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 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 EDTThe sharp estimate of the lifespan for semilinear wave equation with time-dependent dampinghttps://projecteuclid.org/euclid.die/1544497284<strong>Masahiro Ikeda</strong>, <strong>Takahisa Inui</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 1/2, 1--36.</p><p><strong>Abstract:</strong><br/>
We consider the following semilinear wave equation with time-dependent damping. \begin{align*} \left\{ \begin{array}{ll} \partial_t^2 u - \Delta u + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb R^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb R^n, \end{array} \right. \end{align*} where $n \in \mathbb N$, $p > 1$, $\varepsilon>0$, and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$. It is known that small data blow-up occurs when $1 < p < p_F$ and, on the other hand, small data global existence holds when $p > p_F$, where $p_F:=1+2/n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1 < p < p_F$. In the critical case $p=p_F$, the lower estimate of the lifespan was also investigated. Recently, Lai and Zhou [15] obtained the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)=1$. In the present paper, we give the sharp upper estimate of the lifespan when $p=p_F$ and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$ by the Lai--Zhou method.
</p>projecteuclid.org/euclid.die/1544497284_20181210220141Mon, 10 Dec 2018 22:01 ESTNonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey's conjecturehttps://projecteuclid.org/euclid.die/1544497285<strong>Ning-An Lai</strong>, <strong>Hiroyuki Takamura</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 1/2, 37--48.</p><p><strong>Abstract:</strong><br/>
This work is devoted to the nonexistence of global-in-time energy solutions of nonlinear wave equation of derivative type with weak time-dependent damping in the scattering and scale invariant range. By introducing some multipliers to absorb the damping term, we succeed in establishing the same upper bound of the lifespan for the scattering damping as the non-damped case, which is a part of so-called Glassey's conjecture on nonlinear wave equations. We also study an upper bound of the lifespan for the scale invariant damping with the same method.
</p>projecteuclid.org/euclid.die/1544497285_20181210220141Mon, 10 Dec 2018 22:01 ESTExistence and local uniqueness of bubbling solutions for the Grushin critical problemhttps://projecteuclid.org/euclid.die/1544497286<strong>Billel Gheraibia</strong>, <strong>Chunhua Wang</strong>, <strong>Jing Yang</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 1/2, 49--90.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the following Grushin critical problem $$ -\Delta u(x)=\Phi(x)\frac{u^{\frac{N}{N-2}}(x)} {|y|},\,\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb R^{N}, $$ where $x=(y,z)\in\mathbb R^{k}\times \mathbb R^{N-k},N\geq 5,\Phi(x)$ is positive and periodic in its the $\bar{k}$ variables $(z_{1},...,z_{\bar{k}}),1\leq \bar{k} < \frac{N-2}{2}.$ Under some suitable conditions on $\Phi(x)$ near its critical point, we prove that the problem above has solutions with infinitely many bubbles. Moreover, we also show that the bubbling solutions obtained in our existence result are locally unique. Our result implies that some bubbling solutions preserve the symmetry from the potential $\Phi(x).$
</p>projecteuclid.org/euclid.die/1544497286_20181210220141Mon, 10 Dec 2018 22:01 ESTImplication of age-structure on the dynamics of Lotka Volterra equationshttps://projecteuclid.org/euclid.die/1544497287<strong>Antoine Perasso</strong>, <strong>Quentin Richard</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 1/2, 91--120.</p><p><strong>Abstract:</strong><br/>
In this article, we study the behavior of a nonlinear age-structured predator-prey model that is a generalization of Lotka-Volterra equations. We prove global existence, uniqueness and positivity of the solution using a semigroup approach. We make some analytically explicit thresholds that ensure, or not depending of their values, the boundedness of the solution and time asymptotic stability of equilibria. The latter theoretical results and their limits are enlightened by simulations.
</p>projecteuclid.org/euclid.die/1544497287_20181210220141Mon, 10 Dec 2018 22:01 ESTNonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearityhttps://projecteuclid.org/euclid.die/1548212426<strong>Satoshi Masaki</strong>, <strong>Hayato Miyazaki</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 121--138.</p><p><strong>Abstract:</strong><br/>
We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.
