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Existence and asymptotic behavior for L^2-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the L^2-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in H^1. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive.

Let us note that the functional µ [u] ensures that the L 2 norm is preserved along the flow, namely u(t) L 2 = u 0 L 2 , for any t > 0. Hence, µ[u] may be interpreted as a Lagrange multiplier that takes into account the fact that the solution is constrained to stay on a given sphere in L 2 , whose radius is prescribed by the initial datum. In this respect, equation (1) may be viewed as an L 2 gradient flow constrained on a manifold. More precisely, let λ = u 0 L 2 and let us define  where ∇ M is the L 2 −gradient projected onto the tangent space T u M and the energy functional is defined by Nonlocal heat flows arise in geometrical problems where some L p norms, related to specific geometrical quantities, are required to be preserved by the dynamics, see for instance [2], [24] and the references therein. In particular, the Yamabe flow was set up in order to show the existence of a solution to the celebrated Yamabe problem. More precisely, the conformal transformation bringing a metric g 0 , with scalar curvature R 0 , into a metric with constant scalar curvature can be determined by the asymptotic limit for large times of the evolutionary problem where ∆ 0 is the Laplace-Beltrami operator and p = 2 * − 1. Here the Lagrange multiplier s(t) is determined in such a way that the volume, proportional to |u| 2 * , is preserved. We address the interested reader to [28,5] for more details.
Our study is also motivated by some recent numerical works that exploit normalized gradient flow methods to compute the ground states for some models in Bose-Einstein condensation, see [10,11,3] for instance. In this perspective, our results may be seen as a rigorous justification to the methods proposed in those papers, see Theorem 1.7 below.
Equation (1) with g = 0 was already rigorously studied in [8]. Apart from the nonlocal term, equation (1) becomes linear when g = 0. This fact allows for a detailed description of the asymptotic behavior of the solution, which is characterized by the initial condition. In [8], this result is also used to study a singularly perturbed heat flow, with applications to an optimal partition problem [7], see also [12] for a numerical implementation of the method.
In [19], the authors consider equation (1) on a closed manifold (i.e., a compact manifold with no boundary) and g < 0. They show global existence of solutions and study their asymptotic behavior. Let us remark that this choice of g makes the energy functional (3) always non-negative definite.
In our work, we extend the above results by considering g ∈ R and the case when either Ω is a bounded domain or Ω = R d . The positive sign of g implies that the energy does not control theḢ 1 norm of the solution in general. In particular, we show the existence of global in time solutions in the case g > 0 and 0 < σ < 2/d, for arbitrarily large initial data. Let us remark that, under the same assumptions g > 0 and 0 < σ < 2/d, the classical semilinear parabolic equation, i.e. equation (1) with µ = 0, experiences possible finite-time blow-up, see [22,Theorem 17.6] for instance. The L 2 constraint provided by the dynamics (1) then prevents this kind of singularity formation.
Moreover, in the case of σ ∈ 2 d , 2 (d−2) + and g ≥ 0, it is possible to exploit some arguments borrowed from the standard potential well method (see [15,21] for instance) to show global existence of solutions for some initial data. More precisely, we determine a subset of H 1 that is invariant with respect to the dynamics and initial data belonging to that region emanate global solutions. On the other hand, we also identify a set of initial data whose evolution experiences a grow-up behavior, see Theorem 1.5 below.
Our second main goal is to investigate the asymptotic behavior for large times. As in [19], by standard compactness methods it is possible to show that along sequences of times going to infinity, the solution is converging to a stationary solution. On the other hand, when Ω is a bounded domain we can further improve this result by showing that solutions emanating from non-negative initial data converge to the ground state solution, see Theorem 1.7 for a more precise statement.
We now present our main results. First of all, we prove the local well-posedness for the Cauchy problem (1). Let us remark that the usual fixed point argument, see for instance [22,Sect. 16], cannot be applied to (1) in a straightforward way. Indeed, there is a quite delicate interplay between the power-type nonlinearity and the nonlocal term. A general proof of the local well-posedness result appears to be missing so far. For this reason, in proving Theorem 1.1 we are going to adopt different strategies. We refer to the beginning of Section 3 for a more detailed discussion.
Let Ω be a regular bounded domain with C 2 boundary or Ω = R d . Then for any u 0 ∈ H 1 0 (Ω) there exists a maximal time of existence As previously mentioned, in the case of g > 0, the energy functional in (3) is indefinite, hence global well-posedness does not follow straightforwardly from available a priori bounds.
