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Communications on Pure & Applied Analysis

February 2021 , Volume 20 , Issue 2

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Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency
Hongyu Cheng and Shimin Wang
2021, 20(2): 467-494 doi: 10.3934/cpaa.2020222 +[Abstract](356) +[HTML](75) +[PDF](450.54KB)

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators

where \begin{document}$ \lambda \in \mathcal{O} $\end{document}, a closed interval not containing zero, the forcing term \begin{document}$ f $\end{document} is real analytic, and the frequency vector \begin{document}$ \omega \in \mathbb{R}^d \, (d \geq 2) $\end{document} is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency \begin{document}$ \omega\in\mathbb{R}^{d}, $\end{document} which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.

On the Cahn-Hilliard equation with mass source for biological applications
Hussein Fakih, Ragheb Mghames and Noura Nasreddine
2021, 20(2): 495-510 doi: 10.3934/cpaa.2020277 +[Abstract](292) +[HTML](72) +[PDF](1413.88KB)

This article deals with some generalizations of the Cahn–Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn–Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn–Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line
Álvaro Castañeda, Pablo González and Gonzalo Robledo
2021, 20(2): 511-532 doi: 10.3934/cpaa.2020278 +[Abstract](300) +[HTML](75) +[PDF](433.38KB)

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half–line while the nonlinearity has some boundedness and Lipschitz conditions. In addition, we provide new characterizations for the resulting homeomorphisms. When the linear system is asymptotically stable and the nonlinear system has a unique equilibrium, we deduce sharper results for the smoothness of the topological equivalence. Finally, we study the asymptotic stability and its preservation by topological equivalence.

Homogenization and singular perturbation in porous media
Eduard Marušić-Paloka and Igor Pažanin
2021, 20(2): 533-545 doi: 10.3934/cpaa.2020279 +[Abstract](310) +[HTML](100) +[PDF](340.46KB)

We study a Dirichlet problem in periodic porous medium depending on two small parameters, the hydraulic permeability of the porous inclusions \begin{document}$ \delta $\end{document} and the period \begin{document}$ \varepsilon $\end{document}. We study the situation as \begin{document}$ \delta\to 0 \;, \; \varepsilon\to 0 $\end{document} and \begin{document}$ \varepsilon \to 0 \;, \; \delta\to 0 $\end{document} and prove that the two limits do not commute.

A unique continuation property for a class of parabolic differential inequalities in a bounded domain
Guojie Zheng, Dihong Xu and Taige Wang
2021, 20(2): 547-558 doi: 10.3934/cpaa.2020280 +[Abstract](338) +[HTML](73) +[PDF](331.74KB)

This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain \begin{document}$ \Omega $\end{document} prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset \begin{document}$ \omega $\end{document} in \begin{document}$ \Omega $\end{document} at any given positive time \begin{document}$ T $\end{document}. We also derive the quantitative nature of this unique continuation, that is, the estimate of a \begin{document}$ L^2(\Omega) $\end{document} norm of the initial data, which is majorized by that of solution on the bounded open subset \begin{document}$ \omega $\end{document} at terminal moment \begin{document}$ t = T $\end{document}.

Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy
Kuo-Chih Hung and Shin-Hwa Wang
2021, 20(2): 559-582 doi: 10.3934/cpaa.2020281 +[Abstract](314) +[HTML](80) +[PDF](424.61KB)

We study the classification and evolution of bifurcation curves for the porous-medium combustion problem