</p>projecteuclid.org/euclid.die/1548212426_20190122220052Tue, 22 Jan 2019 22:00 ESTLocal and global existence for evolutionary p-Laplacian equation with nonlocal sourcehttps://projecteuclid.org/euclid.die/1548212427<strong>Haifeng Shang</strong>, <strong>Mengmeng Song</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 139--168.</p><p><strong>Abstract:</strong><br/>
This paper examines the existence and nonexistence of solutions on the Cauchy problem for the evolutionary p-Laplacian equation with nonlocal source. By a priori estimates, we establish the local existence, global existence and nonexistence of solutions for that Cauchy problem. In particular, a Fujita's type critical exponent is obtained, which extends several classical results to the problem considered here.
</p>projecteuclid.org/euclid.die/1548212427_20190122220052Tue, 22 Jan 2019 22:00 ESTLarge data global regularity for the $2+1$-dimensional equivariant Faddeev modelhttps://projecteuclid.org/euclid.die/1548212428<strong>Dan-Andrei Geba</strong>, <strong>Manoussos G. Grillakis</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 169--210.</p><p><strong>Abstract:</strong><br/>
This article addresses the large data global regularity for the equivariant case of the $2+1$-dimensional Faddeev model and shows that it holds true for initial data in $H^s\times H^{s-1}(\mathbb R^2)$ with $s>3$.
</p>projecteuclid.org/euclid.die/1548212428_20190122220052Tue, 22 Jan 2019 22:00 ESTKrál type removability results for $k$-Hessian equation and $k$-curvature equationhttps://projecteuclid.org/euclid.die/1548212429<strong>Kazuhiro Takimoto</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 211--222.</p><p><strong>Abstract:</strong><br/>
We consider some removability problem for solutions to the so-called $k$-Hessian equation and $k$-curvature equation. We prove that if a $C^1$ function $u$ is a generalized solution to $k$-Hessian equation $F_k[u]=g(x,u,Du)$ or $k$-curvature equation $H_k[u]=g(x,u,Du)$ in $\Omega \setminus u^{-1}(E)$ for $E \subset \mathbb{R}$, then it is indeed a generalized solution to the same equation in the whole domain $\Omega$, under some hypotheses on $u$, $g$ and $E$.
</p>projecteuclid.org/euclid.die/1548212429_20190122220052Tue, 22 Jan 2019 22:00 ESTOn a class of nonlinear elliptic equations with lower order termshttps://projecteuclid.org/euclid.die/1548212430<strong>A. Alvino</strong>, <strong>M.F. Betta</strong>, <strong>A. Mercaldo</strong>, <strong>R. Volpicelli</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 223--232.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove an existence result for weak solutions to a class of Dirichlet boundary value problems whose prototype is \begin{equation*} \label{pa} \left\{ \begin{array}{lll} -\Delta_p u =\beta |\nabla u|^{q} +c(x)|u|^{p-2}u +f & & \text{in}\ \Omega \\ u=0 & & \text{on}\ \partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded open subset of $\mathbb R^N$, $N\geq 2$, $\Delta_p u={\rm div} \left(|\nabla u|^{p-2}\nabla u\right)$, $1 < p < N$, $ p-1 < q\le p-1+\frac p N$, $\beta $ is a positive constant, $c\in L^{\frac N p}(\Omega)$ with $c\ge 0$, $c\neq 0$ and $f\in L^{(p^*)'}(\Omega).$ We further assume smallness assumptions on $c$ and $f$. Our approach is based on Schauder's fixed point theorem.
</p>projecteuclid.org/euclid.die/1548212430_20190122220052Tue, 22 Jan 2019 22:00 ESTGlobal existence for semilinear damped wave equations in the scattering casehttps://projecteuclid.org/euclid.die/1548212431<strong>Yige Bai</strong>, <strong>Mengyun Liu</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 3/4, 233--248.</p><p><strong>Abstract:</strong><br/>
We study the global existence of solutions to semilinear damped wave equations in the scattering case with power-type nonlinearity on the derivatives, posed on nontrapping asymptotically Euclidean manifolds. The main idea is to shift initial time by local existence. As a result, we could convert the damping term to small enough perturbation and obtain the global existence.
