Differential and Integral Equations Articles (Project Euclid)
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The latest articles from Differential and Integral Equations on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2012 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Fri, 14 Dec 2012 11:51 ESTFri, 14 Dec 2012 11:51 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Decay transference and Fredholmness of differential operators in weighted Sobolev
spaces
http://projecteuclid.org/euclid.die/1355502291
<strong>Patrick J. Rabier</strong><p><strong>Source: </strong>Differential Integral Equations, Volume 21, Number 11-12, 1001--1018.</p><p><strong>Abstract:</strong><br/>
We show that, for some family of weights $\omega ,$ there are corresponding weighted
Sobolev spaces $W_{\omega }^{m,p}$ on $ \mathbb {R}^{N}$ such that whenever
$P(x,\partial)$ is a differential operator with $L^{\infty }$ coefficients and
$P(x,\partial):W^{m,p}\rightarrow L^{p}$ is Fredholm for some $p\in (1,\infty),$ then
$P(x,\partial):W_{\omega }^{m,p}\rightarrow L_{\omega }^{p}$ ($=W_{\omega }^{0,p}$)
remains Fredholm with the same index. We also show that many spectral properties of
$P(x,\partial)$ are closely related, or even the same, in the non-weighted and the
weighted settings. The weights $\omega $ arise naturally from a feature of independent
interest of the Fredholm differential operators in classical Sobolev spaces (``full''
decay transference), proved in the preparatory Section 2. A main virtue of the spaces
$W_{\omega }^{m,p}$ is that they are well suited to handle nonlinearities that may be
ill-defined or ill-behaved in non-weighted spaces. Together with the invariance results of
this paper, this has proved to be instrumental in resolving various bifurcation issues in
nonlinear elliptic PDEs.
</p>projecteuclid.org/euclid.die/1355502291_Fri, 14 Dec 2012 11:51 ESTFri, 14 Dec 2012 11:51 ESTPositive semiclassical states for a fractional Schrödinger-Poisson systemhttp://projecteuclid.org/euclid.die/1487386824<strong>Edwin G. Murcia</strong>, <strong>Gaetano Siciliano</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 3/4, 231--258.</p><p><strong>Abstract:</strong><br/>
We consider a fractional Schrödinger-Poisson system in the whole space
$\mathbb R^{N}$
in presence of a positive potential and depending on a small positive parameter
$\varepsilon.$ We
show that, for suitably small $\varepsilon$ (i.e., in the ``semiclassical
limit'') the
number of positive solutions is estimated below by the
Ljusternick-Schnirelmann category of the set of minima of the potential.
</p>projecteuclid.org/euclid.die/1487386824_20170217220038Fri, 17 Feb 2017 22:00 ESTOn the existence of homoclinic type solutions of inhomogenous Lagrangian systemshttp://projecteuclid.org/euclid.die/1487386825<strong>Jakub Ciesielski</strong>, <strong>Joanna Janczewska</strong>, <strong>Nils Waterstraat</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 3/4, 259--272.</p><p><strong>Abstract:</strong><br/>
We study the existence of homoclinic type solutions for second order
Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$,
where $t\in \mathbb R$, $q\in\mathbb R^n$, $
a\colon\mathbb R\to\mathbb R$ is a continuous positive
bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential
satisfying the Ambrosetti-Rabinowitz superquadratic growth condition
and $f\colon\mathbb R\to\mathbb R^n$
is a continuous bounded square integrable forcing term.
A homoclinic type solution is obtained as limit of $2k$-periodic solutions
of an approximative sequence of second order differential equations.
</p>projecteuclid.org/euclid.die/1487386825_20170217220038Fri, 17 Feb 2017 22:00 ESTOn ordinary differential inclusions with mixed boundary conditionshttp://projecteuclid.org/euclid.die/1487386826<strong>Gabriele Bonanno</strong>, <strong>Antonio Iannizzotto</strong>, <strong>Monica Marras</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 3/4, 273--288.</p><p><strong>Abstract:</strong><br/>
By means of nonsmooth critical point theory, we prove existence of three weak
solutions
for an ordinary differential inclusion of Sturm-Liouville type involving a
general set-valued
reaction term depending on a parameter, and coupled with mixed boundary
conditions.
As an application, we give a multiplicity result for ordinary differential
equations involving discontinuous nonlinearities.
