Abstract
AbstractThis paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on $$M^N$$
M
N
, an N-dimensional Cartan-Hadamard manifold. Namely we consider $$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t\psi = -\Delta _M^2\psi +\beta \Delta _M \psi +\lambda |\psi |^{2\sigma }\psi \quad \text {in }\mathbb {R}\times M,\\ \psi (0,\cdot )=\psi _0\in X, \end{array}\right. } \end{aligned}$$
i
∂
t
ψ
=
-
Δ
M
2
ψ
+
β
Δ
M
ψ
+
λ
|
ψ
|
2
σ
ψ
in
R
×
M
,
ψ
(
0
,
·
)
=
ψ
0
∈
X
,
where $$\beta \ge 0$$
β
≥
0
, $$\lambda =\{-1,1\}$$
λ
=
{
-
1
,
1
}
, $$0<\sigma < 4/(N-4)_+$$
0
<
σ
<
4
/
(
N
-
4
)
+
, $$\Delta _M$$
Δ
M
is the Laplace-Beltrami operator on M and $$X=L^2 (M)$$
X
=
L
2
(
M
)
or $$X=H^2 (M)$$
X
=
H
2
(
M
)
. At first, we focus on the case where M is the hyperbolic space $$\mathbb {H}^N$$
H
N
. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that $$N\ge 4$$
N
≥
4
and $$0<\sigma < 4/N$$
0
<
σ
<
4
/
N
if $$X=L^2 (\mathbb {H}^N)$$
X
=
L
2
(
H
N
)
or $$0<\sigma < 4/(N-4)_+$$
0
<
σ
<
4
/
(
N
-
4
)
+
if $$X=H^2 (\mathbb {H}^N)$$
X
=
H
2
(
H
N
)
. Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.
Funder
Fundação para a Ciência e a Tecnologia
Magnus Ehrnroothin Säätiö
Publisher
Springer Science and Business Media LLC
Reference33 articles.
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