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 ESTScattering 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 ESTPositive solutions of indefinite semipositone problems via sub-super solutionshttps://projecteuclid.org/euclid.die/1526004027<strong>Uriel Kaufmann</strong>, <strong>Humberto Ramos Quoirin</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 497--506.</p><p><strong>Abstract:</strong><br/>
Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem \[ \left\{ \begin{array} [c]{lll} -\Delta u=\lambda m(x)(f(u)-k) & \mathrm{in} & \Omega,\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where $\lambda,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $\lambda$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.
</p>projecteuclid.org/euclid.die/1526004027_20180510220048Thu, 10 May 2018 22:00 EDTLong range scattering for the cubic Dirac equation on $\mathbb R^{1+1}$https://projecteuclid.org/euclid.die/1526004028<strong>Timothy Candy</strong>, <strong>Hans Lindblad</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 507--518.</p><p><strong>Abstract:</strong><br/>
We show that the cubic Dirac equation, also known as the Thirring model, scatters at infinity to a linear solution modulo a phase correction.
</p>projecteuclid.org/euclid.die/1526004028_20180510220048Thu, 10 May 2018 22:00 EDTUniform asymptotic stability of a discontinuous predator-prey model under control via non-autonomous systems theoryhttps://projecteuclid.org/euclid.die/1526004029<strong>E.M. Bonotto</strong>, <strong>J. Costa Ferreira</strong>, <strong>M. Federson</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 519--546.</p><p><strong>Abstract:</strong><br/>
The present paper deals with uniform stability for non-autonomous impulsive systems. We consider a non-autonomous system with impulses in its abstract form and we present conditions to obtain uniform stability, uniform asymptotic stability and global uniform asymptotic stability using Lyapunov functions. Using the results from the abstract theory we present sufficient conditions for a controlled predator-prey model under impulse conditions to be globally uniformly asymptotically stable.
</p>projecteuclid.org/euclid.die/1526004029_20180510220048Thu, 10 May 2018 22:00 EDTGlobal stability in a two-competing-species chemotaxis system with two chemicalshttps://projecteuclid.org/euclid.die/1526004030<strong>Pan Zheng</strong>, <strong>Chunlai M</strong>, <strong>Yongsheng Mi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 547--558.</p><p><strong>Abstract:</strong><br/>
This paper deals with a two-competing-species chemotaxis system with two different chemicals \begin{equation*} \begin{cases} u_{t}=d_{1}\Delta u-\chi_{1}\nabla \cdot(u\nabla v)+\mu_{1} u(1-u-a_{1}w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{2}\Delta v-\alpha_{1}v+\beta_{1}w, & (x,t)\in \Omega\times (0,\infty),\\ w_{t}=d_{3}\Delta w-\chi_{2}\nabla \cdot(w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{4}\Delta z-\alpha_{2}z+\beta_{2}u, & (x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$ with nonnegative initial data $(u_{0},w_{0})\in (C^{0}(\overline{\Omega}))^{2}$ satisfying $u_{0}\not\equiv0$ and $w_{0}\not\equiv 0$, where $\chi_{1},\chi_{2}\geq0$, $a_{1}, a_{2}\in[0,1)$, and the parameters $d_{i}$ ($i=1,2,3,4$) and $\alpha_{j},\beta_{j}, \mu_{j}$ ($j=1,2$) are positive. Based on the approach of eventual comparison, it is shown that under suitable conditions, the system possesses a unique global-in-time classical solution, which converges to the constant steady states.
</p>projecteuclid.org/euclid.die/1526004030_20180510220048Thu, 10 May 2018 22:00 EDTCauchy-Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theoremshttps://projecteuclid.org/euclid.die/1526004031<strong>Loïc Bourdin</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 559--594.</p><p><strong>Abstract:</strong><br/>
The aim of the present paper is to contribute to the development of the study of Cauchy problems involving Riemann-Liouville and Caputo fractional derivatives. First, existence-uniqueness results for solutions of non-linear Cauchy problems with vector fractional multi-order are addressed. A qualitative result about the behavior of local but non-global solutions is also provided. Finally, the major aim of this paper is to introduce notions of fractional state-transition matrices and to derive fractional versions of the classical Duhamel formula. We also prove duality theorems relying left state-transition matrices with right state-transition matrices.