where \begin{document}$ u $\end{document} is the solid temperature, parameters \begin{document}$ \lambda >0 $\end{document}, \begin{document}$ a\geq 0 $\end{document}, and the activation energy parameter \begin{document}$ d>0 $\end{document} is large. We mainly prove that, on the \begin{document}$ (\lambda , ||u||_{\infty }) $\end{document}-plane, the bifurcation curve is S-shaped with exactly two turning points for any\begin{document}$ \ (d, a)\in \Omega \equiv \left \{ (d, a):(0<d<d_{1}, \text{ }a\geq A_{1}(d))\text{ or }(d\geq d_{1}, \text{ }a\geq 0)\right \} $\end{document} for some positive number \begin{document}$ d_{1}\approx 2.225 $\end{document} and a nonnegative, strictly decreasing function \begin{document}$ A_{1}(d) $\end{document} defined on \begin{document}$ (0, d_{1}]. $\end{document} Furthermore, for any\begin{document}$ \ (d, a)\in \Omega , $\end{document} we give a classification and evolution of totally four different S-shaped bifurcation curves. In addition, for any \begin{document}$ d>0 $\end{document} and \begin{document}$ a\geq \tilde{a}\approx 1.704 $\end{document} for some positive \begin{document}$ \tilde{a}, $\end{document} then the bifurcation curve \begin{document}$ S $\end{document} is type 4 S-shaped on the \begin{document}$ (\lambda , \left \Vert u\right \Vert _{\infty }) $\end{document}-plane.

Semilinear Caputo time-fractional pseudo-parabolic equations
Nguyen Huy Tuan, Vo Van Au and Runzhang Xu
2021, 20(2): 583-621 doi: 10.3934/cpaa.2020282 +[Abstract](398) +[HTML](176) +[PDF](508.12KB)

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model
Guillaume Cantin and M. A. Aziz-Alaoui
2021, 20(2): 623-650 doi: 10.3934/cpaa.2020283 +[Abstract](295) +[HTML](68) +[PDF](1813.66KB)

The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements.

Random data theory for the cubic fourth-order nonlinear Schrödinger equation
Van Duong Dinh
2021, 20(2): 651-680 doi: 10.3934/cpaa.2020284 +[Abstract](361) +[HTML](58) +[PDF](465.57KB)

We consider the cubic nonlinear fourth-order Schrödinger equation

on \begin{document}$ \mathbb R^N, N\geq 5 $\end{document} with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Dual spaces of mixed-norm martingale Hardy spaces
Ferenc Weisz
2021, 20(2): 681-695 doi: 10.3934/cpaa.2020285 +[Abstract](242) +[HTML](72) +[PDF](364.07KB)

In this paper, we generalize the Doob's maximal inequality for mixed-norm \begin{document}$ L_{\vec{p}} $\end{document} spaces. We consider martingale Hardy spaces defined with the help of mixed \begin{document}$ L_{{\vec{p}}} $\end{document}-norm. A new atomic decomposition is given for these spaces via simple atoms. The dual spaces of the mixed-norm martingale Hardy spaces is given as the mixed-norm \begin{document}$ BMO_{\vec{r}}(\vec{\alpha}) $\end{document} spaces. This implies the John-Nirenberg inequality \begin{document}$ BMO_{1}(\vec{\alpha}) \sim BMO_{\vec{r}}(\vec{\alpha}) $\end{document} for \begin{document}$ 1<\vec{r}<\infty $\end{document}. These results generalize the well known classical results for constant \begin{document}$ p $\end{document} and \begin{document}$ r $\end{document}.

Elliptic problems with rough boundary data in generalized Sobolev spaces
Anna Anop, Robert Denk and Aleksandr Murach
2021, 20(2): 697-735 doi: 10.3934/cpaa.2020286 +[Abstract](288) +[HTML](74) +[PDF](515.58KB)

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity
Zheng Han and Daoyuan Fang
2021, 20(2): 737-754 doi: 10.3934/cpaa.2020287 +[Abstract](259) +[HTML](63) +[PDF](381.34KB)

We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on \begin{document}$ \mathbb{T}^d $\end{document} with Cauchy data of small amplitude \begin{document}$ \epsilon $\end{document}. We show a lower bound \begin{document}$ \epsilon^{-2N-2} $\end{document} for the existence time with any natural number \begin{document}$ N $\end{document}. The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.