</p>projecteuclid.org/euclid.die/1487386826_20170217220038Fri, 17 Feb 2017 22:00 ESTBifurcation of Space Periodic Solutions in Symmetric Reversible FDEshttp://projecteuclid.org/euclid.die/1487386827<strong>Zalman Balanov</strong>, <strong>Hao-Pin Wu</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 3/4, 289--328.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1487386827_20170217220038Fri, 17 Feb 2017 22:00 ESTDecay estimates for four dimensional Schrödinger, Klein-Gordon and wave
equations with obstructions at zero energyhttp://projecteuclid.org/euclid.die/1489802418<strong>William R. Green</strong>, <strong>Ebru Toprak</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 5/6, 329--386.</p><p><strong>Abstract:</strong><br/>
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, if there is a zero
energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from
$L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free
evolution in spite of the zero energy eigenvalue.
</p>projecteuclid.org/euclid.die/1489802418_20170317220036Fri, 17 Mar 2017 22:00 EDTMultiplicity results and sign changing solutions of non-local equations with
concave-convex nonlinearitieshttp://projecteuclid.org/euclid.die/1489802419<strong>Mousomi Bhakta</strong>, <strong>Debangana Mukherjee</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 5/6, 387--422.</p><p><strong>Abstract:</strong><br/>
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 changing solution.
</p>projecteuclid.org/euclid.die/1489802419_20170317220036Fri, 17 Mar 2017 22:00 EDTUniform bounds for solutions to volume-surface reaction diffusion systemshttp://projecteuclid.org/euclid.die/1489802420<strong>Vandana Sharma</strong>, <strong>Jeff Morgan</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 5/6, 423--442.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1489802420_20170317220036Fri, 17 Mar 2017 22:00 EDTPullback dynamics of non-autonomous wave equations with acoustic boundary
conditionhttp://projecteuclid.org/euclid.die/1489802421<strong>To Fu Ma</strong>, <strong>Thales Maier Souza</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 5/6, 443--462.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1489802421_20170317220036Fri, 17 Mar 2017 22:00 EDTNodal solutions to problem with mean curvature operator in Minkowski
spacehttp://projecteuclid.org/euclid.die/1489802422<strong>Guowei Dai</strong>, <strong>Jun Wang</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 5/6, 463--480.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1489802422_20170317220036Fri, 17 Mar 2017 22:00 EDTHeat equation with a nonlinear boundary condition and growing initial datahttp://projecteuclid.org/euclid.die/1493863391<strong>Kazuhiro Ishige</strong>, <strong>Ryuichi Sato</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 481--504.</p><p><strong>Abstract:</strong><br/>
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 $\partial\Omega$.
</p>projecteuclid.org/euclid.die/1493863391_20170503220324Wed, 03 May 2017 22:03 EDTRemarks on large time behavior of the $L^{2}$-norm of solutions to strongly damped wave
equationshttp://projecteuclid.org/euclid.die/1493863392<strong>Ryo Ikehata</strong>, <strong>Michiaki Onodera</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 505--520.</p><p><strong>Abstract:</strong><br/>
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 ${\bf R}^{3}$.
</p>projecteuclid.org/euclid.die/1493863392_20170503220324Wed, 03 May 2017 22:03 EDTThe initial boundary value problem for some quadratic nonlinear Schrödinger
equations on the half-linehttp://projecteuclid.org/euclid.die/1493863393<strong>Márcio Cavalcante</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 521--554.</p><p><strong>Abstract:</strong><br/>
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].
</p>projecteuclid.org/euclid.die/1493863393_20170503220324Wed, 03 May 2017 22:03 EDTA three critical points result in a bounded domain of a Banach space and
applicationshttp://projecteuclid.org/euclid.die/1493863394<strong>Radu Precup</strong>, <strong>Patrizia Pucci</strong>, <strong>Csaba Varga</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 555--568.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1493863394_20170503220324Wed, 03 May 2017 22:03 EDTA classification of solutions of a fourth order semi-linear elliptic equation in
$\mathbb R^n$http://projecteuclid.org/euclid.die/1493863395<strong>Ridha Chammakhi</strong>, <strong>Abdellaziz Harrabi</strong>, <strong>Abdelbaki Selmi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 569--586.</p><p><strong>Abstract:</strong><br/>
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 $p\in (1,(n+2m)/(n-2m) )$.
</p>projecteuclid.org/euclid.die/1493863395_20170503220324Wed, 03 May 2017 22:03 EDTExact Evolution versus mean field with second-order correction for Bosons interacting
via short-range two-body potentialhttp://projecteuclid.org/euclid.die/1493863396<strong>Elif Kuz</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 587--630.</p><p><strong>Abstract:</strong><br/>
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 $N\rightarrow\infty$.