In the following theorem, we present some conditions under which the solutions constructed above can be extended globally.
Let Ω be a regular bounded domain with C 2 boundary or Ω = R d . Then for any u 0 ∈ H 1 0 (Ω), there exists a unique global solution u ∈ C([0, ∞), H 1 0 (Ω)) to (1). The next theorem will also provide conditions under which the solution is global. In order to state it, let us define and Ω is a bounded domain. Since Ω is a bounded domain, W is a nonempty set, see Section 2.4 and 3.4 for more details. In the spirit of the potential well method, see [21] for instance, we have the following global existence result.
The potential well method can also be used in the whole Euclidean space R d to obtain additional sufficient conditions for global existence, as in [15]. In order to state the next theorem, we introduce the following notations. Let 2 d < σ < 2 (d−2) + and let Q be the unique positive solution to Let K be defined as Once again, we note that the above set is nonempty (see Section 3.4). Then the following holds true: While it is interesting to further extend the range of models for which global well-posedness holds true, another main open problem is to study the possible occurrence of finite-time blow-up in some specific regimes. Let us recall that the usual arguments for nonlinear parabolic equations (see for instance [22,Theorem 17.1]) do not apply here, due to the conservation of the L 2 norm. On the other hand, our next result shows a grow-up scenario for some data, in the case of g ≥ 0 and σ ≥ 2 d . Notice that such conditions complement the hypothesis of Theorem 1.2 where global existence is proven for any data.
After having showed the existence of global solutions, we now turn our attention to their asymptotic behavior. When g = 0 in (1) and Ω is bounded, the eigenspaces associated to the Dirichlet Laplacian are invariant along the dynamics. As a byproduct, the solution asymptotically approaches the eigenspace of least energy, that contains a non-trivial component of the initial datum, see [8] where this is proved by explicit calculations. On the other hand, allowing for g = 0 leads to more complex dynamics. In [19], this problem is addressed in the case of g < 0 for bounded domains where they prove convergence to some stationary state (no uniqueness is given) for sequences of times going to infinity. The next proposition extends this result to arbitrary g = 0 for both Ω bounded and Ω = R d .
Let us emphasize that, in the previous proposition, we may only obtain u ∞ L 2 ≤ u 0 L 2 , because of possible loss of mass at infinity (when Ω = R d ). In fact, this result may be improved when considering a bounded domain. By exploiting further compactness properties, available in this case, it is possible to show the strong convergence of the sequence. Moreover, in the case when g ≤ 0, 0 < σ < 2 (d−2) + or g > 0, 0 < σ < 2/d, it is well known (see Section 2.3) that there exists a unique positive, radially symmetric stationary solution, that is also a minimizer of the energy functional (3) under a constraint on the total mass Such a solution is usually called the ground state solution. The maximum principle then ensures that global solutions emanated by non-negative initial data converge asymptotically to the ground state.
) be the solution to (1). Let Ω ⊂ R d be a bounded domain such that there exists a unique positive solution Q ∈ H 1 0 (Ω) to equation (12) which is also the unique minimizer of the problem (4). Then The problem we address in this paper can also be looked at as an example of a bilinear control system. Indeed, consider the nonlinear heat equation where q is a given smooth function defined on Ω and p ∈ L 2 (0, T ) is a scalar function of our choice. The problem consists of finding, for any initial condition u 0 , a control p which steers the solution to a given target. In [4], this question was addressed for d = 1, with the ground state solution as target. This result was later extended, in [1], to arbitrary eigensolutions of evolution equations of parabolic type in one space dimension, under various boundary conditions. Notice that, in both [4] and [1], q is not allowed to be the identity.
On the other hand, the approach proposed in [8] applies to the above problem in arbitrary dimension, with q(x) ≡ 1 and g = 0, by choosing p(t) = µ[u(t)]. In [8], however, the target is attained asymptotically (as T → ∞) also by an appropriate selection of the initial condition. So, in this paper, we extend the approach by [8] to nonlinear flows (g = 0) restricting the target to the ground state (for u 0 ≥ 0).
Finally, let us also mention that a similar analysis can be also performed on other models, such as the Navier-Stokes equations, see for instance [9,6] where the two-dimensional system is studied under various constraints.
