Recursos de colección
Project Euclid (Hosted at Cornell University Library) (192.977 recursos)
Differential Integral Equations
Differential Integral Equations
Lindqvist, Peter
We study the Evolutionary $p$-Laplace Equation in the
singular case $1 < p < 2$. We prove that a weak solution
has a time derivative (in Sobolev's sense) which
is a function belonging (locally) to a $L^q$-space.
Kato, Isao; Tsugawa, Kotaro
We study the Cauchy problem for the Zakharov system in
spatial dimension $d\ge 4$ with initial datum
$ (u(0), n(0), \partial_t n(0) )\in H^k(\mathbb R^d)\times
\dot{H}^l(\mathbb R^d)\times \dot{H}^{l-1}(\mathbb R^d)$.
According to Ginibre, Tsutsumi and Velo ([9]),
the critical exponent of $(k,l)$ is $ ((d-3)/2,(d-4)/2 ). $
We prove the small data global well-posedness
and the scattering at the critical space.
It seems difficult to get the crucial bilinear estimate only
by applying the $U^2,\ V^2$ type spaces introduced by Koch and
Tataru ([23], [24]).
To avoid the difficulty, we use an intersection space of
$V^2$ type space and the space-time Lebesgue space
$E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz
estimate.
Ardila, Alex H.
In this paper, we consider the logarithmic Schrödinger
equation on a
star graph. By using a compactness method, we construct a unique global solution
of
the associated Cauchy problem in a suitable functional framework. Then we show
the existence
of several families of standing waves. We also prove the existence of ground
states as minimizers
of the action on the Nehari manifold. Finally, we show that the ground states
are orbitally stable
via a variational approach.
Sano, Hiroki; Tanaka, Naoki
The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild
solutions is introduced through a new type of variation of constants formula and the
well-posedness is established under a
semilinear stability condition with respect
to a metric-like functional and a subtangential
condition. The abstract result is applied
to a size-structured model with birth delay.
Lizama, Carlos; Mesquitan, Jaqueline G.; Ponce, Rodrigo; Toon, Eduard
The existence and uniqueness of almost automorphic solutions for linear and
semilinear nonconvolution Volterra equations on time scales is studied. The
existence
of asymptotically almost automorphic solutions is proved. Examples that
illustrate our results are given.
Molica Bisci, Giovanni; Mugnai, Dimitri; Servadei, Raffaella
The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by
$$ \left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \mathbb R^n\setminus \Omega\,,
\end{array} \right.
$$
where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator,
$\lambda$ is a real parameter, $\Omega\subset \mathbb R^n$, $n>2s$, is an open
bounded set with
continuous boundary and nonlinearity $f$ satisfies natural superlinear and
subcritical growth assumptions.
Precisely, along the paper, we prove the existence of at least three non-trivial
solutions for this
problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$.
For this purpose, we employ a variational theorem of mixed type...
Gorni, Gianluca; Zampieri, Gaetano
We give a recipe to generate ``nonlocal'' constants of motion for ODE Lagrangian systems
and we apply the method to find useful constants of motion which permit to prove global
existence and estimates of solutions to dissipative mechanical systems, and to the
Lane-Emden equation.
Kuz, Elif
We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions
of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter
$\beta$ measures the strength of interactions. We compare the exact evolution with an
approximation which considers the evolution of a mean field coupled with an appropriate
description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$, we derive an
error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial,
which implies a specific rate of convergence as $N\rightarrow\infty$.
Chammakhi, Ridha; Harrabi, Abdellaziz; Selmi, Abdelbaki
In this paper, we classify all regular sign changing~solutions~of $$ \Delta ^2 u=u_+^{p}
\,\,\,\mbox {in}\, \, \mathbb R^n\ \ \,\,u_+^{p}\in L^1(\mathbb R^n), $$ where $\Delta ^2$
denotes the biharmonic operator in $\mathbb R^n$, $1 < p\leq \frac{n}{n-4}$ and $n\geq
5$. We prove by using the procedure of moving parallel planes that such solutions are
radially symmetric about some point in $\mathbb R^n$. We also present a sup+inf type
inequality for regular solutions of the following equation: $$ (-\Delta )^m
u=u_+^{p}\,\,\,\mbox{in}\,\,\, \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb
R^n$, $m\geq1$, $n\geq 2m+1$ and $p\in (1,(n+2m)/(n-2m) )$.
Precup, Radu; Pucci, Patrizia; Varga, Csaba
Using the bounded mountain pass lemma and the Ekeland variational principle, we prove a
bounded version of the Pucci-Serrin three critical points result in the intersection of a
ball with a wedge in a Banach space. The localization constraints are overcome by boundary
and invariance conditions. The result is applied to obtain multiple positive solutions for
some semilinear problems.
Cavalcante, Márcio
We prove local well-posedness for the initial-boundary value problem associated to some
quadratic nonlinear Schrödinger equations on the half-line. The results are obtained
in the low regularity setting by introducing an analytic family of boundary forcing
operators, following the ideas developed in [14].
