) for nonlinearities satisfying even a certain critical growth condition. We nonetheless include the case k = 0 discussing also p ∈ (1, ∞), since this seems natural from the point of view of the analysis of (1) | η∈(0,1] . For k = 0, p = 2 the limiting procedure will not give however the result of a capacity similar to the case k = 1, p = 2, due to weaker bounds on the solutions of (2) η∈(0,1] . Hypothesis H k p . Let k ∈ {0, 1} be given and p ∈ (1, ∞). We assume that there are constants and there also exist certain constants ζ > 0, Observe that, due to hypothesis H k p , Duhamel's formula associated for η ∈ (0, 1] with (3) can be solved with the aid of Banach's fixed point theorem, which in turn leads to Theorem 2.1 below. For details, exhibiting in particular the role of the conditions on parameters ε, ρ, γ, ζ, see [3, pp. (3) is locally well posed inḢ k p . Furthermore, if u is a solution of (3) | η∈(0,1] through initial datum u 0 ∈Ḣ k p defined on a maximal interval of existence [0, τ u0 ) and if one of the following conditions holds (c 1 ) hypothesis H k p holds with γ > ρε (c 2 ) hypothesis H k p holds with γ = ρε and arbitrarily small ζ > 0 then u satisfies the blow upḢ k p alternative, that is, either τ u0 = ∞ or otherwise lim sup Theorem 2.1 leads in a natural way to the consideration of critical exponents ρ c (k, p), which describe the maximal growth of the nonlinear term allowed for the local well posedness of (3) η∈(0,1] inḢ k p with k = 1 or k = 0 respectively.
with any ρ > 1 when k = 1 and N ≤ p, and with Furthermore, γ in hypothesis H k p can be chosen strictly bigger than ρε unless ρ = ρ c (k, p) in which case γ = ρε.
In Proposition 4 no growth restriction is actually needed in the case when k = 1 and N < p whereas when k = 1 and 1 < p = N one can even consider the exponential growth due to Trudinger's inequality [1, §8.25].
Concerning the critical exponent ρ c (k, p) the following version of the above proposition holds (see [4,Lemma 3.2]).
Proposition 5. Let k ∈ {0, 1} be given. Hypothesis H k p holds with arbitrarily small ζ > 0 provided that f (z) = h(|z|) for some differentiable real map h : R → R satisfying Note that hypothesis H k p is also satisfied by multiplication operators Q V associated with external potentials V , where Q V is defined for any measurable function Proposition 6. If V : Ω → R is a potential of the class L r and r > N 2 , r ≥ 1 then f e (u) = Q V (u) satisfies hypothesis H k p with k = 1 and k = 0 respectively. Furthermore, γ in hypothesis H k p can be chosen strictly bigger than ρε. For global solvability of approximate equations we restrict our consideration to Hilbert phase spaces as we need to rely on the a priori bounds on the solutions in L 2 and in H 1 respectively. For this we will assume, on the one hand, that Im(f (x, u)ū) = 0 a.e. in Ω, (10) and, on the other, that f can be viewed as the gradient of some suitable functional F , namely We will also use the structure condition with certain Note that we neither assume that C(·) in (12) is negative nor that the bottom spectrum of −∆ − C(·)Id is positive.
Concerning approximate problems we have the following global well posedness result.
Focusing on the case p = 2 we now consider the solutions of (3) η∈(0,1] as approximate solutions of This latter problem involves the equation (2) obtained by passing in (3) to a limit as η → 0.
In the linear case, that is when f = 0 in (3), the following result holds, which in turn comes back to the Trotter-Kato approximation theorem.