</p>projecteuclid.org/euclid.die/1493863396_20170503220324Wed, 03 May 2017 22:03 EDTNonlocal variational constants of motion in dissipative dynamicshttp://projecteuclid.org/euclid.die/1493863397<strong>Gianluca Gorni</strong>, <strong>Gaetano Zampieri</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 7/8, 631--640.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1493863397_20170503220324Wed, 03 May 2017 22:03 EDTOn multiple solutions for nonlocal fractional problems via $\nabla$-theoremshttp://projecteuclid.org/euclid.die/1495850422<strong>Giovanni Molica Bisci</strong>, <strong>Dimitri Mugnai</strong>, <strong>Raffaella Servadei</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 641--666.</p><p><strong>Abstract:</strong><br/>
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 (one of the
so-called $\nabla$-theorems).
</p>projecteuclid.org/euclid.die/1495850422_20170526220033Fri, 26 May 2017 22:00 EDTAlmost automorphic solutions of Volterra equations on time scaleshttp://projecteuclid.org/euclid.die/1495850423<strong>Carlos Lizama</strong>, <strong>Jaqueline G. Mesquitan</strong>, <strong>Rodrigo Ponce</strong>, <strong>Eduard Toon</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 667--694.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1495850423_20170526220033Fri, 26 May 2017 22:00 EDTWell-posedness and flow invariance for semilinear functional differential equations governed by non-densely defined operatorshttp://projecteuclid.org/euclid.die/1495850424<strong>Hiroki Sano</strong>, <strong>Naoki Tanaka</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 695--734.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1495850424_20170526220033Fri, 26 May 2017 22:00 EDTLogarithmic NLS equation on star graphs: Existence and stability of standing waveshttp://projecteuclid.org/euclid.die/1495850425<strong>Alex H. Ardila</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 735--762.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1495850425_20170526220033Fri, 26 May 2017 22:00 EDTScattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensionshttp://projecteuclid.org/euclid.die/1495850426<strong>Isao Kato</strong>, <strong>Kotaro Tsugawa</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 763--794.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1495850426_20170526220033Fri, 26 May 2017 22:00 EDTThe time derivative in a singular parabolic equationhttp://projecteuclid.org/euclid.die/1495850427<strong>Peter Lindqvist</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 9/10, 795--808.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.die/1495850427_20170526220033Fri, 26 May 2017 22:00 EDTPointwise estimates for $G\Gamma$-functions and applicationshttps://projecteuclid.org/euclid.die/1504231274<strong>Alberto Fiorenza</strong>, <strong>Maria Rosaria Formica</strong>, <strong>Jean Michel Rakotoson</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 809--824.</p><p><strong>Abstract:</strong><br/>
Some regularity results in small Lebesgue
spaces for the weak or the very weak solution of the
linear equation $-\Delta u = f$ are given,
improving all the previous results obtained in
the usual Lebesgue spaces.
Some of our results have been derived using borderline
Sobolev embeddings related to the Grand Lebesgue spaces.
So, we provide new Sobolev inclusions using the
Generalized Gamma spaces and generalizing the Fusco-Lions-Sbordone
results.
</p>projecteuclid.org/euclid.die/1504231274_20170831220137Thu, 31 Aug 2017 22:01 EDTUnstable phases for the critical Schrödinger-Poisson system in dimension 4https://projecteuclid.org/euclid.die/1504231275<strong>Pierre-Damien Thizy</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 825--832.</p><p><strong>Abstract:</strong><br/>
We consider, in this note, the critical Schrödinger-Poisson
system
\begin{equation}\label{SP0}
\begin{cases}
\Delta_g u+ \omega^2 u +\varphi u = u^{\frac{n+2}{n-2}}~,\\
\Delta_g \varphi +m_0^2 \varphi = 4\pi q^2 u^2~
\end{cases}
\end{equation}
on a closed Riemannian $n$-dimensional manifold $(M^n,g)$, for $n=4$. If the
scalar curvature
is negative somewhere, we prove that this system admits positive solutions for
small phases
$\omega$ and that $\omega=0$ is an unstable phase (see Definition 1.1. By contrast,
small phases are always stable (see [32]) when $n=4$ and the scalar
curvature is positive
everywhere, and unstable phases never exist when $n\ge 5$ (see [29, 31]).
