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Title and Author(s) |
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Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty) = 0$
Vieri Benci, Carlo R. Grisanti and Anna Maria Micheletti
ABSTRACT.
We study the existence of the ground state
solution of the problem
-\Delta u+V(x)u=f'(u) & x\in R^N,
u(x)>0,
under the assumption that
\lim_{x\to\infty}V(x)=0.
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203
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Global regular nonstationary flow for the Navier-Stokes equations in a cylindrical pipe
Wojciech M. Zajaczkowski
ABSTRACT.
Global existence of regular solutions to the Navier-Stokes equations
describing the motion of a fluid in a cylindrical pipe with large inflow
and outflow in shown. The global existence is proved under the following
conditions:
- small variations of velocity and pressure with respect to the
variable along the pipe,
- inflow and outflow are very close to homogeneous and decay
exponentially with time,
- the external force decays exponentially with time.
Global existence is proved in two steps. First by the Leray-Schauder fixed
point theorem we prove local existence with large existence time which is
inversely proportional to the above smallness restrictions. Next the local
solution is prolonged step by step.
The existence is proved for a solution without any restrictions on the
magnitudes of inflow, outflow, external force and the initial velocity.
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221
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Topological Index for Condensing maps on Finsler Manifolds with Applications to Functional-Differential Equations of Neutral Type
Elena V. Bogacheva and Yuri E. Gliklikh
| ABSTRACT.
The topological index for maps of infinite-dimensional Finsler manifolds,
condensing with respect to internal Kuratowski's measure of non-compactness,
is constructed under the hypothesis that the manifold can be embedded into
a certain Banach linear space as a neighbourhood retract so that the Finsler
norm in tangent spaces and the restriction of the norm from enveloping space
on the tangent spaces are equivalent. It is shown that the index is an
internal topological characteristic, i.e. it does not depend on the choice
of enveloping space, embedding, etc. The total index (Lefschetz number) and
the Nielsen number are also introduced. The developed machinery is applied
to investigation of functional-differential equations of neutral type on
Riemannian manifolds. A certain existence and uniqueness theorem is proved.
It is shown that the shift operator, acting in the manifold of C1-curves,
is condensing, its total index is calculated to be equal to the Euler
characteristic of (compact) finite-dimensional Riemannian manifold where
the equation is given. Some examples of calculating the Nielsen number are
also considered.
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287
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Second Noether-type theorem for the generalized variational principle of Herglotz
Bogdana Georgieva and Ronald B. Guenther
| ABSTRACT.
The generalized variational principle of Herglotz defines the functional,
whose extrema are sought, by a differential equation rather than by
an integral. For such functionals the classical Noether theorems are
not applicable. First and second Noether-type theorems which do apply
to the generalized variational principle of Herglotz were formulated
and proved. These theorems contain the classical first and second Noether
theorems as special cases. We published the first Noether-type theorem
previously in this journal. Here we prove the second Noether-type theorem
and show that it reduces to the classical second Noether theorem when the
Herglotz variational principle reduces to the classical variational principle.
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307
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Continuity of attractors for net-shaped thin domains
Thomas Elsken
| ABSTRACT.
Consider a reaction-diffusion equation u_t=\triangle u+f(u) on a family of net-shaped
thin domains \Omega_\eps converging to a one dimensional set as \eps\downarrow
0. With suitable growth and dissipativeness conditions on f these equations
define global semiflows which have attractors \script{A}_\eps. In [4]
it has been shown that there is a limit problem which also defines a semiflow
having an attractor \script{A}_0, and the family of attractors is
upper-semi-continuous at \eps=0. Here we show that under a stronger
dissipativeness condition the family of attractors \script{A}_\eps,
\eps\ge 0, is
actually continuous at \eps=0.
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315
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Multiple nonnegative solutions for elliptic boundary value problems involving the p-Laplacian
Giovanni Anello
ABSTRACT.
In this paper we present a result concerning the existence of two
nonzero nonnegative solutions for the following Dirichlet problem involving
the p-Laplacian
-\Delta_p u=\lambda f(x,u) &\text{\rm in\ } \Omega,
u=0 &\text{\rm on\ } \partial \Omega,
using variational methods. In particular, we will determine an
explicit real interval \Lambda for which these solutions exist
for every \lambda\in \Lambda. We also point out that our result
improves and extends to higher dimension a recent multiplicity
result for ordinary differential equations.
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355
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Extreme cases of asymptotically positively linear conditions and solvability of Sturm-Liouvill BVPs of Duffing equations
Huang Qi and Dong Yujun
| ABSTRACT.
In this paper we study the solvability of Sturm-Liouville BVPs for Duffing
equations by means of homotopy continuation methods. We propose a new kind
of solvable conditions on the nonlinear function in the equation. This kind
of conditions can be seen as some limiting cases of the well-known
asymptotically positive linear conditions. The obtained results generalize
and unify some previous results by S. Villegas, T. Ma and L. Sanchez, and
Y. Dong, respectively.
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367
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Addendum
A. Nowakowski and A. Rogowski
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385
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Author Index for Volumes 25 and 26
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391
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