</p>projecteuclid.org/euclid.die/1526004031_20180510220048Thu, 10 May 2018 22:00 EDTHomogenization of imperfect transmission problems: the case of weakly converging datahttps://projecteuclid.org/euclid.die/1526004032<strong>Luisa Faella</strong>, <strong>Sara Monsurrò</strong>, <strong>Carmen Perugia</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 595--620.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to describe the asymptotic behavior, as $\varepsilon\to 0$, of an elliptic problem with rapidly oscillating coefficients in an $\varepsilon$-periodic two component composite with an interfacial contact resistance on the interface, in the case of weakly converging data.
</p>projecteuclid.org/euclid.die/1526004032_20180510220048Thu, 10 May 2018 22:00 EDTAn application of a diffeomorphism theorem to Volterra integral operatorhttps://projecteuclid.org/euclid.die/1526004033<strong>Josef Diblík</strong>, <strong>Marek Galewski</strong>, <strong>Marcin Koniorczyk</strong>, <strong>Ewa Schmeidel</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 621--642.</p><p><strong>Abstract:</strong><br/>
Using global diffeomorphism theorem based on duality mapping and mountain geometry, we investigate the properties of the Volterra operator given pointwise for $t\in \left[ 0,1\right] $ by \begin{equation*} V(x)(t)=x(t)+ \int _{0}^{t} v(t,\tau ,x(\tau ))d\tau ,\text{ }x(0)=0. \end{equation*}
</p>projecteuclid.org/euclid.die/1526004033_20180510220048Thu, 10 May 2018 22:00 EDTAn existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ballhttps://projecteuclid.org/euclid.die/1526004034<strong>Maya Chhetri</strong>, <strong>Lakshmi Sankar</strong>, <strong>R. Shivaji</strong>, <strong>Byungjae Son</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 7/8, 643--656.</p><p><strong>Abstract:</strong><br/>
We study the existence of positive radial solutions to the problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u &= \lambda K_1(|x|) f(v) \hspace{.3in}\mbox{in } \Omega_e,\\ -\Delta_p v &= \lambda K_2(|x|) g(u) \hspace{.31in}\mbox{in } \Omega_e, \\u &= v=0 \hspace{.7in} \mbox{ if } |x|=r_0, \\u(x)&\rightarrow 0,v(x)\rightarrow 0 \hspace{.4in} \mbox{as }\left|x \right|\rightarrow\infty, \end{aligned} \right. \end{equation*} where $\Delta_p w:=\mbox{div}(|\nabla w|^{p-2}\nabla w)$, $1 < p < n$, $\lambda$ is a positive parameter, $r_0>0$ and $\Omega_e:=\{x\in\mathbb{R}^n|~|x|>r_0\}$. Here, $K_i:[r_0,\infty)\rightarrow (0,\infty)$, $i=1,2$ are continuous functions such that $\lim_{r \rightarrow \infty} K_i(r)=0$, and $f, g:[0,\infty)\rightarrow \mathbb{R}$ are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of $\lambda$ via degree theory and rescaling arguments.
</p>projecteuclid.org/euclid.die/1526004034_20180510220048Thu, 10 May 2018 22:00 EDTClassification of blow-up limits for the sinh-Gordon equationhttps://projecteuclid.org/euclid.die/1528855434<strong>Aleks Jevnikar</strong>, <strong>Juncheng Wei</strong>, <strong>Wen Yang</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 657--684.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$ \Delta u+h_1e^u-h_2e^{-u}=0 \qquad \mathrm{in}~B_1\subset\mathbb R^2. $$ In particular, we get that the blow-up values are multiple of $8\pi.$ It generalizes the result of Jost, Wang, Ye and Zhou [20] where the extra assumption $h_1 = h_2$ is crucially used.