This work is organized as follows: in Section 2, we will present some useful preliminary results regarding the heat semigroup generated by the Laplacian on L 2 (Ω), the stationary states of system (1), and the potential well method. In Section 3, we will prove our main results (Theorem 1.1 up to 1.4) regarding the local and global existence of solutions to the system (1). In Section 4, we will study the asymptotic behavior and give the proof of Theorem 1.7 and Theorem 1.5.

Preliminaries
where e t∆ is the Dirichlet heat semigroup defined in Section 2.2. c) Weak solution: We define u to be a weak solution if there exists T > 0 such that u ∈ L 2 ([0, T ], H 1 0 (Ω)) and for any t for any φ ∈ H 2 (Ω) ∩ H 1 0 (Ω) and (φ, u(t)) is absolutely continuous in time. The following result states that any mild solution is actually a strong solution, see [22,Example 51.28,Lemma 17.5].
be the corresponding mild solution to (1). Then the energy

2.2.
Heat semigroup. In this subsection we collect some useful properties of the Dirichlet heat semigroup. We refer the reader to [22] for more details.
Let Ω be a bounded domain in R d with C 2 boundary. Let −A denote the Dirichlet Laplacian in L 2 (Ω), that is the Laplacian on L 2 (Ω) subject to homogeneous Dirichlet boundary conditions. Then −A is a nonnegative self-adjoint operator and it generates a C 0 -semigroup e −tA on L 2 (Ω). In this work, we will denote with e t∆ both the semigroup e −tA on L 2 (Ω) and L 2 (R d ). Note that the function u = e t∆ u 0 solves the linear heat equation is the Gaussian heat kernel. Let us notice that this implies the following inequality for any Ω ⊂ R d , x, y ∈ Ω and t ∈ (0, ∞). In view of the discussion above, we can prove the following L p − L q smoothing estimates of the heat semigroup (see [22,Proposition 48.4]): , As a consequence of the L p − L q smoothing property in Proposition 2.3, we also obtain the following space-time estimates: . Then for any f ∈ L 2 , we have Moreover, if (p, s) is another pair satisfying condition (8), then , We next give a useful smoothing estimate for the gradient. The case Ω = R d follows directly from calculations involving the Gaussian heat kernel (7). For a general bounded domain Ω, see [17,Theorem 16.3].

Proposition 2.5.
Let Ω be a regular domain of class C 2 boundary. Let {e t∆ } t>0 be the heat semigroup in R d or the Dirichlet heat semigroup in Ω. Then there exists a constant C > 0 such that for any

Stationary states.
In this subsection we recall some results concerning the stationary solutions of equation (1), namely satisfying Notice that if the problem , then Q solves equation (12) where the constant µ[Q] is a Lagrange multiplier. Moreover we recall the following Pohozaev's identities: (12). Then (14) µ Proof. We take the scalar product of (12) with Q and obtain (14). Similarly, we obtain (15) by taking the scalar product of (12) with x · ∇Q.
As a consequence of (14), we have that In what follows, we denote by B(0, R) = {x ∈ R d ; |x| < R} a ball in R d , centered in zero. We start with the following standard proposition (see [18,23,25]): Then for any α > 0, the minimizing problem (13) admits a nonnegative, radially symmetric minimizer.
We proceed by recalling the result of Gidas, Ni, Nirenberg [14] stating that any nonnegative solution to (12) with Ω = B(0, R) or Ω = R d is radially symmetric: (12) and let Ω = B(0, R) or Ω = R d . Then Q is radially symmetric, strictly positive, and strictly decreasing in |x|.
Next, we recall the result stating uniqueness of positive, radial solutions due to Kwong [16] and Mcleod-Serrin [20]: where Let us notice that Propositions 2.7, 2.8, 2.9 imply the following: . Then for any α > 0, the unique, positive minimizer to the problem is also the unique, strictly positive solution to (16) ∆Q + g|Q| 2σ Q + µ[Q]Q = 0.

Potential well in a bounded domain.
In this subsection, we present some preliminary definitions and results regarding the potential well method in a bounded set. We use this method to provide an alternative sufficient condition for global existence for the Cauchy problem (1), stated in Theorem 1.3. This argument was firstly introduced in [26] and [21] for parabolic problems. We refer the interested reader to [22,Section 19] for more details.
Let Ω be a bounded domain, 0 < σ < 2 d−2 and g > 0. We fix g = 1 without losing generality to facilitate the exposition. We define the functional I by Notice that I can also be written as (17) I where E is the energy functional defined in (3).
