¥mathrmL²-Solutions for Nonlinear Schrödinger Equations and Nonlinear Groups

Introduction and main resultsWe consider the unique global existence of solutions in a weaker class than H¹(Rⁿ) and some properties of the solution operator for the following nonlinear Schrodinger equation:where t₀in R and $lambdain R$ .By $a(n)$ we denote $infty$ if $n=1$ or $n=2$ and $(n+2)/(n-2)$ if $ngeqq 3$ .There are many papers concerning the global existence of solutions for Problem (1.1)?(1.2) (see, e.g., $[1]-[2]$ , $[4]-[7]$ and [9]).Ill [1] Baillon, Cazenave and Figueira show that if $1leqq nleqq 3,1<p<a(n)$ and $lambda>0$ , Problem (1.1)?(1.2) has a unique global strong solution u(t)in C(R;H²(Rⁿ))cap C¹(R;L²(Rⁿ)) for any u₀in H²(Rⁿ) .In [₋2] Ginibre and Velo show that if $1<p<alpha(n)$ and $lambda>0$ or if $1<p<$ $1+4/n$ and $lambda<0$ , Problem (1.1)?(1.2) has a unique global weak solution $ u(t)in$ C(R;H¹(Rⁿ)) for any u₀in H¹(Rⁿ) .In [6] Strauss shows that if $lambda>0$ and $p>1$ , Problem (1.1)?(1.2) has at least one global weak solution u(t)in Lⁱⁿᶠᵗʸ(R;H¹(Rⁿ)cap Lᵖ⁺¹(Rⁿ)) for any u₀in H¹(Rⁿ)cap Lᵖ⁺¹(Rⁿ) (see also [5]).In [10] M. Tsutsumi and N. Hayashi mathrmdmathrmimathrmsmathrmcmathrmu_dotmathrmOmathrms the unique global existence of classical solutions for (1.1)?(1.2) with $lambda>0$ (see also Pecher and von Wahl [4]).Furthermore, in [9]M. Tsutsumi discusses the unique global solution in ovalboxttsmall REJECT(Rⁿ) or in the weighted Sobolev space for (1.1)?(1.2) with $lambda>0$ .In almost all of previous papers the solution of (1.1)?(1.2) has been constructed in a space not larger than the energy space, that is, H¹(Rⁿ) , because the proofs in almost all of previous papers are based on the energy conservation law.However, in [7] Strauss constructs the wave operators from L²(Rⁿ) to L²(Rⁿ) for the equation (1.1) with $p=1+4/n$ (see [7, Theorem 5]).His results are almost equivalent to the construction in L²(Rⁿ) of unique local solutions for (1.1)?(1.2) with $p=1+4/n$ .In this paper we prove that when $1<p<1+4/n$ , we can construct the unique global solution of (1.1)?(1.2) for u₀ in L²(Rⁿ) (but possibly not in H¹(Rⁿ) ).Such a solution is