On the existence of infinite energy solutions for nonlinear Schrodinger equations

We derive new results about existence and uniqueness of local and global solutions for nonlinear Schrodinger equation, including self-similar global solutions. Our analysis is performed in the framework of Marcinkiewicz spaces.


Introduction
We consider the nonlinear Schrödinger equation i∂ t u + ∆u = λ|u| ρ u, x ∈ R n , t ∈ R, (1.1) where u = u(t, x) is a complex valued function, λ is a fixed complex number, and 0 < ρ < ∞. The initial value φ : R n → C is given. The Cauchy problem (1.1)-(1.2) is formally equivalent to the integral equation where S(t) is the unitary group determined by the linear Schrödinger equation If φ ∈ S(R n ) and u is defined by u(t)(ξ) = e −i|ξ| 2 t φ(ξ), for ξ ∈ R n , then u t + i|ξ| 2 u = 0 in R × R n . In this case, the solution of is given by u(t) = S(t)φ = K t * φ, where K t (x) = e −i|ξ| 2 t ˇ.
As a corollary, we show that when the initial data φ is a homogeneous function of degree − 2 ρ , we obtain a self-similar solution, if S(1)φ (ρ+2,∞) is sufficiently small. Moreover, we discuss asymptotic stability of the global solutions, and show that regular perturbations of the linear Schrödinger equations are negligible for large times. We also analyze the behavior of the local solutions as t → 0 in the space L (ρ+2,∞) .
Our approach is different from the methods used in [2,16], where the authors use a Strichartztype inequality in weak-L p and Besov spaces, respectively. Indeed, our existence results are based on bounds for the Schrödinger linear group S(t) in the context of Lorentz spaces. In Lemma 2.1, we state and prove these bounds via real interpolation techniques. They generalize the bounds for usual L p spaces used in [7].
In section 2, we carefully state our results and discuss their improvement in the light of previous results. We prove them in section 3.

Main Results
We first recall some facts about the Lorentz spaces. For more details see, for instance, [1] and [17].
Let 1 < p ≤ ∞ and 1 ≤ q ≤ ∞. A measurable function f defined on R n belongs to Lorentz space L (p,q) (R n ) if the quantity Note that L p (R n ) = L (p,p) (R n ). The spaces L (p,∞) (R n ) are called weak-L p spaces or Marcinkiewicz spaces. Lorentz spaces have the same scaling relation as L p spaces, that is, for all λ > 0 one has Morover, Lorentz spaces can be constructed via real interpolation [1]. Indeed, They have the interpolation property where (·, ·) θ,q stands for the real interpolation spaces constructed via the K-method [1].
We begin by bounding the Schrödinger group S(t) in Lorentz spaces.
Proof. Fix t = 0 and let 1 < p 0 < p < p 1 < 2 such that 1 p ′ = λ p 0 + 1−λ p 1 and 0 < λ < 1. By the well known L p = L (p,p) estimate of Schrödinger group, we have that S(t) : where the operator norms are respectively bounded by Through real interpolation, which is equivalent to (2.1).

3)
with respective norms which are weakly continuous in the sense of distributions at t = 0.
Our main results are , then there exists 0 < T 0 < ∞ and n 0 ∈ N such that, for n ≥ n 0 , the solutions u n and u with respective initial data φ n and φ lie in E T 0 α,β and u n → u in E T 0 α,β . Actually, the solution map φ → u is Lipschitz continuous. 1. If φ is a distribution such that sup −∞<t<∞ |t| α 2 S(t)φ (ρ+2,∞) < ε, for ε > 0 small enough, then the initial value problem (1.1)-(1.2) has a global in time mild solution u(t, x) ∈ E α . This solution is the only one satisfying u α ≤ 2ε.
2. Futhermore, if (φ n ) is a sequence of distributions such that S(t)φ n − S(t)φ Eα → 0 when n → ∞, and u n , u are the solutions with respective initial data φ n and φ, then u n → u in E α .
We compare the theorems above with previous results.
• In the range ρ 0 < ρ < 4 n−2 , Theorem 2.5 extends the global solutions results derived in [7] to the framework of Marcinkiewicz spaces. Our range for ρ is also greater than the one in [2] (see 1.7).
• Theorem 2.4 assures the existence of local in time solutions even for singular initial data is a homogeneous polynomial of degree k. As far as we know, there were no previous existence results covering this case. On the other hand, we were not able to obtain self-similar solutions in E α,β though, since the norm · α,β is not invariant by the scaling relation u µ (t, x) = µ 2 ρ u(µ 2 t, µx).
As a direct consequence of Theorem 2.5, one can show the existence of a self-similar solution.
Corollary 2.6. (self-similar solutions) In addition to the hypothesis of Theorem 2.5, if the initial data φ is a sufficiently small homogeneous function of degree − 2 ρ , then the solution u(t, x) provided by Theorem 2.5 is self-similar, that is, u(t, x) = µ 2 ρ u(µ 2 t, µx) for all µ > 0, almost everywhere for x ∈ R n and t > 0. We also analyze the large time behaviour of the solutions given by Theorem 2.5, and study the behaviour of the solutions given in Theorem 2.4 near to time t = 0. These are the content of the following theorem. Let us comment some improvements produced by Theorem 2.8.
• (Decay rate when t → 0) By bound (3.2), one can see that which implies the bound (2.5) for h < δ. Assuming further regularity for φ − ϕ, the second item of Theorem 2.8 extends this property for the range h > −δ.

Proofs
The following Lemma is important to our ends. For its proof, see [8].
Lemma 3.1. Let 0 < ρ < ∞ and X to be a Banach space with norm · . Suppose B : X → X to be a map satisfying B(0) = 0, and let R > 0 be the unique positive root of equation 2 ρ+1 K(R) ρ −1 = 0. Given 0 < ε < R and y ∈ X, y = 0, such that y ≤ ε, there exists a solution x ∈ X for the equation The solution x is unique in the ball B 2ε := B(0, 2ε). Moreover, the solution depends continuously on y in the following sense: If ỹ ≤ ε,x =ỹ + B(x), and x ≤ 2ε, then Now, we state and prove the necessary estimates in order to apply Lemma 3.1 in our case. If nρ 2 < ρ + 2 ρ + 1 , then there exists a positive constant K α,β such that
The continuity of the solutions with respect to the initial conditions, as well as the continuity of the solutions in the sense of distributions, follow as in the proof of Theorem 2.4.

Proof of Theorem 2.8
Without loss of generality, assume t > 0. Subtracting the integral equations satisfied by u and v, one gets .
Since u α , v α ≤ 2ε, one uses the change of variable s −→ ts, and bound Therefore, Using the assumption on the initial perturbation φ − ϕ, it is not difficult to show that A < ∞. Now, note that lim sup −h ds. Choosing ε > 0 sufficiently small such that ε ρ Γ < 1, one concludes that A = 0. This proves part 1 of the theorem.