Mostrando recursos 1 - 20 de 79

  1. On the number and complete continuity of weighted eigenvalues of measure differential equations

    Zhang, Meirong; Wen, Zhiyuan; Meng, Gang; Qi, Jiangang; Xie, Bing
    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$^*$...

  2. Two-phase eigenvalue problem on thin domains with Neumann boundary condition

    Yachimura, Toshiaki
    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.

  3. Nonexistence of positive solutions for a system of semilinear fractional Laplacian problem

    Dou, Jingbo; Li, Ye
    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...

  4. Critical well-posedness and scattering results for fractional Hartree-type equations

    Herr, Sebastian; Yang, Changhun
    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.

  5. A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity

    Sagawa, Yuji; Sunagawa, Hideaki; Yasuda, Shunsuke
    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].

  6. Classification of blow-up limits for the sinh-Gordon equation

    Jevnikar, Aleks; Wei, Juncheng; Yang, Wen
    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.

  7. An existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ball

    Chhetri, Maya; Sankar, Lakshmi; Shivaji, R.; Son, Byungjae
    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...

  8. An application of a diffeomorphism theorem to Volterra integral operator

    Diblík, Josef; Galewski, Marek; Koniorczyk, Marcin; Schmeidel, Ewa
    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*}

  9. Homogenization of imperfect transmission problems: the case of weakly converging data

    Faella, Luisa; Monsurrò, Sara; Perugia, Carmen
    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.

  10. Cauchy-Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems

    Bourdin, Loïc
    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.

  11. Global stability in a two-competing-species chemotaxis system with two chemicals

    Zheng, Pan; M, Chunlai; Mi, Yongsheng
    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,...

  12. Uniform asymptotic stability of a discontinuous predator-prey model under control via non-autonomous systems theory

    Bonotto, E.M.; Costa Ferreira, J.; Federson, M.
    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.

  13. Long range scattering for the cubic Dirac equation on $\mathbb R^{1+1}$

    Candy, Timothy; Lindblad, Hans
    We show that the cubic Dirac equation, also known as the Thirring model, scatters at infinity to a linear solution modulo a phase correction.

  14. Positive solutions of indefinite semipositone problems via sub-super solutions

    Kaufmann, Uriel; Quoirin, Humberto Ramos
    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.

  15. Existence of entropy solutions to a doubly nonlinear integro-differential equation

    Scholtes, Martin; Wittbold, Petra
    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.

  16. Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition

    Kim, Jae-Myoung; Kim, Yun-Ho; Lee, Jongrak
    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...

  17. Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

    do Ó, João Marcos; de Albuquerque, José Carlos
    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...

  18. A class of differential operators with complex coefficients and compact resolvent

    Behncke, Horst; Hinton, Don
    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.

  19. On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$

    Strütt, David
    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...

  20. On the global well-posedness of 3-d Navier-Stokes equations with vanishing horizontal viscosity

    Abidi, Hammadi; Paicu, Marius
    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.

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