The potential well associated with problem (1) is the set We also define the exterior of the potential well as (21) is attained for some v ∈ H 1 0 (Ω), due to the compactness of the embedding H 1 0 (Ω) ֒→ L 2σ+2 (Ω). By multiplying v for a suitable constant ν, we may suppose that I[v] = 0. Therefore Thus, there exists an element v so that E[v] = p where p is defined as in (20) and To show that such a p is the infimum of the problem (19) we notice that for any Next, we state a sufficient smallness condition for a function in H 1 (Ω) to belong to W: Proof. Notice that the condition ∇f L 2 < √ 2p implies that On the other hand, using (20), (21) and This implies that I[f ] > 0 and consequently that f ∈ W.
2.5. Potential well method in R d . In this subsection, we present some preliminary results regarding the potential well method in the whole Euclidean space R d . This argument was introduced in [15] to prove a sufficient condition for global existence of the Nonlinear Schrödinger equation. We suppose that 2 d < σ < 2 (d−2) + and g > 0. We fix g = 1 without losing generality. We recall that the best constant in the Gagliardo-Nirenberg inequality is given by where Q is the unique positive solution of the elliptic equation In [27], this profile is found as the maximizer of the following problem We define the potential well as the set Finally, we observe that Indeed, from the Pohozaev's identities (14), (15) it follows that This implies (24).

Existence of Solutions
In this Section we are going to prove our local and global well-posedness results. As already mentioned in the Introduction, there is a delicate interplay between (the mathematical difficulties determined by) the power-type nonlinearity and the nonlocal term involving µ[u]. More precisely, it does not seem possible to prove Theorem 1.1 by a general contraction argument. Indeed, the presence of the nonlocal term would require to set up the fixed point in Sobolev spaces, say L ∞ t H 1 0 for instance. On the other hand, the power-type nonlinearity is not locally Lipschitz in Sobolev spaces for low values of σ, say 0 < σ < 1 2 . For this reason, we are going to provide three different proofs for the local well-posedness.
The first one, stated in Theorem 3.1 below, constructs a sequence of approximating solutions by means of an iterative procedure. The solution of the original problem is then found by a compactness argument. For this reason the proof is well suited when dealing with a bounded domain, but it fails in the case Ω = R d , where, in general, the L 2 constraint may not be satisfied. The second proof, stated in Theorem 3.6, is based on a fixed point argument and crucially relies on smoothing properties of the heat semi-group, see Proposition 2.5. This proof requires the nonlinearity to satisfy the condition σ < 1 (d−2) + . The third proof relies on the classical fixed point argument. In order to overcome the difficult interplay between the nonlocal term and the power-type nonlinearity, a lower bound on σ is required. In particular, for σ ≥ 1 2 , the power-type nonlinearity is locally Lipschitz in Sobolev spaces and we we perform a contraction argument using the natural and stronger distance. We remark that unifying the results of the second and third proof, we cover all the energy subcritical range 0 < σ < Finally in subsection 3.4 we globally extend the previous results, under the assumptions on the nonlinear term stated in Theorem 1.2 above. Then we provide addition sufficient conditions on the initial data for global existence, as stated in Theorem 1.3 and Theorem 1.4, using the potential well method.

Local well-posedness for bounded domains.
In this subsection we present a second proof for local well-posedness, that covers the case 0 < σ < 2 (d−2)+ and Ω ⊂ R d bounded. A similar argument was also used in [19].
The proof is based on an iteration argument that constructs a sequence of approximating solutions as follows. Given the initial datum u 0 ∈ H 1 0 (Ω), we start by defining u (0) ≡ u 0 . Let us now assume we already constructed u (k−1) for k ≥ 1, we define u (k) to be the solution to the following nonlinear problem, For this reason, we first state a local well-posedness result for the model problem where the functional µ[u] is replaced by a given, uniformly bounded function of time.
