Math 557
Spring 2010
Assignment 3
Due Thursday April 15, 2010.
Do three of the following.
1. Consider the focusing cubic NLS in R3 ,
i∂t u + ∆u = −|u|2 u,
u(0) = u0 ∈ H 1 .
(1)
Suppose u0 ∈ H 1 is sufficiently small, show that the solution u(t) is global and u(t) ∈
BC([0, ∞), H 1 ).
2. Under the same setting as Problem 1, show that the solution scatters: There is u+ ∈
H 1 so that limt→∞ kS(−t)u(t) − u+ kH 1 = 0.
3. Consider the NLS in Rn with power nonlinearity f (u) = λ|u|p−1 u, λ = ±1 and
1 < p < pmax ,
i∂t u + ∆u = λ|u|p−1 u, u(0) = u0 ∈ Σ.
(2)
Here Σ is the weighted space with the norm kukΣ = (kuk2H 1 + kxuk2L2 )1/2 . Let
Z
φ(t) = |x|2 |u(x, t)|2 dx.
Suppose u(t) ∈ C([0, T ), Σ) and φ ∈ C 2 ([0, T )). Formally verify that
Z
Z
0
00
φ (t) = 4 Im
ūx · ∇u dx, φ (t) = 16E + C1
|u|p+1 dx,
Rn
where E(u) =
R
1
2
2 |∇u|
+
λ
p+1 dx
p+1 |u|
Rn
and C1 =
4n
p+1 λ(p
− pc ).
4. Evolution with L2 -data does not have decay rate. It is shown in class that if
a ∈ L2 (Rn ), then eit∆ a → 0 weakly in L2 in the sense that (eit∆ a, φ) → 0 for any
φ ∈ L2 . Show that for any decreasing continuous function g(t) : [0, ∞) → (0, ∞) with
limt→∞ g(t) = 0, no matter how slowly, we can find a ∈ L2 so that
Z
1
sup
|eit∆ a|2 dx = ∞.
g(t)
t>0
|x|<1