</p>projecteuclid.org/euclid.die/1528855434_20180612220415Tue, 12 Jun 2018 22:04 EDTA sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearityhttps://projecteuclid.org/euclid.die/1528855435<strong>Yuji Sagawa</strong>, <strong>Hideaki Sunagawa</strong>, <strong>Shunsuke Yasuda</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 685--700.</p><p><strong>Abstract:</strong><br/>
Let $T_{\varepsilon}$ be the lifespan for the solution to the Schrödinger equation on $\mathbb R^d$ with a power nonlinearity $\lambda |u|^{2\theta/d}u$ ($\lambda \in \mathbb C$, $0< \theta < 1$) and the initial data in the form $\varepsilon \varphi(x)$. We provide a sharp lower bound estimate for $T_{\varepsilon}$ as $\varepsilon \to +0$ which can be written explicitly by $\lambda$, $d$, $\theta$, $\varphi$ and $\varepsilon$. This is an improvement of the previous result by H. Sasaki [Adv. Diff. Eq., 14 (2009), 1021-1039].
</p>projecteuclid.org/euclid.die/1528855435_20180612220415Tue, 12 Jun 2018 22:04 EDTCritical well-posedness and scattering results for fractional Hartree-type equationshttps://projecteuclid.org/euclid.die/1528855436<strong>Sebastian Herr</strong>, <strong>Changhun Yang</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 701--714.</p><p><strong>Abstract:</strong><br/>
Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.
</p>projecteuclid.org/euclid.die/1528855436_20180612220415Tue, 12 Jun 2018 22:04 EDTNonexistence of positive solutions for a system of semilinear fractional Laplacian problemhttps://projecteuclid.org/euclid.die/1528855437<strong>Jingbo Dou</strong>, <strong>Ye Li</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 715--734.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a system of semilinear equations involving the fractional Laplacian in the Euclidean space $\mathbb{R}^n$: \begin{equation*} \begin{cases} (-\Delta)^{\alpha/2}u(x)=f(x_n)v^p(x)\\ (-\Delta)^{\alpha/2}v(x)=g(x_n)u^q(x) \end{cases} \end{equation*} in the subcritical case $1 < p,q\le \frac{n+\alpha}{n-\alpha}$ where $\alpha \in (0,\,2)$. Instead of investigating the above system directly, we discuss its equivalent integral system: \begin{equation*} \begin{cases} u(x)=\int_{\mathbb{R}^n} G_{\infty}(x,y)f(y_n)v^p(y)dy\\ v(y)=\int_{\mathbb{R}^n} G_{\infty}(x,y)g(x_n)u^q(x)dx , \end{cases} \end{equation*} where $G_{\infty}(x, y)$ is the Green's function associated with the fractional Laplacian in $\mathbb{R}^n$. Under natural structure condition on $f$ and $g$, we indicate the nonexistence of the positive solutions to the above integral system according to the method of moving spheres in integral form and the classic Hardy-Littlewood-Sobolev inequality.
</p>projecteuclid.org/euclid.die/1528855437_20180612220415Tue, 12 Jun 2018 22:04 EDTTwo-phase eigenvalue problem on thin domains with Neumann boundary conditionhttps://projecteuclid.org/euclid.die/1528855438<strong>Toshiaki Yachimura</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 735--760.</p><p><strong>Abstract:</strong><br/>
In this paper, we study an eigenvalue problem with piecewise constant coefficients on thin domains with Neumann boundary condition, and we analyze the asymptotic behavior of each eigenvalue as the domain degenerates into a certain hypersurface being the set of discontinuities of the coefficients. We show how the discontinuity of the coefficients and the geometric shape of the interface affect the asymptotic behavior of the eigenvalues by using a variational approach.
</p>projecteuclid.org/euclid.die/1528855438_20180612220415Tue, 12 Jun 2018 22:04 EDTOn the number and complete continuity of weighted eigenvalues of measure differential equationshttps://projecteuclid.org/euclid.die/1528855439<strong>Meirong Zhang</strong>, <strong>Zhiyuan Wen</strong>, <strong>Gang Meng</strong>, <strong>Jiangang Qi</strong>, <strong>Bing Xie</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 9/10, 761--784.</p><p><strong>Abstract:</strong><br/>
The classical eigenvalue theory for second-order ordinary differential equations (ODE) describes the spatial oscillation of strings whose distributions of masses are absolutely continuous. For general distributions of masses, including completely singular ones, the spatial oscillation can be explained using measure differential equations (MDE). In this paper we will study weighted eigenvalue problems for second-order MDE with general distributions or measures. It will be shown that the numbers of weighted eigenvalues depend on measures and may be finite. Furthermore, it will be proved that weighted eigenvalues and eigenfunctions are completely continuous in measures, i.e., when measures are convergent in the weak$^*$ topology, these eigen-pairs are strongly convergent. The present paper and the work of Meng and Zhang (Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Equations , 254 (2013), 2196-2232, have given an extension of the classical Sturm-Liouville theory to the measure version of ODE.