admits a unique local-in-time solution u ∈ C([0, T ], H 1 0 (Ω)). Proof. The proof follows from the contraction principle. Let T, M > 0, to be chosen later, let ρ = 2σ + 2 and γ ∈ R be such that the pair (γ, ρ) satisfies condition (8) We define the space endowed with the following distance Notice that (Y, d) defines a complete metric space. Given u 0 ∈ H 1 0 (Ω) and λ ∈ L ∞ (R), we consider the map Our goal is to prove that G : Y → Y is a contraction with the distance defined in (28). By using the space-time estimates (9), (10) with (q, r) = (p, s) = (γ, ρ) and then Holder's inequality and Sobolev's embedding, it follows that there exist two constants K 0 > 0 and K(M, λ L ∞ ) > 0 such that Moreover, for any u, v ∈ Y, we deduce that Thus, given any u 0 ∈ H 1 0 (Ω), we choose M = 2K 0 u 0 H 1 and we choose T small enough so that Note that T depends only on u 0 H 1 and µ L ∞ [0,T ) . It follows that G(u) ∈ Y and In particular G is a contraction in Y. By Banach's fixed-point Theorem, G has a unique fixed point u ∈ Y. u solves the integral equation and u ∈ C([0, T ], H 1 0 (Ω)). The following corollary immediately follows from the previous lemma.
be the corresponding local solution to (27). Then there exists Moreover, there exists a constant C > 0, depending only on u 0 H 1 , so that Proof. The time of existence in Lemma 3.2 depends only on u 0 H 1 and λ L ∞ [0,T ) and u ∈ C([0, T ], H 1 0 (Ω)). So by continuity, we can find a time such that (30) is true. By multiplying equation (27) by u and integrating in space and time we obtain Since u ∈ C([0, T ], H 1 0 (Ω)) and the right-hand-side of (33) is a L 1 loc (R) function of time. Hence there exists T 2 ( u 0 H 1 , λ L ∞ [0,T2] ) > 0 such that condition (31) is true.
Next, using an iterative process, we are going to prove that a solution to (1) exists. The uniqueness of solutions will be proved in Proposition 3.5 below. Proof. The proof follows an iterative argument. Let u 0 ∈ H 1 (R d ) and let us define u (0) ≡ u 0 . For any k ∈ N, we define a sequence {u (k) } k∈N as solutions to the following Cauchy problem Applying Lemma 3.2 for any k ∈ N, we find a local solution u (k) ∈ C([0, T k ], H 1 0 (Ω)) to (35). We claim that there exists a timeT > 0, independent on k ∈ N, so that for any k ∈ N, u (k) ∈ C([0,T ], H 1 0 (Ω)) and Assume that it is true up to k ∈ N. In particular, we have This argument implies that we can chooseT = T * ( u 0 H 1 , C( u 0 H 1 )) to conclude that for any k ∈ N, Hence the sequence {u (k) } ⊂ C([0,T ], H 1 0 (Ω)) is uniformly bounded. Now we are going to prove that {u (k) } is uniformly bounded also in L 2 ((0,T ), H 2 (Ω)). This will allow us to exploit an Aubin-Lions type argument to extract a converging subsequence. In order to prove this further bound, we multiply equation (26) by ∆u (k) , and integrate in space, to get Integrating this equation in time leads to , By the Hölder's inequality, it follows that .
Since the proof of Proposition 3.4 uses a compactness method to show the existence of solutions, their uniqueness does not follow from the previous arguments. Proof. Let u 0 ∈ H 1 0 (Ω) so that u 0 H 1 = M and suppose that u, v ∈ C([0, T ], H 1 0 (Ω)) are two different strong solutions to (1). Here we choose T > 0 so that if T u , T v > 0 are the maximal times of existence of u, v then T < min(T u , T v ). In particular, there exists a constant C = C(M ) > 0 such that By multiplying the equation and integrating in space, we obtain We start by estimating the contribution given by the power-type nonlinearities. We suppose that d ≥ 2. Notice that . We choose the exponent p so that and consequently By Gagliardo-Nirenberg and Young's inequalities, for any δ > 0, we obtain Thus, it follows that for any δ > 0, we have Next, we deal with the nonlocal term. We observe that By Young's inequality, for any δ > 0 we have From (39) and the triangular inequality we also obtain that, for any δ > 0, In particular, from (38), (40) and (39) we have For δ < 1 2C(M) , we obtain by Gronwall's lemma that  (1). Moreover, either T max = ∞, or T max < ∞ and we have We shall need the following lemma, where the restriction on σ is justified by direct application of Sobolev's embedding in (43) below.
Proof. The first term on the left-hand-side of (42) is bounded by To obtain the bound on the second term on the left-hand-side of (42), we observe that The condition 0 < σ < 1 (d−2) + implies that 2 < 4σ + 2 < 2d (d−2) + . Thus Sobolev's embedding theorem implies that there exists a constant C(M ) > 0 such that Next, we obtain the bound in (43). Suppose that d > 2. Then, by Hölder's inequality, we get Let us choose .
