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A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions Differ. Integral Equ. (IF 1.4) Pub Date : 2021-01-12 Yoshikazu Giga, Fumihiko Onoue, Keisuke Takasao
We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and some uniform boundedness property of
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Existence of ground state solution of Nehari-Pohožaev type for a quasilinear Schrödinger system Differ. Integral Equ. (IF 1.4) Pub Date : 2021-01-12 Jianqing Chen, Qian Zhang
This paper is concerned with the following quasilinear Schrödinger system in the entire space $\mathbb R^{N}$($N\geq3$): $$ \begin{cases} -\Delta u+A(x)u-\frac{1}{2} \triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta} |u|^{\alpha-2}u|v|^{\beta},\\ -\Delta v+Bv-\frac{1}{2}\triangle(v^{2}) v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha} |v|^{\beta-2}v. \end{cases} $$ By establishing a suitable constraint set and
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Well-posedness of the initial-boundary value problem for the Schrödinger-Boussinesq system Differ. Integral Equ. (IF 1.4) Pub Date : 2020-11-12 Boling Guo, Rudong Zheng
In the paper, we establish the local well-posedness of the Schödinger-Boussinesq system on the half line with data of low regularity. The proof is based on the explicit solution formula of the linear boundary problem and the restricted norm method. Besides, we prove that the nonlinearity is smoother than the initial data. Our result match the known result on the full line in [9].
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Nontrivial solutions for a quasilinear elliptic system with weight functions Differ. Integral Equ. (IF 1.4) Pub Date : 2020-11-12 Xiyou Cheng, Zhaosheng Feng, Lei Wei
In this paper, we consider nontrivial solutions of a quasilinear elliptic system with the weight functions $h(x)$ and $f_i(x)$ $(i = 1, 2)$ by applying the Nehari manifold method along with the fibrering maps and the minimization method. We analyze the effect of the parameters and weight functions on the existence and multiplicity of nontrivial solutions for the quasilinear elliptic system. When $(\lambda_1
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Regularity results of nonlinear perturbed stable-like operators Differ. Integral Equ. (IF 1.4) Pub Date : 2020-11-12 Anup Biswas, Mitesh Modasiya
We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of lower order Lévy measures. Such operators do not have a global scaling property. We establish Hölder regularity, Harnack inequality and boundary Harnack
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The fifth order KP–II equation on the upper half–plane Differ. Integral Equ. (IF 1.4) Pub Date : 2020-11-12 M.B. Erdoğan, T.B. Gürel, N. Tzirakis
In this paper, we study the fifth order Kadomtsev–Petviashvili II (KP–II) equation on the upper half–plane $U=\{(x,y)\in \mathbb R^2: y>0\}$. In particular, we obtain low regularity local well–posedness using the restricted norm method of Bourgain and the Fourier–Laplace method of solving initial and boundary value problems. Moreover, we prove that the nonlinear part of the solution is in a smoother
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Gaussian fields and stochastic heat equations Differ. Integral Equ. (IF 1.4) Pub Date : 2020-09-15 S.V. Lototsky, A. Shah
The objective of the paper is to characterize the Gaussian free field as a stationary solution of the heat equation with additive space-time white noise. In the whole space, the investigation leads to other types of Gaussian fields, as well as interesting phenomena in dimensions one and two.