Single species population dynamics in seasonal environment with short reproduction period
Attila Dénes and Gergely Röst
2021, 20(2): 755-762 doi: 10.3934/cpaa.2020288 +[Abstract](287) +[HTML](83) +[PDF](587.42KB)

We present a periodic nonlinear scalar delay differential equation model for a population with short reproduction period. By transforming the equation to a discrete dynamical system, we reduce the infinite dimensional problem to one dimension. We determine the basic reproduction number not merely as the spectral radius of an operator, but as an explicit formula and show that is serves as a threshold parameter for the stability of the trivial equilibrium and for permanence.

Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation
Erica Ipocoana and Andrea Zafferi
2021, 20(2): 763-782 doi: 10.3934/cpaa.2020289 +[Abstract](244) +[HTML](60) +[PDF](363.11KB)

The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial \begin{document}$ L^{\infty} $\end{document} estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

The anisotropic fractional isoperimetric problem with respect to unconditional unit balls
Andreas Kreuml
2021, 20(2): 783-799 doi: 10.3934/cpaa.2020290 +[Abstract](195) +[HTML](54) +[PDF](362.47KB)

The minimizers of the anisotropic fractional isoperimetric inequality with respect to a convex body \begin{document}$ K $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document} are shown to be equivalent to star bodies whenever \begin{document}$ K $\end{document} is strictly convex and unconditional. From this a Pólya-Szegő principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.

The boundedness of multi-linear and multi-parameter pseudo-differential operators
Liang Huang and Jiao Chen
2021, 20(2): 801-815 doi: 10.3934/cpaa.2020291 +[Abstract](234) +[HTML](63) +[PDF](347.22KB)

In this paper, we establish the boundedness on \begin{document}$ L^r(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}) $\end{document} of bilinear and bi-parameter pseudo-differential operators whose symbols \begin{document}$ \sigma(x,\xi,\eta)\in S^{(0,0)}_{(1,1),(\delta_1,\delta_2)} $\end{document}   for \begin{document}$ x,\xi,\eta\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2} $\end{document} and \begin{document}$ 0\leq\delta_1,\delta_2<1 $\end{document}, which extends the result of Dai and Lu in [8].

Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity
Chungen Liu and Huabo Zhang
2021, 20(2): 817-834 doi: 10.3934/cpaa.2020292 +[Abstract](237) +[HTML](60) +[PDF](394.47KB)

In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem

where \begin{document}$ a, b,\kappa $\end{document} are positive parameters, \begin{document}$ \alpha\in(\frac{3}{4},1),\beta\in(0,1) $\end{document}, and \begin{document}$ 2^{\ast}_{\alpha} = \frac{6}{3-2\alpha} $\end{document}, \begin{document}$ (-\Delta)^{\alpha} $\end{document} stands for the fractional Laplacian. By the nodal Nehari manifold method, for each \begin{document}$ b>0 $\end{document}, we obtain a ground state nodal solution \begin{document}$ u_{b} $\end{document} and a ground-state solution \begin{document}$ v_b $\end{document} to this problem when \begin{document}$ \kappa\gg 1 $\end{document}, where the nonlinear function \begin{document}$ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R} $\end{document} is a Carathéodory function. We also give an analysis on the behavior of \begin{document}$ u_{b} $\end{document} as the parameter \begin{document}$ b\to 0 $\end{document}.

Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian
Zaizheng Li and Qidi Zhang
2021, 20(2): 835-865 doi: 10.3934/cpaa.2020293 +[Abstract](264) +[HTML](60) +[PDF](490.8KB)

We prove a Hopf's lemma in the point-wise sense for fractional \begin{document}$ p $\end{document}-Laplacian. The essential technique is to prove \begin{document}$ (-\Delta)^s_p u(x) $\end{document} is uniformly bounded in the unit ball \begin{document}$ B_1\subset\mathbb{R}^n $\end{document}, where \begin{document}$ u(x) = (1-|x|^2)^s_{+} $\end{document}. Also we study the global Hölder continuity of bounded positive solutions for \begin{document}$ (-\Delta)^s_p u(x) = f(x,u). $\end{document}

Multiple positive solutions for coupled Schrödinger equations with perturbations
Haoyu Li and Zhi-Qiang Wang
2021, 20(2): 867-884 doi: 10.3934/cpaa.2020294 +[Abstract](254) +[HTML](61) +[PDF](463.51KB)

For coupled Schrödinger equations with nonhomogeneous perturbations we give several results on the existence of multiple positive solutions. In particular in one case we consider perturbations of the permutation symmetry.