</p>projecteuclid.org/euclid.die/1504231275_20170831220137Thu, 31 Aug 2017 22:01 EDTConnected sets of solutions for a nonlinear Neumann problemhttps://projecteuclid.org/euclid.die/1504231276<strong>Anna Gołębiewska</strong>, <strong>Joanna Kluczenko</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 833--852.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to study connected, unbounded sets of
solutions of the non-cooperative
elliptic system of equations with Neumann boundary conditions. The existence of
such sets is obtained by proving
the bifurcation from infinity.
To this end we apply the degree for $G$-invariant strongly indefinite
</p>projecteuclid.org/euclid.die/1504231276_20170831220137Thu, 31 Aug 2017 22:01 EDTNew distributional travelling waves for the nonlinear Klein-Gordon equationhttps://projecteuclid.org/euclid.die/1504231277<strong>C.O.R. Sarrico</strong>, <strong>A. Paiva</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 853--878.</p><p><strong>Abstract:</strong><br/>
The present paper concerns the study of distributional
travelling waves in
models ruled by the nonlinear Klein-Gordon equation $u_{tt}-c^{2}u_{xx}
=\phi(u)$, where $c>0$ is a real number and $\phi$ is an entire function which
takes real values on the real axis. For this purpose, we use a product of
distributions that extends the meaning of $\phi(u)$ to certain distributions
$u$ and that allows us to define a solution concept consistent with the
classical solution concept. The phi-four equation and the sine-Gordon equation
are examined as particular cases.
</p>projecteuclid.org/euclid.die/1504231277_20170831220137Thu, 31 Aug 2017 22:01 EDTA weak Harnack estimate for supersolutions to the porous medium equationhttps://projecteuclid.org/euclid.die/1504231278<strong>Pekka Lehtelä</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 879--916.</p><p><strong>Abstract:</strong><br/>
In this work, we prove a weak Harnack estimate for the weak supersolutions to the porous medium equation.
The proof is based on a priori estimates for the supersolutions and measure theoretical arguments.
</p>projecteuclid.org/euclid.die/1504231278_20170831220137Thu, 31 Aug 2017 22:01 EDTA note on positive radial solutions of $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$ with exactly quadratic growth at infinityhttps://projecteuclid.org/euclid.die/1504231279<strong>Trinh Viet Duoc</strong>, <strong>Quôc Anh Ngô</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 917--928.</p><p><strong>Abstract:</strong><br/>
Of interest in this note is the following geometric equation,
$\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$. It was found by Choi--Xu (J.
Differential Equations, 246, 216–234)
and McKenna–Reichel (Electron. J. Differential Equations, 37 (2003))
that the condition $q > 1$ is necessary and any positive radially symmetric
solution grows at
least linearly and at most quadratically at infinity for any $q > 1$. In addition,
when $q > 3$ any
positive radially symmetric solution is either exactly linear growth or exactly
quadratic growth at infinity.
Recently, Guerra (J. Differential Equations, {253}, 3147–3157) has shown
that the equation always
admits a unique positive radially symmetric solution of exactly given linear
growth at infinity
for any $q > 3$ which is also necessary. In this note, by using the phase-space
analysis, we show
the existence of infinitely many positive radially symmetric solutions of
exactly given quadratic
growth at infinity for any $q > 1$, hence completing the picture of positive
radially
symmetric solutions of the equation.
</p>projecteuclid.org/euclid.die/1504231279_20170831220137Thu, 31 Aug 2017 22:01 EDTExistence and boundary behaviour of solutions for a nonlinear Dirichlet problem in the annulushttps://projecteuclid.org/euclid.die/1504231280<strong>Sonia Ben Makhlouf</strong>, <strong>Malek Zribi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 929--946.</p><p><strong>Abstract:</strong><br/>
In this paper, we mainly study the following semilinear Dirichlet problem $
-\Delta u=q(x)f(u),\; u > 0,\;x\in \Omega ,$ $u_{|\partial \Omega }=0,$ where $
\Omega $ is an annulus in $\mathbb{R}^{n},\;\big( n\geq 2\big) .$ The
function $f$ is nonnegative in $\mathcal{C}^{1}(0,\infty )$ and $q\in
\mathcal{C}_{loc}^{\gamma }(\Omega ),\;(0 < \gamma < 1),$ is positive and
satisfies some required hypotheses related to Karamata regular variation
theory. We establish the existence of a positive classical solution to this
problem. We also give a global boundary behavior of such solution.