</p>projecteuclid.org/euclid.die/1548212431_20190122220052Tue, 22 Jan 2019 22:00 ESTLife-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent casehttps://projecteuclid.org/euclid.die/1554256866<strong>Ziheng Tu</strong>, <strong>Jiayun Lin</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 249--264.</p><p><strong>Abstract:</strong><br/>
The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be: $T(\varepsilon)\leqslant \exp(C\varepsilon^{-p(p-1)})$ when $p=p_S(n+\mu)$ for $0 < \mu < \frac{n^2+n+2}{n+2}$. This result completes our previous study [9] on the sub-Strauss type exponent $p < p_S(n+\mu)$. Different from the work of M. Ikeda and M. Sobajima [5], we construct the suitable test function by introducing the modified Bessel function of second type. We note this method can be easily extended to some other scale-invariant wave models even with the Laplacian of variable coefficients.
</p>projecteuclid.org/euclid.die/1554256866_20190402220118Tue, 02 Apr 2019 22:01 EDTRegularity and uniqueness for the rough solutions of the derivative nonlinear Schrödinger equationhttps://projecteuclid.org/euclid.die/1554256867<strong>Yuanyuan Dan</strong>, <strong>Yongsheng Li</strong>, <strong>Cui Ning</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 265--290.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain the unconditional uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation $$ i\partial_t u + \partial^2_{x} u =i\partial_{x}(|u|^2u) $$ in $C([0,T];H^s(\mathbb R))$, $s\in(\frac{2}{3},1]$. The arguments used here are the normal form argument, resonant decomposition and the Bourgain argument. The main ingredient in the proof is to improve the regularity of the solution by iteration method and finally show that the solution belongs to some Bourgain space.
</p>projecteuclid.org/euclid.die/1554256867_20190402220118Tue, 02 Apr 2019 22:01 EDTExistence and uniqueness of $\bf C^{1+\alpha}$-strict solutions for integro-differential equations with state-dependent delayhttps://projecteuclid.org/euclid.die/1554256868<strong>Eduardo Hernández</strong>, <strong>Jianhong Wu</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 291--322.</p><p><strong>Abstract:</strong><br/>
We study the existence and uniqueness of strict and $\bf C^{1+\alpha}$-strict solutions for a general class of abstract integro-differential equations with state-dependent delay. Some examples concerning partial integro-differential equations with state dependent delay are presented.
</p>projecteuclid.org/euclid.die/1554256868_20190402220118Tue, 02 Apr 2019 22:01 EDTThe stationary Navier-Stokes equations in the scaling invariant Triebel-Lizorkin spaceshttps://projecteuclid.org/euclid.die/1554256869<strong>Hiroyuki Tsurumi</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 323--336.</p><p><strong>Abstract:</strong><br/>
We consider the stationary Navier-Stokes equations in $\mathbb{R}^n$ for $n\ge 3$. We show the existence and uniqueness of solutions in the homogeneous Triebel-Lizorkin space $\dot F^{-1+\frac{n}{p}}_{p,q}$ with $1 < p\leq n$ for small external forces in $\dot F^{-3+\frac{n}{p}}_{p,q}$. Our method is based on the boundedness of the Riesz transform, the para-product formula, and the embedding theorem in homogeneous Triebel-Lizorkin spaces. Moreover, it is proved that under some additional assumption on external forces, our solutions actually have more regularity.
</p>projecteuclid.org/euclid.die/1554256869_20190402220118Tue, 02 Apr 2019 22:01 EDTOn the global existence and stability of 3-D viscous cylindrical circulatory flowshttps://projecteuclid.org/euclid.die/1554256870<strong>Huicheng Yin</strong>, <strong>Zhang Lin</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 337--358.</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the global existence and stability of a 3-D perturbed viscous circulatory flow around an infinite long cylinder. This flow is described by 3-D compressible Navier-Stokes equations. By introducing some suitably weighted energy spaces and establishing a priori estimates, we show that the 3-D cylindrical symmetric circulatory flow is globally stable in time when the corresponding initial states are perturbed suitably small.