2,η contains λ ≥ 0 and the associated resolvent operators converge in L(L 2 ), that is The associated linear semigroups converge as well. Actually, given any u 0 ∈ L 2 and a bounded time interval J we have, uniformly for t ∈ J, In what follows we will show that such an approximation procedure also applies in a nonlinear case. Given k ∈ {0, 1} and passing to the limit as η → 0 we will assume the following k-condition which is satisfied in many situations as shown in Example 4 below. Definition 2.3. Let k ∈ {0, 1} be given. We say that k-condition holds if where g 1 = g 1 (y 1 ), g 2 = g 2 (y 1 , y 2 , y 3 ) are nonnegative functions such that g 1 is bounded on bounded subsets of [0, ∞) and lim y3→0 g 2 (y 1 , y 2 , y 3 ) = 0 uniformly for (y 1 , y 2 ) in bounded subsets of [0, ∞) × [0, ∞).
(see the proof of [5, Lemma 2.2]) so that k-condition in Definition 2.3 holds with k = 1. Actually, no growth restriction is needed when k = 1 and N = 1 whereas in the case k = 1 and N = 2 the growth can even be exponential. ii) If V is an external potential as in Proposition 6, then f e (u) = V u satisfies k-condition with k = 1 (see Subsection 3.3.2). iii) Due to i)-ii) above typical nonlinearities as in Example 1 satisfy k-condition with k = 1. iv) A nonlinearity in Example 2 satisfies k-condition with k = 0 (see Section 3.3.3). v) A critically growing nonlinearity defined in (15) of Example 3 satisfies k-condition with k = 1 (see Section 3.3.4).
We will look for a solution of (16) satisfying variation of constants formula as in Definition 2.4 below.
Note that in (20) the nonlinear term f e (u(s)) will in general belong to a function spaceḢ s for some s < 0. Hence the linear semigroup appearing therein has to be suitably extended from L 2 to these larger spaces in which we follow the ideas of [2]. Namely, combining Proposition 3 together with [ 2,η (which we denote the same) is an infinitesimal generator of the semigroup of contractions inḢ σ for any σ ∈ [−2, 0] and the resolvent set of −A ± 2,η inḢ σ coincides with the one in L 2 . Using the concept of parabolic approximation we then prove the existence of global solutions of (16).
Recall Remark 2 and note that, in particular, Theorem 2.5 applies with k = 1 to subcritical nonlinearities (14) as in Example 1 ii). 2 On the other hand note that Theorem 2.5 applies with k = 1 to critically growing nonlinearities as in Example 3 (see Example 4 iii)-v)).
With additional assumptions one can obtain further properties of the limit solution as in Propositions 9, 10 below (see also Remarks 6,7). (12) is such that the bottom spectrum of the operator A C = −∆ − C(·)I in L 2 is strictly positive then in Theorem 2.5 we will also have that Proposition 10. If the assumptions of Theorem 2.5 hold with k = 1 then the following hold. i) (charge conservation) The solution will satisfy the equality ii) (energy inequality) If, in addition, holds forsome s < 1 with a function g = g(y 1 , y 2 , y 3 ) such that lim y3→0 g(y 1 , y 2 , y 3 ) = 0 uniformly for (y 1 , iii) (uniqueness) The solution will even be unique if similarly as in [6, Corollary 3.3.11] one assumes that given r > 0 there exists L(r) > 0 such that The above mentioned results will be proved in the following two sections.
3.1. Generalities concerning operators A ± p,η := e ±iθη A p with η ∈ (0, 1]. Given p ∈ (1, ∞), η ∈ (0, 1] and Re(λ) ≤ 0 we have thatλ = λe ±iθη ∈ ρ(A p ) because σ(A p ) consists of strictly positive eigenvalues separated from zero. Therefore the which proves Proposition 2; in particular, A ± p,η is a sectorial operator in X p := L p . Given η ∈ (0, 1] the initial boundary value problem for the approximate equations (3) can be thus viewed as an abstract Cauchy problem with A ± p,η = e ±iθη A p and f e η,± = e ±iθη f e . By [20, Theorem 4.9.1] (see also [11]) for each η ∈ (0, 1] A ± p,η possesses bounded imaginary powers, that is, and the domains of fractional powers X α p,η, Although we have assumed that Ω is a smooth domain let us remark that [20, Theorem 4.9.1] requires ∂Ω to be of the class C ∞ . The latter can be weakened following [11], where in the case of the second order operators it is required that ∂Ω is of the class C 2 .
