This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .
AbstractIn the paper we prove the weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int _{\mathbb {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$
∫
R
N
V
φ
2
μ
(
x
)
d
x
≤
∫
R
N
|
∇
φ
|
2
μ
(
x
)
d
x
+
K
∫
R
N
φ
2
μ
(
x
)
d
x
,
for functions $$\varphi $$
φ
in a weighted Sobolev space $$H^1_\mu $$
H
μ
1
, for a wider class of potentials V than inverse square potentials and for weight functions $$\mu $$
μ
of a quite general type. The case $$\mu =1$$
μ
=
1
is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators $$\begin{aligned} Lu=\varDelta u+\frac{\nabla \mu }{\mu }\cdot \nabla u \end{aligned}$$
L
u
=
Δ
u
+
∇
μ
μ
·
∇
u
perturbed by singular potentials.
The article presents the results of study the existence of the solution of nonlinear problem for elliptic by Petrovsky system in unbounded domain. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Applying the fixed point and embedding theorems the solvability of nonlinear problem for elliptic by Petrovsky system in weighted Sobolev space are established.
A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$