</p>projecteuclid.org/euclid.die/1504231280_20170831220137Thu, 31 Aug 2017 22:01 EDTA priori bounds and positive solutions for non-variational fractional elliptic systemshttps://projecteuclid.org/euclid.die/1504231281<strong>Edir Junior Ferreira Leite</strong>, <strong>Marcos Montenegro</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 947--974.</p><p><strong>Abstract:</strong><br/>
In this paper, we study strongly coupled elliptic systems in
non-variational form
involving fractional Laplace operators. We prove Liouville type theorems and,
by mean of the blow-up method, we establish a priori bounds of positive
solutions for
subcritical and superlinear nonlinearities in a coupled sense.
By using those latter, we then derive the existence of positive solutions
through topological methods.
</p>projecteuclid.org/euclid.die/1504231281_20170831220137Thu, 31 Aug 2017 22:01 EDTExponential decay for waves with indefinite memory dissipationhttps://projecteuclid.org/euclid.die/1504231282<strong>Higidio Portillo Oquendo</strong>, <strong>Bianca Morelli Rodolfo Calsavara</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 975--988.</p><p><strong>Abstract:</strong><br/>
In this work, we deal with the following wave equation with localized
dissipation given by a memory term
$$
u_{tt} -u_{xx} + \partial_x \Big\{ a(x)\int_{0}^{t} g(t-s)u_{x}(x,s)ds \Big\}=0.
$$
We consider that this dissipation is indefinite due to sign changes of the
coefficient $a$ or by sign changes of the memory kernel $g$.
The exponential decay of solutions is proved when the average of coefficient $a$
is positive and the memory kernel $g$ is small.
</p>projecteuclid.org/euclid.die/1504231282_20170831220137Thu, 31 Aug 2017 22:01 EDTNontrivial solutions for impulsive Sturm-Liouville differential equations with nonlinear derivative dependencehttps://projecteuclid.org/euclid.die/1504231283<strong>Giuseppe Caristi</strong>, <strong>Massimiliano Ferrara</strong>, <strong>Shapour Heidarkhani</strong>, <strong>Yu Tiani</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 30, Number 11/12, 989--1010.</p><p><strong>Abstract:</strong><br/>
In this paper, based on variational methods and critical point
theory, we investigate the existence of nontrivial solutions for a
class of impulsive Sturm-Liouville differential equations with
nonlinear derivative dependence.
</p>projecteuclid.org/euclid.die/1504231283_20170831220137Thu, 31 Aug 2017 22:01 EDTSteady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticityhttps://projecteuclid.org/euclid.die/1509041399<strong>Henry David</strong>, <strong>Silvia Sastre-Gomez</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 1--26.</p><p><strong>Abstract:</strong><br/>
In this article, we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution.
</p>projecteuclid.org/euclid.die/1509041399_20171026141014Thu, 26 Oct 2017 14:10 EDTOn the lack of compactness and existence of maximizers for some Airy-Strichartz inequalitieshttps://projecteuclid.org/euclid.die/1509041400<strong>Luiz G. Farah</strong>, <strong>Henrique Versieux</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 27--56.</p><p><strong>Abstract:</strong><br/>
This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb R)$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.
</p>projecteuclid.org/euclid.die/1509041400_20171026141014Thu, 26 Oct 2017 14:10 EDTSome improvements for a class of the Caffarelli-Kohn-Nirenberg inequalitieshttps://projecteuclid.org/euclid.die/1509041401<strong>Megumi Sano</strong>, <strong>Futoshi Takahashi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 57--74.</p><p><strong>Abstract:</strong><br/>
In this paper, we concern a weighted version of the Hardy inequality, which is a special case of the more general Caffarelli-Kohn-Nirenberg inequalities. We improve the inequality on the whole space or on a bounded domain by adding various remainder terms. On the whole space, we show the existence of a remainder term which has the form of ratio of two weighted integrals. Also we give a simple derivation of the remainder term involving a distance from the manifold of the “virtual extremals”. Finally, on a bounded domain, we prove the existence of remainder terms involving the gradient of functions.
</p>projecteuclid.org/euclid.die/1509041401_20171026141014Thu, 26 Oct 2017 14:10 EDTSymmetry breaking for an elliptic equation involving the Fractional Laplacianhttps://projecteuclid.org/euclid.die/1509041402<strong>Pablo L. Nápoli</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 75--94.</p><p><strong>Abstract:</strong><br/>
We study the symmetry breaking phenomenon for an elliptic equation involving the fractional Laplacian in a large ball. Our main tool is an extension of the Strauss radial lemma involving the fractional Laplacian, which might be of independent interest; and from which we derive compact embedding theorems for a Sobolev-type space of radial functions with power weights.