On the other hand note that the discussion concerning boundedness of imaginary powers can be avoided if p = 2 as in Proposition 3 or if one considers the interpolation scale instead of the fractional power scale. We do not pursue this here focusing on the main aspects of the parabolic approximation procedure, thus using fractional powers as the natural tools of the theory.
Following [2] operators A ± p,η can be considered as closed operators on the extrapolated space X −1 p,η,± being the completion of the normed space (L p , (A ± p,η ) −1 · L p ). We then have where p, p are Hölder's conjugate exponents and X −1 p =: Y p denotes the extrapolated space of (X p , A p ). Also, the closed extension of A ± p,η to X −1 p,η,± (for which we use the same notation) has the same resolvent set as A ± p,η in X p,η,± , belongs to a class of linear isomorphisms from X p,η,± into X −1 p,η,± and generates a strongly continuous analytic semigroup (see [ p,η ) the fractional power scale {Y α p,η,± : α ≥ 0} and obtain by duality argument (see [20, §1.11.3 and 3.2. Proof of Theorem 2.1. In the proof of local well posedness of (3) | η∈(0,1] inḢ k p we rely on the formulation of the problem as in (27) and on the approach developed in [3]. Hence all what needs to be shown is that, under hypothesis H k p the Lipschitz type condition holds for k = 1 and k = 0 respectively with certain constants c > 0, ε ∈ (0, 1 ρ ) and With the set up as in Section 3.1 condition (32) follows from (30) and from hypothesis H k p . Then [3, Corollary 1] ensures that (3) is locally well posed inḢ k p . When (c 1 ) or (c 2 ) holds we also have the blow up alternative (6) (see [3,4]). Following [3,14] we additionally have that for γ as in (32)-(33) and for any θ ∈ [0, 1) the solution u constructed above satisfies that is, 3.3. Sample nonlinearities. We exhibit here properties of sample nonlinearities which appeared in Section 1.
In the remaining case when k = 1 and N ≤ p we have that H k+2ε p → L ∞ . Hence after using in (35) Hölder's inequality with any conjugate exponents we will have the right hand side bounded by the right hand side of (5). In this latter case hypothesis H k p is thus easily satisfied and any triple (ρ, ε, γ) such that ρ > 1, ε ∈ (0, 1 2ρ ] and γ ∈ [ερ, 1 2 ] is admissible triple. Having proved Proposition 4 we now observe that if for h : R → R condition (8) is assumed then and, consequently, The proof of Proposition 5 follows thus the lines of the proof of Proposition 4.

Critically growing map satisfying assumption of Theorem 2.2 and 1-condition.
We exhibit here properties of the map f h defined in Example 3. First note that condition (10) is straightforward as f (x, u)ū = −a(x)|u|h(|u|) and a is real. Since a, h are nonnegative, (12)-(13) hold even with C = D = 0.
Next, as in [6, p. 60], we write and using (38) we get for any ζ > 0 and some C ζ > 0 that Consequently, for f in (15) we have where c = 2 a L ∞ , ρ c (1, 2) = N +2 N −2 , N ≥ 3, and repeating the proof of Proposition 4 we conclude that hypothesis H 1 2 holds with γ = ρ c (1, 2)ε and arbitrarily small ζ > 0. We now define H(x, s) = −a(x) s 0 h(r)dr and consider a functional Note that such F is well defined for ψ ∈ H 1 because, due to (38) and boundedness of a, |H(x, s)| is bounded from above by a multiple of 1 + |s| ρc(1,2)+1 whereas H 1 → L ρc(1,2)+1 . As in the proof of [6, Proposition 3.2.5 (i)] we obtain that for a.e. x ∈ Ω 1 Hence, using dominated convergence theorem we infer that This with f as in (15) implies (41) is Gâteaux differentiable for each u ∈ H 1 and F = f . Since, due to (40), f ∈ C(H 1 , H −1 ) we get (11). Concerning validity of k-condition with k = 1 we remark that by (40) |f (x, z)| is bounded from above by a multiple of 1+|s| ρc (1,2) . Hence, recalling that It thus remains to show (19) for which we use that H 2 → L r and L r → H −2 for every r > 1, N −2 we thus obtain (19) with k = 1.