Inequality (43) follows from Sobolev's embedding theorem. Finally, observe that the case d ≤ 2 follows from a similar and easier proof.
Proof of Theorem 3.6: Fix T, M > 0 to be chosen later and consider the set Clearly, (X , d) is a complete metric space. Given Our goal is to prove that the map First of, we all observe that, for u ∈ X , by Gagliardo-Nirenberg's inequality there exists a constant C(M ) > 0 such that ). Observe that, for 0 < σ < 1 (d−2) + , we have 2 < 4σ + 2 < 2d (d−2) + . Thus, by Sobolev's Embedding theorem it follows that there exists a constant C(M ) > 0 such that By the smoothing property of the heat semigroup for the gradient stated in (11), we infer that Notice that, since σ < We choose M = u 0 H 1 and T ≤ M C(M) . This choice implies that F : X → X . Next we show that F is a contraction map on X , namely there exists 0 < K < 1 such that, for any u, v ∈ X , we have From the smoothing effect of the heat semigroup (11) we get (47) As a consequence of inequality (42) we have that, for any u, v ∈ X there exists a constant C(M ) > 0 such that From (48) we obtain that Moreover, from the inequality (43) we obtain From inequalities (47),(49),(50) we have By choosing 0 < T such that T 1 2 C(M ) < 1 we then conclude that F is a contraction in X . Thus F admits a fixed point u ∈ C([0, T ], H 1 0 (Ω)), which is a solution to (51) ∂ t u = ∆u + g|u| 2σ u + ω[u]u, u(0) = u 0 ∈ H 1 0 (Ω).
By multiplying equation (51) for u and integrating in space, we obtain that Since u(0) = u 0 , the L 2 -norm of u(t) is conserved in the whole interval [0, T ), and so system (51) is equivalent to (1).
Observe that the uniqueness follows from the fact that the map F is a contraction in X . Moreover, notice that we can extend the local solution until the H 1 -norm of u(t) is bounded. Hence we obtain the blow-up alternative, that is, if T max > 0 is the maximal time of existence, then T max = ∞, or lim t→Tmax ∇u(t) L 2 = ∞.

3.3.
Local well-posedness for σ ≥ 1 2 . In this subsection we present our third proof for the local well-posedness result, based on a fixed point argument. We state the following theorem for the case Ω = R d , however it is straightforward to see that the same proof applies also to the case of bounded domains. The necessity of the restriction σ ≥ 1 2 is further commented in the remark below the proof of the next theorem. Proof. Fix M, T > 0, to be chosen later, let r = 2σ + 2 and q be such that the pair (q, r) satisfies condition (8). Consider the set Clearly (E, d) is a complete metric space. Consider now u, v ∈ E and observe that the inequality Moreover, using the embedding H 1 R d ֒→ L r R d and Hölder's inequality, it is true that Using Hölder's inequality in time, we deduce from the above estimates that Next, given u 0 ∈ H 1 R d , u 0 = 0, let ω[u] be defined as in (44). Let us observe that the inequality From the embedding H 1 R d ֒→ L r R d we deduce that there exists a constant C(M ) > 0 such that Let H(u) be defined by It follows from Proposition 2.4, (46) and the estimates above that and there exists a constant K(M ) > 0 such that We set M = 2C u 0 H 1 and we choose T small enough so that These choices imply that H maps E into itself and it is a contraction. This is enough to conclude that there exists a local in time mild solution to (51). The uniqueness and the blow-up alternative (25) follow from standard arguments. Finally, the equivalence between equation (51) and (1) follows with the same argument as in the proof of Theorem 3.6.
Remark. Let us emphasize why we need the condition σ ≥ 1 2 . On one hand, in the absence of the nonlocal term µ, we could use the contraction principle in the set E with the weaker distance Indeed, it is standard to prove that (E, g) is a complete metric space. On the other hand, the presence of µ requires a stronger distance induced by the L ∞ t H 1 x norm, as it is clear from inequality (53). This implies that we have to use the distance d instead of g. But for σ < 1 2 , this is not possible because the power-type nonlinearity is not locally Lipschitz continuous in Sobolev spaces (inequality (52) fails).

3.4.
Global well-posedness. From Proposition 2.2 we obtain that for any u 0 ∈ H 1 0 (Ω) and t ∈ [0, T max (u 0 )], the energy of the corresponding solution satisfies and thus it is non-increasing in time. The global existence follows from a classical argument which combines the blow-up alternative (25) and the bound on the energy.
