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Communications on Pure & Applied Analysis

February 2020 , Volume 19 , Issue 2

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Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates
Qingsong Gu, Jiaxin Hu and Sze-Man Ngai
2020, 19(2): 641-676 doi: 10.3934/cpaa.2020030 +[Abstract](311) +[HTML](48) +[PDF](579.83KB)
Abstract:

We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than \begin{document}$ 2 $\end{document} and are closely related to the spectral dimension of fractal Laplacians.

Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space
Md. Rabiul Haque, Takayoshi Ogawa and Ryuichi Sato
2020, 19(2): 677-697 doi: 10.3934/cpaa.2020031 +[Abstract](525) +[HTML](49) +[PDF](389.59KB)
Abstract:

We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space \begin{document}$ L^r_{{\rm uloc}, \rho}( \Omega) $\end{document}, where the solution is not decaying at \begin{document}$ |x|\to \infty $\end{document}. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local \begin{document}$ L^r $\end{document} spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [10] is also valid for the uniformly local function class.

(1+2)-dimensional Black-Scholes equations with mixed boundary conditions
Junkee Jeon and Jehan Oh
2020, 19(2): 699-714 doi: 10.3934/cpaa.2020032 +[Abstract](240) +[HTML](43) +[PDF](402.46KB)
Abstract:

In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximum-quanto lookback options and derive the pricing formula of this options.

Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry
Pak Tung Ho
2020, 19(2): 715-722 doi: 10.3934/cpaa.2020033 +[Abstract](255) +[HTML](41) +[PDF](307.41KB)
Abstract:

Using the flow method, we prove an existence result for the problem of prescribing the \begin{document}$ Q $\end{document}-curvature on the even dimensional sphere \begin{document}$ S^n $\end{document}. More precisely, we prove that there exists a conformal metric on \begin{document}$ S^n $\end{document} such that its \begin{document}$ Q $\end{document}-curvature is \begin{document}$ f $\end{document}, when \begin{document}$ f $\end{document} possesses certain symmetry.

The scalar curvature problem on four-dimensional manifolds
Hichem Chtioui, Hichem Hajaiej and Marwa Soula
2020, 19(2): 723-746 doi: 10.3934/cpaa.2020034 +[Abstract](184) +[HTML](43) +[PDF](412.6KB)
Abstract:

We study the problem of existence of conformal metrics with prescribed scalar curvatures on a closed Riemannian \begin{document}$ 4 $\end{document}-manifold not conformally diffeomorphic to the standard sphere \begin{document}$ S^{4} $\end{document}. Using the critical points at infinity theory of A.Bahri [6] and the positive mass theorem of R.Schoen and S.T.Yau [32], we prove compactness and existence results under the assumption that the prescribed function is flat near its critical points. These are the first results on the prescribed scalar curvature problem where no upper-bound condition on the flatness order is assumed.

Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain
George J. Bautista and Ademir F. Pazoto
2020, 19(2): 747-769 doi: 10.3934/cpaa.2020035 +[Abstract](220) +[HTML](42) +[PDF](207.57KB)
Abstract:

In this paper we are concerned with a Boussinesq system for small-amplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.

Liouville theorems for an integral equation of Choquard type
Phuong Le
2020, 19(2): 771-783 doi: 10.3934/cpaa.2020036 +[Abstract](240) +[HTML](35) +[PDF](355.56KB)
Abstract:

We establish sharp Liouville theorems for the integral equation

where \begin{document}$ 0<\alpha, \beta<n $\end{document} and \begin{document}$ p>1 $\end{document}. Our results hold true for positive solutions under appropriate assumptions on \begin{document}$ p $\end{document} and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive \begin{document}$ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $\end{document} solutions of the higher fractional order Choquard type equation

Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping
Mirelson M. Freitas, Alberto L. C. Costa and Geraldo M. Araújo
2020, 19(2): 785-809 doi: 10.3934/cpaa.2020037 +[Abstract](294) +[HTML](39) +[PDF](577.04KB)
Abstract:

This paper is devoted to study the asymptotic behavior of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. We prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Moreover, we prove the upper-semicontinuity of pullback attractors with respect to non-autonomous perturbations.

Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise
Caibin Zeng, Xiaofang Lin, Jianhua Huang and Qigui Yang
2020, 19(2): 811-834 doi: 10.3934/cpaa.2020038 +[Abstract](167) +[HTML](40) +[PDF](393.23KB)
Abstract:

The fBm-driving rough stochastic lattice dynamical system with a general diffusion term is investigated. First, an area element in space of tensor is desired to define the rough path integral using the Chen-equality and fractional calculus. Under certain conditions, the considered equation is proved to possess a unique local mild path-area solution.

A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data
Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao and Xiongtao Zhang
2020, 19(2): 835-882 doi: 10.3934/cpaa.2020039 +[Abstract](163) +[HTML](39) +[PDF](531.66KB)
Abstract:

We present a two-dimensional coupled system for flocking particle-compressible fluid interactions, and study its global solvability for the proposed coupled system. For particle and fluid dynamics, we employ the kinetic Cucker-Smale-Fokker-Planck (CS-FP) model for flocking particle part, and the isentropic compressible Navier-Stokes (N-S) equations for the fluid part, respectively, and these separate systems are coupled through the drag force. For the global solvability of the coupled system, we present a sufficient framework for the global existence of classical solutions with large initial data which can contain vacuum using the weighted energy method. We extend an earlier global solvability result [20] in the one-dimensional setting to the two-dimensional setting.

Dynamics in a diffusive predator-prey system with stage structure and strong allee effect
Yuying Liu, Yuxiao Guo and Junjie Wei
2020, 19(2): 883-910 doi: 10.3934/cpaa.2020040 +[Abstract](183) +[HTML](39) +[PDF](720.42KB)
Abstract:

In this paper, we consider the dynamics of a diffusive predator-prey system with stage structure and strong Allee effect. The upper-lower solution method and the comparison principle are used in proving the nonnegativity of the solutions. Then the stability and the attractivity basin of the boundary equilibria are obtained, by which we investigated the bistable phenomena. The existence and local stability of the positive constant steady-state are investigated, and the existence of Hopf bifurcation is studied by analyzing the distribution of eigenvalues. On the center manifold, we studied the criticality of the Hopf bifurcation by the normal form theory. Some numerical simulations are carried out for illustrating the theoretical results.

Asymptotic profile of solutions to a certain chemotaxis system
Rafał Celiński and Andrzej Raczyński
2020, 19(2): 911-922 doi: 10.3934/cpaa.2020041 +[Abstract](168) +[HTML](36) +[PDF](304.6KB)
Abstract:

We consider a Cauchy problem for a two-dimensional model of chemotaxis and we show that large time behavior of solution is given by a multiple of the heat kernel.

Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity
Xiumei Deng and Jun Zhou
2020, 19(2): 923-939 doi: 10.3934/cpaa.2020042 +[Abstract](219) +[HTML](37) +[PDF](356.11KB)
Abstract:

This paper concerns with a semilinear heat equation with singular potential and logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, the existence of global solutions and infinite time blow-up solutions are obtained. The results of this paper indicate that the polynomial nonlinearity is a critical condition of existence of finite time blow-up solutions to semilinear heat equation with singular potential.

Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term
Zhiqing Liu and Zhong Bo Fang
2020, 19(2): 941-966 doi: 10.3934/cpaa.2020043 +[Abstract](208) +[HTML](41) +[PDF](286.28KB)
Abstract:

A transmission problem for Kirchhoff-type wave equations with nonlinear damping and delay term in the internal feedback is considered under a memory condition on one part of the boundary. By virtue of multiplier method, Faedo-Galerkin approximation and energy perturbation technique, we establish the appropriate conditions to guarantee the existence of global solution, and derive a general decay estimate of the energy, which includes exponential, algebraic and logarithmic decay etc.