</p>projecteuclid.org/euclid.die/1554256870_20190402220118Tue, 02 Apr 2019 22:01 EDTExplicit solutions for a system of nonlinear Schrödinger equations with delta functions as initial datahttps://projecteuclid.org/euclid.die/1554256871<strong>Kazuyuki Doi</strong>, <strong>Shoji Shimizu</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 5/6, 359--367.</p><p><strong>Abstract:</strong><br/>
We study a system of nonlinear Schrödinger equations with delta functions as initial data. We seek its special solution and show that it has an explicit solution. Additionally, the global behavior of the solution can be understood by making use of the explicit expression.
</p>projecteuclid.org/euclid.die/1554256871_20190402220118Tue, 02 Apr 2019 22:01 EDTEndpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaceshttps://projecteuclid.org/euclid.die/1556762422<strong>Ze Li</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 7/8, 369--408.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove that the Kato smoothing effects for magnetic half wave operators can yield the endpoint Strichartz estimates for linear wave equations with magnetic potentials on the two dimensional hyperbolic spaces. As a corollary, we obtain the endpoint Strichartz estimates in the case of small potentials. This result serves as a cornerstone for the author's work [27] and collaborative work [29] in the study of asymptotic stability of harmonic maps for wave maps from $ \mathbb R\times \mathbb H^2$ to $ \mathbb H^2$.
</p>projecteuclid.org/euclid.die/1556762422_20190501220038Wed, 01 May 2019 22:00 EDTFinite energy weak solutions to some Dirichlet problems with very singular drifthttps://projecteuclid.org/euclid.die/1556762423<strong>Lucio Boccardo</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 7/8, 409--422.</p><p><strong>Abstract:</strong><br/>
In this paper, the boundary problems (1.1) and (3.1) are studied. The main results are the existence of a bounded weak solution of (1.1) under the minimal assumption (1.3) on $E$, and of the quasilinear problem (Hamilton-Jacobi equation) (3.1).
</p>projecteuclid.org/euclid.die/1556762423_20190501220038Wed, 01 May 2019 22:00 EDTStructure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvaturehttps://projecteuclid.org/euclid.die/1556762424<strong>Ali Hyder</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 7/8, 423--454.</p><p><strong>Abstract:</strong><br/>
In this article, we study the nonlocal equation $$ (-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb R$}, \quad\int_{\mathbb R}e^{nu}dx < \infty, $$ which arises in the conformal geometry. Inspired by the previous work of C.S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three, we classify all solutions to the above equation in terms of their behavior at infinity.
</p>projecteuclid.org/euclid.die/1556762424_20190501220038Wed, 01 May 2019 22:00 EDTOn nonlinear damped wave equations for positive operators. I. Discrete spectrumhttps://projecteuclid.org/euclid.die/1556762425<strong>Michael Ruzhansky</strong>, <strong>Niyaz Tokmagambetov</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 7/8, 455--478.</p><p><strong>Abstract:</strong><br/>
In this paper, we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on the interplay between the bottom of the operator's spectrum and the mass term. Consequently, we prove global in time well-posedness results for semilinear and for more general nonlinear equations with small data. Examples are given for nonlinear damped wave equations for the harmonic oscillator, for the twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compact manifolds.
</p>projecteuclid.org/euclid.die/1556762425_20190501220038Wed, 01 May 2019 22:00 EDTLocal regularity for strongly degenerate elliptic equations and weighted sum operatorshttps://projecteuclid.org/euclid.die/1556762426<strong>G. Di Fazio</strong>, <strong>M.S. Fanciullo</strong>, <strong>P. Zamboni</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 7/8, 479--492.</p><p><strong>Abstract:</strong><br/>
We extend some previously known results to the case of weighted sum operators. We prove that local weak solutions of elliptic equations with very strong degeneracy are smooth.