</p>projecteuclid.org/euclid.die/1509041402_20171026141014Thu, 26 Oct 2017 14:10 EDTRemovable singularities for elliptic equations with $(p,q)$-growth conditionshttps://projecteuclid.org/euclid.die/1509041403<strong>Yuliya V. Namlyeyeva</strong>, <strong>Igor I. Skrypnik</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 95--110.</p><p><strong>Abstract:</strong><br/>
For solutions of a class of divergence type quasilinear elliptic equations with $(p,q)$-growth conditions, we establish the condition for removability of singularity on manifolds.
</p>projecteuclid.org/euclid.die/1509041403_20171026141014Thu, 26 Oct 2017 14:10 EDTLocal well-posedness of the NLS equation with third order dispersion in negative Sobolev spaceshttps://projecteuclid.org/euclid.die/1509041404<strong>Tomoyuki Miyaji</strong>, <strong>Yoshio Tsutsumi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 111--132.</p><p><strong>Abstract:</strong><br/>
We show the time local well-posedness in $H^s$ of the reduced NLS equation with third order dispersion (r3NLS) on $\mathbf{T}$ for $s > -1/6$. Our proof is based on the nonlinear smoothing effect, which is similar to that for mKdV. However, when (r3NLS) is considered in Sobolev spaces of negative indices, the unconditional uniqueness of solutions, that is, the uniqueness of solutions without auxiliary spaces breaks down in marked contrast to mKdV.
</p>projecteuclid.org/euclid.die/1509041404_20171026141014Thu, 26 Oct 2017 14:10 EDTStability of the solution semigroup for neutral delay differential equationshttps://projecteuclid.org/euclid.die/1509041405<strong>Richard Fabiano</strong>, <strong>Catherine Payne</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 133--156.</p><p><strong>Abstract:</strong><br/>
We derive a new condition for delay-independent stability of systems of linear neutral delay differential equations. The method applies ideas from linear semigroup theory, and involves renorming the underlying Hilbert space to obtain a dissipative inequality on the infinitesimal generator of the solution semigroup. The new stability condition is shown to either improve upon or be independent of existing stability conditions.
</p>projecteuclid.org/euclid.die/1509041405_20171026141014Thu, 26 Oct 2017 14:10 EDTErrata to the paper “Minimization of a Ginzburg-Landau type energy with potential having a zero of infinite order”https://projecteuclid.org/euclid.die/1509041406<strong>Rejeb Hadiji</strong>, <strong>Itai Shafrir</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 1/2, 157--159.</p><p><strong>Abstract:</strong><br/>
As was kindly pointed to one of us by an anonymous referee, there is a gap in the argument in [1] whose origin is in the statement and proof of Lemma 2.2. This error can be easily corrected, and after this correction all the main results in [1] remain valid, as explained below.
</p>projecteuclid.org/euclid.die/1509041406_20171026141014Thu, 26 Oct 2017 14:10 EDTGlobal stability of an SIS epidemic model with a finite infectious periodhttps://projecteuclid.org/euclid.die/1513652421<strong>Yukihiko Nakata</strong>, <strong>Gergely Röst</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 161--172.</p><p><strong>Abstract:</strong><br/>
Assuming a general distribution for the sojourn time in the infectious class, we consider an SIS type epidemic model formulated as a scalar integral equation. We prove that the endemic equilibrium of the model is globally asymptotically stable whenever it exists, solving the conjecture of Hethcote and van den Driessche (1995) for the case of nonfatal diseases.
</p>projecteuclid.org/euclid.die/1513652421_20171218220030Mon, 18 Dec 2017 22:00 ESTOn the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noisehttps://projecteuclid.org/euclid.die/1513652422<strong>Igor Kukavica</strong>, <strong>Kerem Uğurlu</strong>, <strong>Mohammed Ziane</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 173--186.</p><p><strong>Abstract:</strong><br/>
We investigate the convergence of the Galerkin approximations for the stochastic Navier-Stokes equations in an open bounded domain $\mathcal{O}$ with the non-slip boundary condition. We prove that \begin{equation*} \mathbb{E} \Big [ \sup_{t \in [0,T]} \phi_1(\lVert (u(t)-u^n(t)) \rVert^2_V) \Big ] \rightarrow 0, \end{equation*} as $n \rightarrow \infty$ for any deterministic time $T > 0$ and for a specified moment function $\phi_1$ where $u^n(t)$ denotes the Galerkin approximations of the solution $u(t)$. Also, we provide a result on uniform boundedness of the moment $\mathbb{E} [ \sup_{t \in [0,T]} \phi(\lVert u(t) \rVert^2_V) ] $ where $\phi$ grows as a single logarithm at infinity. Finally, we summarize results on convergence of the Galerkin approximations up to a deterministic time $T$ when the $V$-norm is replaced by the $H$-norm.