ALEXANDRE N. CARVALHO AND JAN W. CHOLEWA
Remark 3. In Lemma 3.2 instead of (12)-(13) one can assume alternatively that with some C ∈ L r , r > N 2 , r ≥ 1 and D ∈ L 1 . However the proof of Lemma 3.2 below indicates that to derive H 1 bound on the solutions (46) is not as suitable as (12).
Recall now from Proposition 8 that a closed extension of −A ± 2,η , which we denote the same, is an infinitesimal generator of the semigroup of contractions inḢ σ for any σ ∈ [−2, 0] and the resolvent set of −A ± 2,η inḢ σ coincides with the one in L 2 (Ω).
Thus, with a similar argument as in the proof of Proposition 7 we obtain the following convergence result.

4.2.
Proof of Theorem 2.5. Suppose that k ∈ {0, 1} is given, η n → 0 and let u ηn ± be the solution of (3) η=ηn through u 0 ∈Ḣ k as in Theorem 2.1. Rewriting (3) as we infer from k-condition and Lemmas 3.1, 3.2 that the sequence {u ηn 2 )} for each T > 0. Following regularity properties of approximate solutions expressed in (34) and applying Arzela-Ascoli theorem (see [12, §7.5]) we then have, choosing a subsequence which is still denoted the same, that and (54) Using properties of weak limits we also infer that Furthermore, u ± being continuous in Y With the aid of functions u ± we now define u as u(t) = u + (t) for t ≥ 0 and u(t) = u − (−t) for t < 0.

Proof of Proposition 9.
If the solutions of the linear problem (50) in L 2 (Ω) are asymptotically decaying then, due to (51), (52) we will have in the proof of ). This and (57) will then lead to (21).
We finally remark that if (25) is assumed then the uniqueness result follows by a standard application of Gronwall's lemma (see [6,Corollary 3.3.11] for details). (14) assuming that a, b ∈ L ∞ (R N ), V ∈ L r (R N ) are real valued functions, r > N 2 , r ≥ 1, 1 <ρ < ρ and ρ < N +2 N −2 when N ≥ 3 (see [6, Lemma 3.3.7]). 5. Some comments concerning a critical regime. Critical exponents appear naturally when proving well posedness of (1) η∈(0,1] . Hence, the approach to NLSlike equations via parabolic approximation procedure naturally includes the case of a critical regime. Focusing on the energy spaceḢ 1 , note that some limitations of the approach come from the fact that in the critical regime, that is when hypothesis H 1 2 holds with γ = ρε, without suitable smallness assumption on the parameter ζ in (5) it is generally unknown whether the H 1 -estimate of the solutions to (1) η∈(0,1] guarantees their global existence. However, once the global existence is established, then validity of the k-condition with k = 1 leads to the construction of a limit solution satisfying Duhamel's formula (20).
Note that another condition can sometimes be used to determine whether the solutions of (1) η∈(0,1] exist globally in time. Namely, if under hypothesis H 1 2 , u is a solution through u 0 ∈Ḣ 1 defined on a maximal interval of existence [0, τ u0 ) then τ u0 = ∞ if and only if  [21]). Nonetheless, verification of (67) for problems in the critical regime has not been satisfactorily addressed as yet in the literature (see e.g. [4, (4.19)-(4.24) and Corollary 4.2] for some results in that matter).
Appendix A. Auxiliary results. We include here some useful results, which we adapt from [9].