On the quotient quantum graph with respect to the regular representation
Gökhan Mutlu
2021, 20(2): 885-902 doi: 10.3934/cpaa.2020295 +[Abstract](208) +[HTML](53) +[PDF](331.92KB)

Given a quantum graph \begin{document}$ \Gamma $\end{document}, a finite symmetry group \begin{document}$ G $\end{document} acting on it and a representation \begin{document}$ R $\end{document} of \begin{document}$ G $\end{document}, the quotient quantum graph \begin{document}$ \Gamma /R $\end{document} is described and constructed in the literature [1,2,18]. In particular, it was shown that the quotient graph \begin{document}$ \Gamma/\mathbb{C}G $\end{document} is isospectral to \begin{document}$ \Gamma $\end{document} by using representation theory where \begin{document}$ \mathbb{C}G $\end{document} denotes the regular representation of \begin{document}$ G $\end{document} [18]. Further, it was conjectured that \begin{document}$ \Gamma $\end{document} can be obtained as a quotient \begin{document}$ \Gamma/\mathbb{C}G $\end{document} [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph \begin{document}$ \Gamma $\end{document} and a finite symmetry group \begin{document}$ G $\end{document} acting on it, the quotient quantum graph \begin{document}$ \Gamma / \mathbb{C}G $\end{document} is not only isospectral but rather identical to \begin{document}$ \Gamma $\end{document} for a particular choice of a basis for \begin{document}$ \mathbb{C}G $\end{document}. Furthermore, we prove that, this result holds for an arbitrary permutation representation of \begin{document}$ G $\end{document} with degree \begin{document}$ |G| $\end{document}, whereas it doesn't hold for a permutation representation of \begin{document}$ G $\end{document} with degree greater than \begin{document}$ |G|. $\end{document}

Inequalities of Hermite-Hadamard type for higher order convex functions, revisited
Tomasz Szostok
2021, 20(2): 903-914 doi: 10.3934/cpaa.2020296 +[Abstract](236) +[HTML](55) +[PDF](455.31KB)

In this paper we present a very short proof of inequalities of Hermite-Hadamard type obtained by M. Bessenyei and Zs. Páles. This proof is based on the recently developed method connected with use of stochastic orderings of random variables. In the second part of the paper we present a way to extend these known inequalities. Namely, we describe completely the possible inequalities of Hermite-Hadamard type for longer expression than it was the case in the results of Bessenyei and Páles.

The degenerate Monge-Ampère equations with the Neumann condition
Juhua Shi and Feida Jiang
2021, 20(2): 915-931 doi: 10.3934/cpaa.2020297 +[Abstract](223) +[HTML](47) +[PDF](372.98KB)

In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampère equation \begin{document}$ \det D^{2}u = f(x) $\end{document} with the Neumann boundary value condition, which are independent of \begin{document}$ \inf f $\end{document}. Based on these uniform estimates, the existence and uniqueness of the global \begin{document}$ C^{1,1} $\end{document} solution to the Neumann problem of the degenerate Monge-Ampère equation are established under the assumption \begin{document}$ f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}) $\end{document}.

Ground states for a class of quasilinear Schrödinger equations with vanishing potentials
Zhouxin Li and Yimin Zhang
2021, 20(2): 933-954 doi: 10.3934/cpaa.2020298 +[Abstract](232) +[HTML](57) +[PDF](466.6KB)

In this paper, we study a class of quasilinear Schrödinger equation of the form

where \begin{document}$ V $\end{document}, \begin{document}$ K $\end{document} are smooth functions and \begin{document}$ V $\end{document} may vanish at infinity, \begin{document}$ 2<q<2(2^*) $\end{document}. We prove the existence of a positive ground state solution which possesses a unique local maximum and decays exponentially.

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