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Fast diffusion equations on Riemannian manifolds Differ. Integral Equ. (IF 1.4) Pub Date : 2020-09-15 Sümeyye Bakim, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Ismail Kömbe
In the present paper, we first study the nonexistence of positive solutions of the following nonlinear parabolic problem \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\Delta_g( u^m)+V(x)u^m+\lambda u^q & \text{in}\quad \Omega \times (0, T ), \\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T). \end{cases} \end{equation*} Here, $\Omega$
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Stability of the line soliton of the Kadomtsev–Petviashvili-I equation with the critical traveling speed Differ. Integral Equ. (IF 1.4) Pub Date : 2020-09-15 Yohei Yamazaki
We consider the orbital stability of line solitons of the Kadomtsev–Petviashvili-I equation in $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$. Zakharov [40] and Rousset–Tzvetkov [31] proved the orbital instability of the line solitons of the Kadomtsev–Petviashvili-I equation on $\mathbb R^2$. The orbital instability of the line solitons on $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$ with the traveling
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A Discrete Stochastic Interpretation of the Dominative $p$-Laplacian Differ. Integral Equ. (IF 1.4) Pub Date : 2020-09-15 Karl K. Brustad, Peter Lindqvist, Juan J. Manfredi
We build a discrete stochastic process adapted to the (nonlinear) dominative $p$-Laplacian $$ \mathcal{D}_p u(x):=\Delta u + (p-2)\lambda_{N} , $$ where $\lambda_{N}$ is the largest eigenvalue of $D^2 u$ and $p > 2$. We show that the discrete solutions of the Dirichlet problems at scale $\varepsilon$ tend to the solution of the Dirichlet problem for $\mathcal{D}_p$ as $\varepsilon\to 0$. We assume
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Blowup solutions for the nonlinear Schrödinger equation with complex coefficient Differ. Integral Equ. (IF 1.4) Pub Date : 2020-09-15 Shota Kawakami, Shuji Machihara
We construct finite positive time blow up solutions for the nonlinear Schrödinger equation with the power nonlinearity whose coefficient is complex number. We also observe that those solutions exist time globally for the negative time. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin-Lions lemma for the compactness argument for its
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A weak solution for a point mass camphor motion Differ. Integral Equ. (IF 1.4) Pub Date : 2020-07-14 Jishan Fan, Masaharu Nagayama, Gen Nakamura, Mamoru Okamoto
The model system of equations which describes the self-propelled motion of point mass objects driven by camphor is a diffusion equation coupled with a system of nonlinear ordinary differential equations. If the objects have masses, then the motion of objects becomes very complicated when some of the objects hit the boundary of a water surface or collide each other. To avoid such complexity and try
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Quasi-invariance of fractional Gaussian fields by the nonlinear wave equation with polynomial nonlinearity Differ. Integral Equ. (IF 1.4) Pub Date : 2020-07-14 Philippe Sosoe, William J. Trenberth, Tianhao Xian
We prove quasi-invariance of Gaussian measures $\mu_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order in dimension $d=2$ and sub-quintic nonlinearities in dimension $d=3$, for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov and Gunaratnam-Oh-Tzvetkov-Weber who proved this result for a cubic
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The Cauchy problem of plasma equations modelling magnetic-curvature-driven Rayleigh–Taylor instability in 3D Differ. Integral Equ. (IF 1.4) Pub Date : 2020-07-14 Boling Guo, Xinglong Wu
Recently, S. Kondo and A. Tani in SIAM J. Math. Anal. (see [9]) investigated the existence and uniqueness of the strong solution to the initial boundary value problem (IBVP) of electromagnetic fluid equations (1.4) with the magnetic-curvature-driven Rayleigh–Taylor instability on bounded domain in 3D. The present paper will improve and extend the results from bounded domain to $\mathbb{R}^3$. First
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Trace formulae of potentials for degenerate parabolic equations Differ. Integral Equ. (IF 1.4) Pub Date : 2020-07-14 Mukhtar Karazym, Durvudkhan Suragan
In this paper, we analyze main properties of the volume and layer potentials as well as the Poisson integral for a multi-dimensional degenerate parabolic equation. As consequences, we obtain trace formulae of the heat volume potential and the Poisson integral which solve Kac's problem for degenerate parabolic equations in cylindrical domains.
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Nonhomogeneous systems involving critical or subcritical nonlinearities Differ. Integral Equ. (IF 1.4) Pub Date : 2020-07-14 Mousomi Bhakta, Souptik Chakraborty, Patrizia Pucci
This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.