</p>projecteuclid.org/euclid.die/1556762426_20190501220038Wed, 01 May 2019 22:00 EDTIncreasing convergent and divergent solutions to nonlinear delayed differential equationshttps://projecteuclid.org/euclid.die/1565661619<strong>Josef Diblík</strong>, <strong>Radoslav Chupáč</strong>, <strong>Miroslava Růžičková</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 9/10, 493--516.</p><p><strong>Abstract:</strong><br/>
The paper is concerned with a nonlinear system of delayed differential equations as a generalization of an equation describing a simple model of the fluctuation of biological populations. The dependence of the behavior of monotone solutions on the coefficients and delays is studied and optimal sufficient conditions are derived for the existence of increasing and unbounded solutions and for the existence of increasing and convergent solutions. Inequalities estimating such solutions with some given increasing functions are derived as well. The results are compared with the linear case illustrated by examples, and open problems are formulated.
</p>projecteuclid.org/euclid.die/1565661619_20190812220045Mon, 12 Aug 2019 22:00 EDTSobolev type time fractional differential equations and optimal controls with the order in $(1,2)$https://projecteuclid.org/euclid.die/1565661620<strong>Yong-Kui Chang</strong>, <strong>Rodrigo Ponce</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 9/10, 517--540.</p><p><strong>Abstract:</strong><br/>
This paper is mainly concerned with controlled time fractional differential equations of Sobolev type in Caputo and Riemann-Liouville fractional derivatives with the order in $(1,2)$ respectively. By properties on some corresponding fractional resolvent operators family, we first establish sufficient conditions for the existence of mild solutions to these controlled time fractional differential equations of Sobolev type. Then, we present the existence of optimal controls of systems governed by corresponding time fractional differential equations of Sobolev type via setting up approximating minimizing sequences of suitable functions twice.
</p>projecteuclid.org/euclid.die/1565661620_20190812220045Mon, 12 Aug 2019 22:00 EDTVariational reduction for semi-stiff Ginzburg-Landau vorticeshttps://projecteuclid.org/euclid.die/1565661621<strong>Rémy Rodiac</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 9/10, 541--582.</p><p><strong>Abstract:</strong><br/>
Let $\Omega$ be a smooth bounded domain in $\mathbb R^2$. For $\varepsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $$ -\Delta u=\frac{1}{\varepsilon^2}(1-|u|^2)u \ \text{ in $\Omega$} $$ such that on $\partial \Omega$ u satisfies $|u|=1$ and $u\wedge \partial_\nu u=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions, we use a variational reduction method very similar to the one used in [12]. We obtain the exact same result as the authors of the aforementioned article obtained for the Neumann problem. This is because the renormalized energy for the Neumann problem and for the semi-stiff problem are the same. In particular, if $\Omega$ is simply connected a solution with degree one on the boundary always exists and if $\Omega$ is not simply connected, then for any $k\geq 1$ a solution with $k$ vortices of degree one exists.
</p>projecteuclid.org/euclid.die/1565661621_20190812220045Mon, 12 Aug 2019 22:00 EDTRemarks on eigenfunction expansions for the p-Laplacianhttps://projecteuclid.org/euclid.die/1565661624<strong>Wei-Chuan Wang</strong>. <p><strong>Source: </strong>Differential and Integral Equations, Volume 32, Number 9/10, 583--594.</p><p><strong>Abstract:</strong><br/>
The one-dimensional $p$-Laplacian eigenvalue problem \begin{equation*} \begin{cases} -(|y'|^{p-2}y')'=(p-1)(\lambda -q(x))|y|^{p-2}y,\\ y(0)=y(1)=0, \end{cases} \end{equation*} is considered in this paper. We derive its normalized eigenfunction expansion by using a Prüfer-type substitution. Employing some theories in Banach spaces, we discuss the basis property related to these eigenfunctions as an application.
</p>projecteuclid.org/euclid.die/1565661624_20190812220045Mon, 12 Aug 2019 22:00 EDT