American Institute of Mathematical Sciences

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem

\begin{document}$ \begin{equation*} -(\varepsilon^{2}a+ \varepsilon b\int _{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v+V(x)v = P(x)f(v)\; \; {\rm{in}}\; \mathbb{R}^{3}, \end{equation*} $\end{document}

where \begin{document}$ \varepsilon $\end{document} is a small positive parameter, \begin{document}$ a, b>0 $\end{document} and \begin{document}$ V, P\in C^{1}(\mathbb{R}^{3}, \mathbb{R}) $\end{document}. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential \begin{document}$ P $\end{document} is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.

This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.

We prove multiplicity results for solutions, both with positive and negative energy, for a class of singular quasilinear Schrödinger equations in the entire \begin{document}$ \mathbb {R}^N $\end{document} involving a critical term, nontrivial weights and positive parameters \begin{document}$ \lambda $\end{document}, \begin{document}$ \beta $\end{document}, covering several physical models, coming from plasma physics as well as high-power ultra short laser in matter. Also the symmetric setting is investigated. Our proofs relay on variational tools, including concentration compactness principles because of the delicate situation of the double lack of compactness. In addition, a necessary reformulation of the original problem in a suitable variational setting, produces a singular function, delicate to be managed.

We present a weighted \begin{document}$ L_p $\end{document}-theory of parabolic systems on a half space \begin{document}$ {\mathbb{R}}^d_+ $\end{document}. The leading coefficients are assumed to be only measurable in time \begin{document}$ t $\end{document} and have small bounded mean oscillations (BMO) with respect to the spatial variables \begin{document}$ x $\end{document}, and the lower order coefficients are allowed to blow up near the boundary.

We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.

In this paper, we analyze a nonlinear equation modeling the mechanical replication of the DNA molecule based on a Kolmogorov-Jhonson-Mehl-Avrami (KJMA) type model inspired on the mathematical analogy between the DNA replication process and the crystal growth. There are two different regions on the DNA molecule deep into a duplication process, the connected regions where the base pairs have been already duplicated, called eyes or islands and the regions not yet duplicated, called holes. The Cauchy problem associated with this model will be analyzed, where some dependences and nonlinearities on the replication velocity and the origins of replication are introduced.

Rencently Chow, Huang, Li and Zhou proposed discrete forms of the Fokker-Planck equations on a finite graph. As a primary step, they constructed Riemann metrics on the graph by endowing it with some kinds of weight. In this paper, we reveal the relation between these Riemann metrics and the Euclidean metric, by showing that they are locally equivalent. Moreover, various Riemann metrics have this property provided the corresponding weight satisfies a bounded condition. Based on this, we prove that the two-side Łojasiewicz inequality holds near the Gibbs distribution with Łojasiewicz exponent \begin{document}$ \frac{1}{2} $\end{document}. Then we use it to prove the solution of the discrete Fokker-Planck equation converges to the Gibbs distribution with exponential rate. As a corollary of Łojasiewicz inequality, we show that the two-side Talagrand-type inequality holds under different Riemann metrics.

We study the existence of the solution to a semilinear higher-order elliptic system

\begin{document}$ \mathcal{L}v(t, \cdot) = F_{S, T}\circ G(v)(t, \cdot), \quad \forall t\in [0, \tau], $\end{document}

with the homogeneous Dirichlet boundary conditions. Here, \begin{document}$ \mathcal{L} = (-\Delta)^m $\end{document} is a harmonic operator of order \begin{document}$ m $\end{document}, \begin{document}$ v = v(t, x):[0, \tau]\times\Omega\rightarrow \mathbb{R}^n $\end{document} is the unknown, \begin{document}$ t $\end{document} is a parameter, \begin{document}$ F_{S, T} $\end{document} is a function related to given functions \begin{document}$ S $\end{document} and \begin{document}$ T $\end{document}, and \begin{document}$ G(v)(t, x) $\end{document} is defined by the solution \begin{document}$ y^v(s;t, x) $\end{document} of an ODE-IVP \begin{document}$ {\rm d}y/\mathrm{d}s = v(s, y), \quad y(t) = x $\end{document}. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.

In this paper, we consider the multiplicity of nodal solutions for the following Kirchhoff type equations:

\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2M\left(\varepsilon^{2-N}||\nabla u||^2_{L^2}\right)\Delta u+u = f\left(x\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ N\geq 4 $\end{document}, \begin{document}$ \varepsilon>0 $\end{document} is a small parameter, \begin{document}$ M\left(t\right) = at+b\left(a,b>0\right) $\end{document} and \begin{document}$ 2<p<2^* = \frac{2N}{N-2} $\end{document}. We assume that the weight function \begin{document}$ f\in C\left(\mathbb{R}^N,\mathbb{R}^+\right) $\end{document} has \begin{document}$ k $\end{document} maximum points in \begin{document}$ \mathbb{R}^N $\end{document}. By using a novel constraint approach as well as the barycenter map, \begin{document}$ k^2 $\end{document} nodal solutions are obtained when \begin{document}$ N\geq4 $\end{document} for \begin{document}$ \varepsilon,a $\end{document} sufficiently small.