Dissipative nonlinear schrödinger equations for large data in one space dimension
Gaku Hoshino
2020, 19(2): 967-981 doi: 10.3934/cpaa.2020044 +[Abstract](155) +[HTML](40) +[PDF](342.59KB)
Abstract:

In this study, we consider the global Cauchy problem for the nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension. In particular, we show the global existence, smoothing effect and asymptotic behavior for solutions to the nonlinear Schrödinger equations with data which belong to \begin{document}$ \mathcal{F}H^\gamma, $\end{document} \begin{document}$ 1/2<\gamma\leq 1. $\end{document} In the proof of main theorem, we introduce a priori estimate for \begin{document}$ H^\gamma $\end{document}-type norm and the condition \begin{document}$ \mathcal{F}H^1 $\end{document} for data relaxed into \begin{document}$ \mathcal{F}H^\gamma, $\end{document} \begin{document}$ 1/2<\gamma\leq1. $\end{document}

The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem
Francesca Colasuonno and Fausto Ferrari
2020, 19(2): 983-1000 doi: 10.3934/cpaa.2020045 +[Abstract](143) +[HTML](35) +[PDF](357.81KB)
Abstract:

We consider the \begin{document}$ p $\end{document}-Laplacian equation \begin{document}$ -\Delta_p u = 1 $\end{document} for \begin{document}$ 1<p<2 $\end{document}, on a regular bounded domain \begin{document}$ \Omega\subset\mathbb R^N $\end{document}, with \begin{document}$ N\ge2 $\end{document}, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature \begin{document}$ H $\end{document} of \begin{document}$ \partial\Omega $\end{document} is constant, then \begin{document}$ \Omega $\end{document} is a ball and the unique solution of the Dirichlet \begin{document}$ p $\end{document}-Laplacian problem is radial. The main tools used are integral identities, the \begin{document}$ P $\end{document}-function, and the maximum principle.

Asymptotic behavior of coupled inclusions with variable exponents
Peter E. Kloeden, Jacson Simsen and Petra Wittbold
2020, 19(2): 1001-1016 doi: 10.3934/cpaa.2020046 +[Abstract](154) +[HTML](45) +[PDF](383.35KB)
Abstract:

This work concerns the study of asymptotic behavior of the solutions of a nonautonomous coupled inclusion system with variable exponents. We prove the existence of a pullback attractor and that the system of inclusions is asymptotically autonomous.

Elliptic approximation of forward-backward parabolic equations
Fabio Paronetto
2020, 19(2): 1017-1036 doi: 10.3934/cpaa.2020047 +[Abstract](144) +[HTML](49) +[PDF](371.01KB)
Abstract:

In this note we give existence and uniqueness result for some elliptic problems depending on a small parameter and show that their solutions converge, when this parameter goes to zero, to the solution of a mixed type equation, elliptic-parabolic, parabolic both forward and backward. The aim is to give an approximation result via elliptic equations of a changing type equation.

Local uniqueness problem for a nonlinear elliptic equation
Miao Chen, Youyan Wan and Chang-Lin Xiang
2020, 19(2): 1037-1055 doi: 10.3934/cpaa.2020048 +[Abstract](147) +[HTML](41) +[PDF](381.21KB)
Abstract:

In this paper, we consider the following nonlinear Schrödinger equation

where \begin{document}$ N\ge3 $\end{document} and \begin{document}$ 2<p<2N/(N-2) $\end{document}. Under mild assumptions on the function \begin{document}$ K $\end{document} and using the local Pohozaev identity method developed by Deng, Lin and Yan [10], we show that multi-peak solutions to the above equation are unique for \begin{document}$ \varepsilon>0 $\end{document} sufficiently small.

Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions
Francesca R. Guarguaglini
2020, 19(2): 1057-1087 doi: 10.3934/cpaa.2020049 +[Abstract](148) +[HTML](41) +[PDF](422.35KB)
Abstract:

In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and nonhomogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.

Dynamics of a multigroup SIRS epidemic model with random perturbations and varying total population size
Qun Liu and Daqing Jiang
2020, 19(2): 1089-1110 doi: 10.3934/cpaa.2020050 +[Abstract](151) +[HTML](41) +[PDF](351.06KB)
Abstract:

In this paper, we analyze a multigroup SIRS epidemic model with random perturbations and varying total population size. By utilizing the stochastic Lyapunov function method, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Since our model is multidimensional, it is extremely difficult to construct an appropriate stochastic Lyapunov function to prove the existence of the stationary distribution, which implies stochastic weak stability. Then we establish sufficient conditions for extinction of the diseases. These conditions are related to the basic reproduction number in its corresponding deterministic system.

Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres
Xiaoxue Ji, Pengcheng Niu and Pengyan Wang
2020, 19(2): 1111-1128 doi: 10.3934/cpaa.2020051 +[Abstract](157) +[HTML](41) +[PDF](349.19KB)
Abstract:

In this article, we consider the cooperative semi-linear fractional system

where \begin{document}$ 0<\alpha <2 $\end{document}, \begin{document}$ \vec u $\end{document} and \begin{document}$ \vec h $\end{document} stand for \begin{document}$ k $\end{document}-dimentional vector-valued functions, and \begin{document}$ \vec {h}(x,\vec {u}(x)) $\end{document} is locally Lipschitz in \begin{document}$ \vec {u} $\end{document}. We first establish two narrow region principles for different cases. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions of the above system in bounded star-shaped domains and the whole space.

Optimal global asymptotic behavior of the solution to a singular monge-ampère equation
Zhijun Zhang
2020, 19(2): 1129-1145 doi: 10.3934/cpaa.2020053 +[Abstract](180) +[HTML](55) +[PDF](358.22KB)
Abstract:

This paper is mainly concerned with the optimal global asymptotic behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation \begin{document}$ {\rm det} \ D^2 u = b(x)g(-u), \ u<0, \ x \in \Omega, \ u|_{\partial \Omega} = 0, $\end{document} where \begin{document}$ \Omega $\end{document} is a strict convex and bounded smooth domain in \begin{document}$ \mathbb R^n $\end{document} with \begin{document}$ n\geq 2 $\end{document}, \begin{document}$ g\in C^1((0,\infty)) $\end{document} is positive and decreasing in \begin{document}$ (0, \infty) $\end{document} with \begin{document}$ \lim_{s \rightarrow 0^+}g(s) = \infty $\end{document}, \begin{document}$ b \in C^{\infty}(\Omega) $\end{document} is positive in \begin{document}$ \Omega $\end{document}, but may vanish or blow up on the boundary properly. Our approach is based on the construction of suitable sub- and super-solutions.

Boundary value problems for harmonic functions on domains in Sierpinski gaskets
Shiping Cao and Hua Qiu
2020, 19(2): 1147-1179 doi: 10.3934/cpaa.2020054 +[Abstract](142) +[HTML](36) +[PDF](969.32KB)
Abstract:

We study boundary value problems for harmonic functions on certain domains in the level-\begin{document}$ l $\end{document} Sierpinski gaskets \begin{document}$ \mathcal{SG}_l $\end{document}(\begin{document}$ l\geq 2 $\end{document}) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of \begin{document}$ \mathcal{SG}_l $\end{document} and the upper and lower domains generated by horizontal cuts of \begin{document}$ \mathcal{SG}_l $\end{document} are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.

Multiple solutions for a nonlinear Schrödinger systems
Fengshuang Gao and Yuxia Guo
2020, 19(2): 1181-1204 doi: 10.3934/cpaa.2020055 +[Abstract](233) +[HTML](54) +[PDF](442.38KB)
Abstract:

This paper is concerned with the following nonlinear Schrödinger systems:

where \begin{document}$ N\geq3 $\end{document} and \begin{document}$ 2<p<\frac{2N}{N-2} = 2^{\ast} $\end{document}, \begin{document}$ \beta\in \mathbb{R} $\end{document} is a coupling constant. \begin{document}$ a(x) $\end{document} is a \begin{document}$ \mathcal{C}^1 $\end{document} potential function. In the repulsive case, i.e. \begin{document}$ \beta<0 $\end{document}, under some suitable decay assumptions but without any symmetric assumptions on the potential \begin{document}$ a(x) $\end{document}, we prove the existence of infinitely many solutions for the problem \begin{document}$ (P). $\end{document}

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