Proof. Let u 0 ∈ H 1 0 (Ω) and let u ∈ C([0, T max ), H 1 0 (Ω)) be the corresponding local solution to (1) given by Theorems 3.1, 3.6 or 3.8. Then the L 2 -norm of the solution is constant and the energy is non increasing in time. Thus for g ≤ 0, 0 < σ < 2 (d−2) + , we have , while for g > 0, σ < 2 d , we use the Gagliardo-Nirenberg's inequality and dσ < 2 to conclude that In particular, in both cases the H 1 -norm of the solution is uniformly bounded in time. From the blow-up alternative (25) we conclude that the solution is global.
In what follows, we will provide an alternative sufficient condition for global existence in bounded domains, as stated in Theorem 1.3. The proof is based on the potential well method which was introduced in Section 2.4. We start with the following lemma where we use the notations introduced in Section 2.4: Lemma 3.10. Let g > 0 and 0 < σ < 2 (d−2) + . If u 0 ∈ W (respectively u 0 ∈ Z), then the corresponding solution u ∈ C([0, T max (u 0 )), H 1 (Ω)) to (1) is so that u(t) ∈ W (respectively u(t) ∈ Z) for all t ∈ [0, T max (u 0 )).
Proof. Fix g = 1 without losing generality and suppose that u 0 ∈ W. Let T = T max (u 0 ) be the maximal time of existence of the corresponding solution to (1). Equation (6) implies that E[u(t)] ≤ E[u 0 ] < p, for all t ∈ [0, T ).
Since I[u 0 ] > 0, it follows by the continuity of the flow associated with (1) that I[u(t)] > 0 for all t ∈ [0, T ). Indeed, by contradiction, suppose that there exists a time t 0 > 0 such that I[u(t 0 )] = 0 and E[u(t 0 )] < p. Notice that this contradicts the definition of p. Hence u(t) ∈ W.
The same argument applies for the case when u 0 ∈ Z.
Lemma 3.10 implies that the sets W and Z are invariant under the flow of equation (1). As a consequence, we obtain the following Theorem 3.11. Let g > 0 and 0 < σ < 2 (d−2) + . If u 0 ∈ W, then the corresponding Proof. We fix g = 1 without losing generality. The condition u 0 ∈ W and Lemma 3.10 imply that u(t) ∈ W for every t ∈ [0, T max (u 0 )) and consequently that Hence the H 1 -norm of u is uniformly bounded by (2σ+2)p σ . From the blow-up alternative (25), we conclude that T max (u 0 ) = ∞.
Remark. Proposition 2.12 provides in particular a sufficient smallness condition on the initial datum for global existence. Indeed, if u 0 ∈ H 1 (Ω) is so that ∇u 0 2 L 2 ≤ 2p, then u 0 ∈ W.
Remark. Notice that when Ω is a bounded domain, the set W is not empty because ∇u L 2 < √ 2p implies u ∈ W while the set Z is not empty since if E[u] ≤ 0, then from (17), I[u] < 0 and u ∈ Z.
On the other hand, when Ω = R d , we have p = 0 and W is empty. Indeed, take any u = 0 such that I[u] = 0. Notice that E[u] > 0. Let v(x) = λ 1 σ u(λx). Then Thus, for λ → 0, we obtain that E[v] → 0. As a consequence, from (17), the set W is empty when Ω = R d . and Theorem 1.3 does not provide any additional sufficient conditions for the global existence in H 1 (R d ).
Finally, if u ∈ H 1 (R d ) is such that E[u] < 0, then u ∈ Z so Z is not empty. Note that (14) and (15) imply that a stationary state Q, solution to (12) in R d , satisfies , and thus, when σ < 2 d , Q belongs to Z.
We proceed by providing additional sufficient conditions for global existence in the whole space R d , as stated in Theorem 1.4. We use the same notations introduced in Section 2.5. We start by proving that the set K is invariant under the flow of equation (1): Proof. We suppose that g = 1 without losing generality. By Gagliardo-Nirenberg's inequality (23) it follows that By taking the derivative of f , we notice that it has two critical points f ′ (0) = 0, and f ′ (x 1 ) = 0, and is a local maximum. From (24), we can write (23) is an equality for Q, we have (55) and (6) we obtain that for any t ∈ [0, T max (u 0 )), the following inequality is true which is in contradiction with (56). Thus we can conclude that the set K is invariant under the flow of equation (1).