Using a Lyapunov-Krasovskii functional, new results concerning the global stability, boundedness of solutions, existence and non-existence of \begin{document}$ T $\end{document}-periodic solutions for a kind of delayed equation for a \begin{document}$ \varphi $\end{document}-Laplacian operator are obtained. An application is given for the well known sunflower equation.

In this paper we explore the theory of fractional powers of positive operators, more precisely, we use the Balakrishnan formula to obtain parabolic approximations of (damped) wave equations in bounded smooth domains in \begin{document}$ \mathbb{R}^N $\end{document}. We also explicitly calculate the fractional powers of wave operators in terms of the fractional Laplacian in bounded smooth domains and characterize the spectrum of these operators.

Considered herein is the initial boundary value problem associated with a sixth-order nonlinear parabolic equation in a bounded domain. We first establish a new global Carleman estimate for the sixth-order parabolic operator. Based on this estimate, we obtain the local exact controllability to the trajectories and the unique continuation property of the parabolic equation.

In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on \begin{document}$ \mathbb{S}^n $\end{document}. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in \begin{document}$ \mathbb{S}^n $\end{document}, we first employ the Mobius transform between \begin{document}$ \mathbb{S}^n $\end{document} and \begin{document}$ \mathbb{R}^n $\end{document}, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on \begin{document}$ \mathbb{S}^n $\end{document} is equivalent to some integral equation in \begin{document}$ \mathbb{R}^n $\end{document}. Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on \begin{document}$ \mathbb{S}^n $\end{document}. As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on \begin{document}$ \mathbb{S}^n $\end{document}. Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on \begin{document}$ \mathbb{S}^n $\end{document}.

In this paper we investigate the existence and the properties for the minimisers of a special Helfrich functional for surfaces of revolution with Dirichlet boundary value conditions. Removing the even restriction for the admissible functions in [5], we prove that the minimiser is even and smooth, the minimal increases as the boundary value increases, and the minimiser is no less than the boundary value which answers an open question in [5] partly. We also obtain the existence and regularity for (general) Helfrich functional when the boundary value is large.

We study the fractional Cucker-Smale (in short, CS) model under general network topologies. In [15], the authors introduced the fractional CS model to see the interplay of memory effect and the flocking dynamics in the all-to-all network topology. As an extension of the previous work, we investigate under which network topologies flocking still emerges. Specifically, we first consider the symmetric network case and show that the existence of a leader guarantees the emergence of flocking. Furthermore, we present a framework for the non-symmetric network case where we can observe the flocking. We also conduct numerical simulations to support our theoretical results and see whether our framework gives necessary and sufficient conditions for the emergence of flocking.

In this paper, we study the following high-order Hardy-H\begin{document}$ \acute{e} $\end{document}non type system:

\begin{document}$ \begin{cases} \ (-\Delta)^{\frac{\alpha}{2}}u(x) = |x|^{a}v^{p}(x) ,\\ \ (-\Delta)^{\frac{\beta}{2}}v(x) = |x|^{b}u^{q}(x) ,\\ \end{cases} $\end{document}

where \begin{document}$ 0<\alpha = s_{1}+2<n $\end{document}, \begin{document}$ 0<\beta = s_{2}+2<n $\end{document}, \begin{document}$ 0<s_{1},s_{2}<2 $\end{document}, \begin{document}$ a>-s_{1} $\end{document}, \begin{document}$ b>-s_{2} $\end{document}, \begin{document}$ p,q>0 $\end{document}. There are two cases to be considered. The first one is where the domain is the whole space \begin{document}$ \mathbb{R}^{n} $\end{document}, and the second one is where the domain is bounded. First of all, we consider the above system in the whole space \begin{document}$ \mathbb{R}^{n} $\end{document}, we show that the above system are equivalent to the integral system:

\begin{document}$ \begin{cases} \ u(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{a}v^{p}(y)}{|x-y|^{n-\alpha}}dy,\\[1.5mm] \ v(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{b}u^{q}(y)}{|x-y|^{n-\beta}}dy.\\ \end{cases} $\end{document}

Then by using the method of moving planes in integral forms, we prove that there are no positive solutions for the above integral system. In addition, while in the subcritical case \begin{document}$ 1<p<\frac{n+\alpha+2a}{n-\alpha} $\end{document}, \begin{document}$ 1<q<\frac{n+\alpha+2b}{n-\alpha} $\end{document} with \begin{document}$ \alpha = \beta $\end{document} in the above elliptic system, we prove the nonexistence of a positive solution for the above system in \begin{document}$ \mathbb{R}^{n} $\end{document}. Then, through the \begin{document}$ Doubling\ Lemma $\end{document} we obtain the singularity estimates of the positive solutions on a bounded domain \begin{document}$ \Omega $\end{document}.