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Noise-vanishing concentration and limit behaviors of periodic probability solutions Differ. Integral Equ. (IF 1.4) Pub Date : 2020-05-16 Min Ji, Weiwei Qi, Zhongwei Shen, Yingfei Yi
The present paper is devoted to the investigation of noisy impacts on the dynamics of periodic ordinary differential equations (ODEs). To do so, we consider a family of stochastic differential equations resulting from a periodic ODE perturbed by small white noises, and study noise-vanishing behaviors of their “steady states” that are characterized by periodic probability solutions of the associated
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Asymptotic expansion of oscillatory bifurcation curves of ODEs with nonlinear diffusion Differ. Integral Equ. (IF 1.4) Pub Date : 2020-05-16 Tetsutaro Shibata
We consider the nonlinear eigenvalue problem $$ [D(u)u']' + \lambda f(u) = 0, \ \ u(t) > 0, \ \ t \in I := (0,1), \ \ u(0) = u(1) = 0, $$ where $D(u) = u^p$, $f(u) = u^{q} + \sin (u^n)$ and $\lambda > 0$ is a bifurcation parameter. Here, $p \ge 0$, $n > 0$ and $q > 0$ are given constants and $k:=(p + q + 1)/2 \in \mathbb{N}$. This equation is motivated by the mathematical model of animal dispersal
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Optimal decay rate of solutions for nonlinear Klein-Gordon systems of critical type Differ. Integral Equ. (IF 1.4) Pub Date : 2020-05-16 Satoshi Masaki, Koki Sugiyama
We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearities. It will turn out that the decay rate which is previously found in some models is optimal.
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Existence and multiplicity of positive solutions of a critical Kirchhoff type elliptic problem in dimension four Differ. Integral Equ. (IF 1.4) Pub Date : 2020-05-16 Daisuke Naimen, Masataka Shibata
In this paper, we study a Kirchhoff type elliptic problem, \[ \begin{cases} \displaystyle - \Big ( 1+\alpha \int_{\Omega}|\nabla u|^2dx \Big ) \Delta u =\lambda u^q+u^3,\ u > 0 \text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \] where $\Omega\subset \mathbb{R}^4$ is a bounded domain with smooth boundary $\partial \Omega$ and we assume $\alpha,\lambda > 0$ and $1\le q < 3$. For $q=1$
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On uniqueness for Schrödinger maps with low regularity large data Differ. Integral Equ. (IF 1.4) Pub Date : 2020-05-16 Ikkei Shimizu
We prove that the solutions to the initial-value problem for the 2-dimensional Schrödinger maps are unique in $$ C_tL^\infty_x \cap L^\infty_t (\dot{H}^1_x\cap\dot{H}^2_x) . $$ For the proof, we follow McGahagan's argument with improving its technical part, combining Yudovich's argument.
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Fractional integro-differential equations with dual anti-periodic boundary conditions Differ. Integral Equ. (IF 1.4) Pub Date : 2020-03-21 Bashir Ahmad, Ymnah Alruwaily, Ahmed Alsaedi, Juan J. Nieto
In this paper, we introduce a new concept of dual anti-periodic boundary conditions. One of these conditions relates to the end points of an interval of arbitrary length, while the second one involves two nonlocal positions within the interval. Equipped with these conditions, we present the criteria for the existence of solutions for a fractional integro-differential equation involving two Caputo fractional
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Langevin equation involving two fractional orders with three-point boundary conditions Differ. Integral Equ. (IF 1.4) Pub Date : 2020-03-21 Ahmed Salem, Faris Alzahrani, Balqees Alghamdi
In the current manuscript, we examine the existence and uniqueness of solution for generalized Langevin equation involving two distinct fractional orders with a three-point boundary value problem. We implement the notions of fractional calculus simultaneously with immutable point types to create the existence and uniqueness results. To explore our problem, we apply the Banach contraction principle
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On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity Differ. Integral Equ. (IF 1.4) Pub Date : 2020-03-21 Rosanna Manzo
We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution
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Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions Differ. Integral Equ. (IF 1.4) Pub Date : 2020-03-21 Jeff Morgan, Vandana Sharma
We consider reaction diffusion systems where some components react and diffuse on the surface, and others diffuse inside the domain and react with surface components through dynamic boundary conditions. Under reasonable hypotheses, we establish the existence of componentwise non-negative global solutions.