Proof. Since K is invariant under the flow of equation (1), it follows from the conservation of the L 2 norm that for any t ∈ [0, T max (u 0 )), and thus the H 1 -norm of the solution is uniformly bounded in time. The blow-up alternative (25) implies that the solution is global in time.

Asymptotic Behavior
In the following proposition, Ω can be either a bounded domain or the whole R d : (Ω) and let u ∈ C([0, ∞), H 1 0 (Ω)) be the solution to (1). Then there exists a sequence {t n } n∈N , t n → ∞ as n → ∞, such that where u ∞ solves the stationary equation where the last convergence is in a weak H −1 (Ω) sense. In particular, we get that By taking the scalar product of this equation with u ∞ , we obtain that µ ∞ = µ[u ∞ ].
The result of the Corollary 4.2 can be further improved by using the Maximum Principle. In this case, we shall work in a bounded domain Ω where there exists a unique positive solution to the stationary equation (12) which is also the unique minimizer of the problem (4). For example, we can take Ω = B(0, R) and g > 0.
Then we can prove that if u 0 ≥ 0, the corresponding solution converges strongly to the ground state. Theorem 4.3. Let g ≤ 0, σ < 2 (d−2) + or g > 0, σ < 2 d . Let u 0 ∈ H 1 0 (Ω), u 0 ≥ 0, u 0 ≡ 0 and let u ∈ C([0, ∞), H 1 0 (Ω)) be the solution to (1). Let Ω ⊂ R d be a bounded domain such that there exists a unique positive solution Q ∈ H 1 0 (Ω) to equation (12) which is also the unique minimizer of the problem (4). Then u(t) → Q in H 1 0 (Ω) as t → ∞. Proof. Let {t n } n∈N , t n → ∞ as n → ∞, be such that u(t n ) → u ∞ in H 1 0 (Ω) where u ∞ satisfies (12). By the Maximum Principle (see [13,Ch. 2,Sec. 4]), u(t n ) ≥ 0 which implies u ∞ ≥ 0 and u ∞ ≡ 0 since u ∞ L 2 = u 0 L 2 > 0. From the uniqueness of the positive solution to (12), we conclude that u ∞ = Q where Q is the unique minimizer of the energy under the constraint Q L 2 = u 0 L 2 . Equation (6) implies that the energy is a continuous, decreasing function of time. In particular there exists the limit as t → ∞. This limit E ∞ is actually the energy of the ground state because Now we show that u(t) → Q in H 1 0 (Ω). Suppose, by contradiction, that this convergence is not true. In particular there exists a time sequence {t k } k∈N such that u(t k ) L 2 = u 0 L 2 , E[u(t k )] → E[Q] and there exists ε > 0 so that u(t k ) − Q H 1 ≥ ε for all k ∈ N.
Remark. We would like to notice that in general, there exist initial data not converging to the ground state. Indeed if g = 1, σ < 2 d and u 0 ∈ Z, where Z is defined in (22), then u ∞ where u ∞ ∈ Z is a stationary state with µ[u ∞ ] < 0. Similarly, if u 0 ∈ W where W is defined in (18) then u ∞ ∈ W and µ[u ∞ ] > 0. Thus, at least in one of the above cases, the solution does not converge to the ground state.
Finally, we show that in the whole space R d and when σ ≥ 2 d there exists a set of initial data whose solutions satisfy the grow-up condition. Specifically, if E[u 0 ] < 0, then there exists a sequence {t k } k∈N , t k → T max such that lim k ∇u(t k ) L 2 = ∞. Proof. Suppose that T max (u 0 ) = ∞ and that there exists a constant C > 0 such that sup t≥0 ∇u L 2 ≤ C.
Gagliardo-Nirenberg's inequality and the conservation of the L 2 -norm imply that Thus the energy is bounded from below. Since E[u(t)] is a continuous, decreasing function, there exists the limit In the same way, we can see that µ[u(t)] is bounded from below and from above. Then there exists a sequence {t n } n∈N , t n → ∞ as n → ∞ such that      u(t n ) ⇀ u ∞ in H 1 0 (Ω), ∂ t u(t n ) → 0 in L 2 (Ω), µ[u(t n )] → µ ∞ , where the last convergence is in a weak H −1 (Ω) sense. In particular, we get that ∆u ∞ + |u ∞ | 2σ u ∞ + µ ∞ u ∞ = 0, and so u ∞ is a stationary state. From (54) On the other hand by the weak lower semicontinuity which is a contradiction.