</p>projecteuclid.org/euclid.die/1528855439_20180612220415Tue, 12 Jun 2018 22:04 EDTOn a weighted Trudinger-Moser type inequality on the whole space and related maximizing problemhttps://projecteuclid.org/euclid.die/1537840869<strong>Van Hoang Nguyen</strong>, <strong>Futoshi Takahashi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 785--806.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.
</p>projecteuclid.org/euclid.die/1537840869_20180924220150Mon, 24 Sep 2018 22:01 EDTExistence and regularity of minimizers for nonlocal energy functionalshttps://projecteuclid.org/euclid.die/1537840870<strong>Mikil D. Foss</strong>, <strong>Petronela Radu</strong>, <strong>Cory Wright</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 807--832.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.
</p>projecteuclid.org/euclid.die/1537840870_20180924220150Mon, 24 Sep 2018 22:01 EDTNorm inflation for equations of KdV type with fractional dispersionhttps://projecteuclid.org/euclid.die/1537840871<strong>Vera Mikyoung Hur</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 833--850.</p><p><strong>Abstract:</strong><br/>
We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.
</p>projecteuclid.org/euclid.die/1537840871_20180924220150Mon, 24 Sep 2018 22:01 EDTGlobal solvability for two-dimensional filtered Euler equations with measure valued initial vorticityhttps://projecteuclid.org/euclid.die/1537840872<strong>Takeshi Gotoda</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 851--870.</p><p><strong>Abstract:</strong><br/>
We study the filtered Euler equations that are the regularized Euler equations derived by filtering the velocity field. The filtered Euler equations are a generalization of two well-known regularizations of incompressible inviscid flows, the Euler-$\alpha$ equations and the vortex blob method. We show the global existence of a unique weak solution for the two-dimensional (2D) filtered Euler equations with initial vorticity in the space of Radon measure that includes point vortices and vortex sheets. Moreover, a sufficient condition for the global well-posedness is described in terms of the filter and thus our result is applicable to various filtered models. We also show that weak solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the regularization parameter provided that initial vorticity belongs to the space of bounded functions.
</p>projecteuclid.org/euclid.die/1537840872_20180924220150Mon, 24 Sep 2018 22:01 EDTNot every conjugate point of a semi-Riemannian geodesic is a bifurcation pointhttps://projecteuclid.org/euclid.die/1537840873<strong>Giacomo Marchesi</strong>, <strong>Alessandro Portaluri</strong>, <strong>Nils Waterstraat</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 871--880.</p><p><strong>Abstract:</strong><br/>
We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which shows that the claim in our title is nevertheless true.
</p>projecteuclid.org/euclid.die/1537840873_20180924220150Mon, 24 Sep 2018 22:01 EDTUniqueness of singular self-similar solutions of a semilinear parabolic equationhttps://projecteuclid.org/euclid.die/1537840874<strong>Pavol Quittner</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 881--892.</p><p><strong>Abstract:</strong><br/>
We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $ u_t-\Delta u = u^p, $ $ x\in\mathbb R^n,\ t>0, $ where $p\geq(n+2)/(n-2)_+$.
</p>projecteuclid.org/euclid.die/1537840874_20180924220150Mon, 24 Sep 2018 22:01 EDTWeak-renormalized solutions for a simplified $k-\varepsilon$ model of turbulencehttps://projecteuclid.org/euclid.die/1537840875<strong>Pitágoras Pinheiro de Carvalho</strong>, <strong>Enrique Fernández-Cara</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 31, Number 11/12, 893--908.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to prove the existence of a weak-renormalized solution to a simplified model of turbulence of the $k-\varepsilon$ kind in spatial dimension $N=2$. The unknowns are the average velocity field and pressure, the mean turbulent kinetic energy and an appropriate time dependent variable. The motion equation and the additional PDE are respectively solved in the weak and renormalized senses.
</p>projecteuclid.org/euclid.die/1537840875_20180924220150Mon, 24 Sep 2018 22:01 EDT