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On the relativistic pendulum-type equation Differ. Integral Equ. (IF 1.4) Pub Date : 2020-03-21 Antonio Ambrosetti, David Arcoya
In the first part of this paper, we consider the equation $$ \Big ( \frac{u'}{\sqrt{1-u'^2}} \Big )'+F'(u)=0 $$ modeling, if $F'(u)=\sin u$, the motion of the free relativistic planar pendulum. Using critical point theory for non-smooth functionals, we prove the existence of non-trivial $T$ periodic solutions provided $T$ is sufficiently large. In the second part, we show the existence of periodic
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On the error of Fokker-Planck approximations of some one-step density dependent processes Differ. Integral Equ. (IF 1.4) Pub Date : 2020-02-06 Dávid Kunszenti-Kovács
Using operator semigroup methods, we show that Fokker-Planck type second-order PDEs can be used to approximate the evolution of the distribution of a one-step process on $N$ particles governed by a large system of ODEs. The error bound is shown to be of order $O(1/N)$, surpassing earlier results that yielded this order for the error only for the expected value of the process through mean-field approximations
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Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation Differ. Integral Equ. (IF 1.4) Pub Date : 2020-02-06 Baowei Feng, Abdelaziz Soufyane
This paper concerns a nonlinear viscoelastic wave equation with time-dependent delay. Under suitable relation between the weight of the delay and the weight of the term without delay, we prove the global existence of weak solutions by the combination of the Galerkin method and potential well theory. In addition, by assuming the minimal conditions on the $L^1(0,\infty)$ relaxation function $g$, namely
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Hénon elliptic equations in $\mathbb R^2$ with subcritical and critical exponential growth Differ. Integral Equ. (IF 1.4) Pub Date : 2020-02-06 João Marcos do Ó, Eudes Mendes Barboza
We study the Dirichlet problem in the unit ball $B_1$ of $\mathbb R^2$ for the Hénon-type equation of the form \begin{equation*} \begin{cases} -\Delta u =\lambda u+|x|^{\alpha}f(u) & \mbox{in } B_1, \\ \quad \ \ u = 0 & \mbox{on } \partial B_1, \end{cases} \end{equation*} where $f(t)$ is a $C^1$-function in the critical growth range motivated by the celebrated Trudinger-Moser inequality. Under suitable
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The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Gianpaolo Piscitelli
We analyze the limiting problem for the anisotropic $p$-Laplacian ($p\rightarrow\infty$) on convex sets, with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szegö-Weinberger type inequality.
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On the Cauchy problem for the periodic fifth-order KP-I equation Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Tristan Robert
The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation $$ { \partial_t} u - { \partial_x}^5 u -{ \partial_x}^{-1} { \partial_y}^2u + u{ \partial_x} u = 0, ~(t,x,y)\in\mathbb R\times\mathbb T^2 . $$ We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbf E = \{u\in L^2,~{ \partial_x}^2 u \in L^2, ~{ \partial_x}^{-1}
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The lifespan of solutions of semilinear wave equations with the scale-invariant damping in one space dimension Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Masakazu Kato, Hiroyuki Takamura, Kyouhei Wakasa
The critical constant $\mu$ (see (1.1)) of time-decaying damping in the scale-invariant case is recently conjectured. It also has been expected that the lifespan estimate is the same as for the associated semilinear heat equations if the constant is in the “heat-like” domain. In this paper, we point out that this is not true if the total integral of the sum of initial position and speed vanishes. In
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Global existence and blow-up of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Hua Chen, Jing Wang, Huiyang Xu
In this paper, we study the initial-boundary value problem for a class of infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity $$ u_{tt}-\triangle_{X} u=u\log | u | , $$ where $X= (X_1,X_2,...,X_m)$ is an infinitely degenerate system of vector fields, and $$ {\triangle_X} = \sum\limits_{j = 1}^m {X_j^2} $$ is an infinitely degenerate elliptic operator. By using the logarithmic
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Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Motohiro Sobajima
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial _t^2u-\Delta u + \partial _tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\ \partial \Omega\times (0,T), \\ u(0)=u_0, \partial _tu(0)=u_1 & \text{in}\ \Omega \end{cases} \end{equation*} in an exterior domain $\Omega$ in $\mathbb R^N$ $(N\geq
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Chaotic dynamics in a periodically perturbed Liénard system Differ. Integral Equ. (IF 1.4) Pub Date : 2019-10-22 Duccio Papini, Gabriele Villari, Fabio Zanolin
We prove the existence of infinitely many periodic solutions, as well as the presence of chaotic dynamics, for a periodically perturbed planar Liénard system of the form $\dot{x} = y - F(x) + p(\omega t),\; \dot{y} = - g(x)$. We consider the case in which the perturbing term is not necessarily small. Such a result is achieved by a topological method, that is by proving the presence of a horseshoe structure
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Remarks on eigenfunction expansions for the p-Laplacian Differ. Integral Equ. (IF 1.4) Pub Date : 2019-08-13 Wei-Chuan Wang
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
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Variational reduction for semi-stiff Ginzburg-Landau vortices Differ. Integral Equ. (IF 1.4) Pub Date : 2019-08-13 Rémy Rodiac
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
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Sobolev type time fractional differential equations and optimal controls with the order in $(1,2)$ Differ. Integral Equ. (IF 1.4) Pub Date : 2019-08-13 Yong-Kui Chang, Rodrigo Ponce
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
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Increasing convergent and divergent solutions to nonlinear delayed differential equations Differ. Integral Equ. (IF 1.4) Pub Date : 2019-08-13 Josef Diblík, Radoslav Chupáč, Miroslava Růžičková
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
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Local regularity for strongly degenerate elliptic equations and weighted sum operators Differ. Integral Equ. (IF 1.4) Pub Date : 2019-05-02 G. Di Fazio, M.S. Fanciullo, P. Zamboni
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.