</p>projecteuclid.org/euclid.die/1513652422_20171218220030Mon, 18 Dec 2017 22:00 ESTGlobal well posedness for a two-fluid modelhttps://projecteuclid.org/euclid.die/1513652423<strong>Yoshikazu Giga</strong>, <strong>Slim Ibrahim</strong>, <strong>Shengyi Shen</strong>, <strong>Tsuyoshi Yoneda</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 187--214.</p><p><strong>Abstract:</strong><br/>
We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray , and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) [8] ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.
</p>projecteuclid.org/euclid.die/1513652423_20171218220030Mon, 18 Dec 2017 22:00 ESTA nondivergence parabolic problem with a fractional time derivativehttps://projecteuclid.org/euclid.die/1513652424<strong>Mark Allen</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 215--230.</p><p><strong>Abstract:</strong><br/>
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Hölder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative $\alpha \to 1$.
</p>projecteuclid.org/euclid.die/1513652424_20171218220030Mon, 18 Dec 2017 22:00 ESTQuasilinear elliptic systems convex-concave singular terms and $\Phi$-Laplacian operatorhttps://projecteuclid.org/euclid.die/1513652425<strong>José V. Gonçalves</strong>, <strong>Marcos L. Carvalho</strong>, <strong>Carlos Alberto Santos</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 231--256.</p><p><strong>Abstract:</strong><br/>
This paper deals with the existence of positive solutions for a class of quasilinear elliptic systems involving the $\Phi$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbation techniques and comparison arguments in the setting of Orlicz-Sobolev spaces.
</p>projecteuclid.org/euclid.die/1513652425_20171218220030Mon, 18 Dec 2017 22:00 ESTNodal solutions for Lane-Emden problems in almost-annular domainshttps://projecteuclid.org/euclid.die/1513652426<strong>Anna Lisa Amadori</strong>, <strong>Francesca Gladiali</strong>, <strong>Massimo Grossi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 257--272.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{ in } \Omega , \\ u= 0 & \text{ on } \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in $\mathbb R^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1 < p_2 < \cdots$ with $\lim\limits_{k\rightarrow+\infty}p_k=+\infty$ such that for any real number $p > 1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as $p\to 1$.
</p>projecteuclid.org/euclid.die/1513652426_20171218220030Mon, 18 Dec 2017 22:00 ESTExponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domainhttps://projecteuclid.org/euclid.die/1513652427<strong>César Augusto Bortot</strong>, <strong>Wellington José Corrêa</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 273--300.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the exponential stability for the semilinear defocusing Schrödinger equation with locally distributed damping on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$. The proofs are based on a result of unique continuation property due to Cavalcanti et al. [15] and on a forced smoothing effect due to Aloui [2] combined with ideas from Cavalcanti et. al. [15], [16] adapted to the present context.
</p>projecteuclid.org/euclid.die/1513652427_20171218220030Mon, 18 Dec 2017 22:00 ESTHorizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamicshttps://projecteuclid.org/euclid.die/1513652428<strong>Kazuo Yamazaki</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 3/4, 301--328.</p><p><strong>Abstract:</strong><br/>
We derive a Biot-Savart law type identity for the horizontal components of the solution to the fluid system of equations with incompressibility in general dimension. Along with another new decomposition of non-linear terms, we give its application to derive two regularity criteria for the four-dimensional magneto-hydrodynamics system, in particular a criteria in terms of two velocity field components, two magnetic field components and two partial derivatives of the other two magnetic field components in a scaling-invariant norm. It is an open problem to obtain a criterion in terms of just two velocity field components and two partial derivatives of two magnetic field components in a scaling-invariant norm; an analogous criterion in the three-dimensional case has already been established.
</p>projecteuclid.org/euclid.die/1513652428_20171218220030Mon, 18 Dec 2017 22:00 ESTOn the global well-posedness of 3-d Navier-Stokes equations with vanishing horizontal viscosityhttps://projecteuclid.org/euclid.die/1516676426<strong>Hammadi Abidi</strong>, <strong>Marius Paicu</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 329--352.</p><p><strong>Abstract:</strong><br/>
We study, in this paper, the axisymmetric $3$-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data in Lebesgue spaces.