Ikehata, Ryo; Onodera, Michiaki
We reconsider the asymptotic behavior as $t \to +\infty$ of the $L^{2}$-norm of solutions
to strongly damped wave equations in ${\bf R}^{n}$ with weighted initial data in the
$n=1,2$ dimensional case. As an application, we shall apply those results to the analysis
of the linearized compressible Navier-Stokes equations in ${\bf R}^{3}$.
Ishige, Kazuhiro; Sato, Ryuichi
We discuss the solvability and the comparison principle for the heat equation with a
nonlinear boundary condition $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u, &
x\in\Omega,\,t > 0, \\ \nabla u\cdot\nu(x)=u^p,\qquad &x\in\partial\Omega,\,\,t
> 0, \\ u(x,0)=\varphi(x)\ge 0, & x\in\Omega, \end{array} \right. $$ where $N\ge
1$, $p > 1$, $\Omega$ is a smooth domain in ${\bf R}^N$ and $\varphi(x)=O(e^{\lambda
d(x)^2})$ as $d(x)\to\infty$ for some $\lambda\ge 0$. Here,
$d(x)=\mbox{dist}\,(x,\partial\Omega)$. Furthermore, we obtain the lower estimates of the
blow-up time of solutions with large initial data by use of the behavior of the initial
data near the boundary $\partial\Omega$.
Dai, Guowei; Wang, Jun
This paper is devoted to investigate the existence and multiplicity of radial
nodal solutions for the following Dirichlet problem with mean curvature operator
in Minkowski space \begin{eqnarray} \begin{cases} -\text{div} \Big (\frac{\nabla
v}{\sqrt{1-\vert \nabla v\vert^2}} \Big ) = \lambda f(\vert x\vert,v)\,\,
&\text{in}\,\, B_R(0),\\ v=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\,
\partial B_R(0). \end{cases} \nonumber \end{eqnarray} By bifurcation approach,
we determine the interval of parameter $\lambda$ in which the above problem has
two or four radial nodal solutions which have exactly $n-1$ simple zeros in
$(0,R)$ according to linear/sublinear/ superlinear nonlinearity at zero. The
asymptotic behaviors of radial nodal solutions as $\lambda \to +\infty$ and $n
\to +\infty$ are also studied.
Ma, To Fu; Souza, Thales Maier
This paper is concerned with a class of wave equations with acoustic boundary
condition subject to non-autonomous external forces. Under some general
assumptions, the problem generates a well-posed evolution process. Then, we
establish the existence of a minimal pullback attractor within a universe of
tempered sets defined by the forcing terms. We also, study the upper
semicontinuity of attractors as the non-autonomous perturbation tends to zero.
Our results allow unbounded external forces and nonlinearities with critical
growth.
Sharma, Vandana; Morgan, Jeff
We consider reaction-diffusion systems where some components react and diffuse on
the boundary of a region, while other components diffuse in the interior and
react with those on the boundary through mass transport. We establish criteria
guaranteeing that solutions are uniformly bounded in time.
Bhakta, Mousomi; Mukherjee, Debangana
In this paper, we prove the existence of infinitely many nontrivial solutions of
the following equations driven by a nonlocal integro-differential operator
$\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet
boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda
|u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0
\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a
smooth bounded domain in $ \mathbb R^N $, $N > 2s$, $s\in(0, 1)$, $0 < q
< 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to
the fractional laplacian operator $-(-\Delta)^s $, $p=\frac{N+2s}{N-2s}$,
$\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $ \lambda =1$, we
find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one
sign...
Green, William R.; Toprak, Ebru
We investigate dispersive estimates for the Schrödinger operator $H=-\Delta
+V$ with $V$ is a real-valued decaying potential when there are zero energy
resonances and eigenvalues in four spatial dimensions. If there is a zero energy
obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H)
P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2
\\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb
R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between
logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators,
further the operators are independent of time. We show that similar expansions
are valid for the solution operators to Klein-Gordon and wave equations.
Finally, we show that under certain orthogonality conditions,...
Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to
study bifurcation of periodic
solutions (of constant period) in symmetric networks of reversible FDEs.
Such a bifurcation occurs when eigenvalues of linearization move along the
imaginary axis
(without change of stability of the trivial solution and possibly without
$1:k$ resonance).
Physical examples motivating considered settings are related to stationary
solutions to
PDEs with non-local interaction: reversible mixed delay differential equations
(MDDEs)
and integro-differential equations (IDEs). In the case of $S_4$-symmetric
networks
of MDDEs and IDEs, we present exact computations of full equivariant
bifurcation invariants.
Algorithms and computational procedures used in this paper are also
included.
Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to
study bifurcation of periodic
solutions (of constant period) in symmetric networks of reversible FDEs.
Such a bifurcation occurs when eigenvalues of linearization move along the
imaginary axis
(without change of stability of the trivial solution and possibly without
$1:k$ resonance).
Physical examples motivating considered settings are related to stationary
solutions to
PDEs with non-local interaction: reversible mixed delay differential equations
(MDDEs)
and integro-differential equations (IDEs). In the case of $S_4$-symmetric
networks
of MDDEs and IDEs, we present exact computations of full equivariant
bifurcation invariants.
Algorithms and computational procedures used in this paper are also
included.