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On nonlinear damped wave equations for positive operators. I. Discrete spectrum Differ. Integral Equ. (IF 1.4) Pub Date : 2019-05-02 Michael Ruzhansky, Niyaz Tokmagambetov
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
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Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature Differ. Integral Equ. (IF 1.4) Pub Date : 2019-05-02 Ali Hyder
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
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Finite energy weak solutions to some Dirichlet problems with very singular drift Differ. Integral Equ. (IF 1.4) Pub Date : 2019-05-02 Lucio Boccardo
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).
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Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces Differ. Integral Equ. (IF 1.4) Pub Date : 2019-05-02 Ze Li
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
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Explicit solutions for a system of nonlinear Schrödinger equations with delta functions as initial data Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Kazuyuki Doi, Shoji Shimizu
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.
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On the global existence and stability of 3-D viscous cylindrical circulatory flows Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Huicheng Yin, Zhang Lin
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
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The stationary Navier-Stokes equations in the scaling invariant Triebel-Lizorkin spaces Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Hiroyuki Tsurumi
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
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Existence and uniqueness of $\bf C^{1+\alpha}$-strict solutions for integro-differential equations with state-dependent delay Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Eduardo Hernández, Jianhong Wu
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.
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Regularity and uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Yuanyuan Dan, Yongsheng Li, Cui Ning
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
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Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case Differ. Integral Equ. (IF 1.4) Pub Date : 2019-04-03 Ziheng Tu, Jiayun Lin
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
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Global existence for semilinear damped wave equations in the scattering case Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 Yige Bai, Mengyun Liu
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.
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On a class of nonlinear elliptic equations with lower order terms Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 A. Alvino, M.F. Betta, A. Mercaldo, R. Volpicelli
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
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Král type removability results for $k$-Hessian equation and $k$-curvature equation Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 Kazuhiro Takimoto
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
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Large data global regularity for the $2+1$-dimensional equivariant Faddeev model Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 Dan-Andrei Geba, Manoussos G. Grillakis
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 $s>3$.
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Local and global existence for evolutionary p-Laplacian equation with nonlocal source Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 Haifeng Shang, Mengmeng Song
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
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Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity Differ. Integral Equ. (IF 1.4) Pub Date : 2019-01-23 Satoshi Masaki, Hayato Miyazaki
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
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Implication of age-structure on the dynamics of Lotka Volterra equations Differ. Integral Equ. (IF 1.4) Pub Date : 2018-12-11 Antoine Perasso, Quentin Richard
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
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Existence and local uniqueness of bubbling solutions for the Grushin critical problem Differ. Integral Equ. (IF 1.4) Pub Date : 2018-12-11 Billel Gheraibia, Chunhua Wang, Jing Yang
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
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Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey's conjecture Differ. Integral Equ. (IF 1.4) Pub Date : 2018-12-11 Ning-An Lai, Hiroyuki Takamura
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
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The sharp estimate of the lifespan for semilinear wave equation with time-dependent damping Differ. Integral Equ. (IF 1.4) Pub Date : 2018-12-11 Masahiro Ikeda, Takahisa Inui
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 $b(t) \approx (t+1)^{-\beta}$