</p>projecteuclid.org/euclid.die/1516676426_20180122220100Mon, 22 Jan 2018 22:01 ESTOn a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$https://projecteuclid.org/euclid.die/1516676430<strong>David Strütt</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 353--374.</p><p><strong>Abstract:</strong><br/> We study the system of linear partial differential equations given by \[ dw+a\wedge w=f, \] on open subsets of $\mathbb R^n$, together with the algebraic equation \[ da\wedge u=\beta, \] where $a$ is a given $1$-form, $f$ is a given $(k+1)$-form, $\beta$ is a given $k+2$-form, $w$ and $u$ are unknown $k$-forms. We show that if $\text{rank}[da]\geq 2(k+1)$ those equations have at most one solution, if $\text{rank}[da] \equiv 2m \geq 2(k+2)$ they are equivalent with $\beta=df+a\wedge f$ and if $\text{rank}[da]\equiv 2 m\geq2(n-k)$ the first equation always admits a solution. Moreover, the differential equation is closely linked to the Poincaré lemma. Nevertheless, as soon as $a$ is nonexact, the addition of the term $a\wedge w$ drastically changes the problem. </p>projecteuclid.org/euclid.die/1516676430_20180122220100Mon, 22 Jan 2018 22:01 ESTA class of differential operators with complex coefficients and compact resolventhttps://projecteuclid.org/euclid.die/1516676435<strong>Horst Behncke</strong>, <strong>Don Hinton</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 375--402.</p><p><strong>Abstract:</strong><br/>
We consider the problem of the a second order singular differential operator with complex coefficients in the discrete spectrum case. The Titchmarsh-Weyl m-function is constructed without the use of nesting circles, and it is then used to give a representation of the resolvent operator. Under conditions on the growth of the coefficients, the resolvent operator is proved to be Hilbert-Schmidt and the root subspaces are shown to be complete in the associated Hilbert space. The operator is considered on both the half line and whole line cases.
</p>projecteuclid.org/euclid.die/1516676435_20180122220100Mon, 22 Jan 2018 22:01 ESTCoupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growthhttps://projecteuclid.org/euclid.die/1516676436<strong>João Marcos do Ó</strong>, <strong>José Carlos de Albuquerque</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 403--434.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \begin{cases} (-\Delta)^{ \frac 12 } u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},\\ (-\Delta)^{ \frac 12 } v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{cases} $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.
</p>projecteuclid.org/euclid.die/1516676436_20180122220100Mon, 22 Jan 2018 22:01 ESTExistence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-conditionhttps://projecteuclid.org/euclid.die/1516676437<strong>Jae-Myoung Kim</strong>, <strong>Yun-Ho Kim</strong>, <strong>Jongrak Lee</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 435--464.</p><p><strong>Abstract:</strong><br/>
We are concerned with the following elliptic equations with variable exponents \begin{equation*} -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u) \quad \text{in} \quad \mathbb R^{N}, \end{equation*} where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \mathbb R^{N} \to (1,\infty)$, $V: \mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $f: \mathbb R^{N}\times \mathbb R \to \mathbb R$ satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of $\lambda$'s for which our problem admits a nontrivial solution with simple assumptions in some sense.
</p>projecteuclid.org/euclid.die/1516676437_20180122220100Mon, 22 Jan 2018 22:01 ESTExistence of entropy solutions to a doubly nonlinear integro-differential equationhttps://projecteuclid.org/euclid.die/1516676439<strong>Martin Scholtes</strong>, <strong>Petra Wittbold</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 5/6, 465--496.</p><p><strong>Abstract:</strong><br/>
We consider a class of doubly nonlinear history-dependent problems associated with the equation $$ \partial_{t}k\ast(b(v)- b(v_{0})) = \text{div}\, a(x,Dv) + f . $$ Our assumptions on the kernel $k$ include the case $k(t) = t^{-\alpha}/\Gamma(1-\alpha)$, in which case the left-hand side becomes the fractional derivative of order $\alpha\in (0,1)$ in the sense of Riemann-Liouville. Existence of entropy solutions is established for general $L^{1}-$data and Dirichlet boundary conditions. Uniqueness of entropy solutions has been shown in a previous work.
</p>projecteuclid.org/euclid.die/1516676439_20180122220100Mon, 22 Jan 2018 22:01 EST