American Institute of Mathematical Sciences

Long time behavior of a semilinear wave equation with variable coefficients with nonlinear boundary dissipation is considered. It is shown that the existence of global and compact attractors depends on the curvature properties of a Riemannian metric given by the variable coefficients.

In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the semi-implicit scheme for the temporal discretization. With reasonable stability conditions, it is shown that the proposed schemes are energy stable. Furthermore, by defining a new projection operator, we deduce the optimal \begin{document}$ L^2 $\end{document} error estimates. Some numerical experiments are presented to confirm the theoretical predictions and the efficiency of the proposed schemes.

In this paper, we show the existence of even solutions with prescribed asymptotic behavior at infinity. Instead of using the integrability of the Toda system, the novel idea here is a degree argument approach. Perturbation theory has also been used in our study. Our method can be generalized to explore non-integrable systems with exponential type nonlinearities.

The Hadamard semidifferential calculus preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the \begin{document}$ n $\end{document}-dimensional Euclidean space \begin{document}$ \operatorname{\mathbb R}^n $\end{document} to subsets of topological vector spaces. This includes most function spaces used in Optimization and the Calculus of Variations, the metric groups used in Shape and Topological Optimization, and functions defined on submanifolds.

Certain set-parametrized functions such as the characteristic function \begin{document}$ \chi_A $\end{document}of a set \begin{document}$ A $\end{document}, the distance function \begin{document}$ d_A $\end{document} to \begin{document}$ A $\end{document}, and the oriented (signed) distance function \begin{document}$ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $\end{document} can be used to identify a space of subsets of \begin{document}$ \operatorname{\mathbb R}^n $\end{document} with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of \begin{document}$ b_A $\end{document} and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.

The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.

We are concerned with ground state solutions of the fractional problems with dipole-type potential and critical exponent. Under certain conditions on the dipole-type potential and the parameter, we show that the structure of the Palais-Smale sequence goes to zero weakly, and establish the existence of ground state solution to the above problems by using a new analytical method not involving the concentration-compactness principle.

Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.

This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period \begin{document}$ T $\end{document} of traveling wave when the small positive parameter \begin{document}$ \tau\rightarrow 0 $\end{document} are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.

We study the asymptotic spreading properties and periodic traveling wave solutions of a time periodic and diffusive SI epidemic model with demographic structure (follows the logistic growth). Since the comparison principle is not applicable to the full system, we analyze the asymptotic spreading phenomena for susceptible class and infectious class by comparing with respective relevant periodic equations with KPP-type. By applying fixed point theorem to a truncated problem on a finite interval, combining with limit idea, the existence of periodic traveling wave solutions are derived. The results show that the minimal wave speed exactly equals to the spreading speed of infectious class when susceptible class is abundant.

The paper concerns the controllability and stabilization of surface water waves in a two-dimensional rectangular basin under the forces of gravity and surface tension. The surface waves are generated by a wave-maker placed at the left side-boundary and it is physical relevant to see whether the surface waves are controllable or can be stabilized using appropriate motion of the wave-maker. Due to the surface tension, an edge condition must be imposed at the contact point between the free surface and a solid boundary. Two types of wave-makers are considered: "flexible" or "rigid". It is shown that the surface waves are approximately controllable, but not exactly controllable, for both "flexible" and "rigid" wave-makers. In addition, under a static feedback to control a "rigid" wave-maker, the strong stability of feedback control system is obtained.

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We investigate Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. The first one involves a polynomial nonlinearity and the second one involves a gradient nonlinearity. Namely, we derive sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data, for which the considered problems admit no nontrivial local weak solutions, i.e., an instantaneous blow-up occurs.

In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space

\begin{document}$ \begin{align*} \begin{cases} - \Delta {u_i}(y) = \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_1}({u_j}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_2}({u_i}(y)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_2}({u_i}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_1}({u_j}(y)), \ y \in \mathbb{R}_ + ^N, \hfill \\ \frac{{\partial {u_i}}} {{\partial \nu }}(\bar x, 0) = \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_2}({u_i}(\bar x, 0)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_1}({u_j}(\bar x, 0)), \quad \quad(\bar x, 0) \in \partial \mathbb{R}_ + ^N, \end{cases} \end{align*} $\end{document}

where \begin{document}$ \mathbb{R}_+^N = \{x\in{\mathbb{R}^N}: x_N > 0\}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $\end{document} are some nonlinear functions. Under some assumptions on the nonlinear functions \begin{document}$ F, G, f, g $\end{document}, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.

In this paper we study a family of the Gagliardo-Nirenberg-Sobolev interpolation inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. The inequalities being equivalent to some minimization problems, we also analyse the set of solutions of the Euler-Lagrange equations satisfied by extremal functions, or equivalently, by minimizers.

This paper is concerned with second-order elliptic operators whose diffusion coefficients degenerate at the boundary in first order. In this borderline case, the behavior strongly depends on the size and direction of the drift term. Mildly inward (or outward) pointing and strongly outward pointing drift terms were studied before. Here we treat the intermediate case equipped with Dirichlet boundary conditions, and show generation of an analytic positive \begin{document}$ C_0 $\end{document}-semigroup. The main result is a precise description of the domain of the generator, which is more involved than in the other cases and exhibits reduced regularity compared to them.

This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.

We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.

In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift \begin{document}$ \gamma_l(t) $\end{document} that depends on the coordinate \begin{document}$ l $\end{document} transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of \begin{document}$ \gamma $\end{document} can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of \begin{document}$ p $\end{document}-\begin{document}$ q $\end{document} type and singular nonlinearities

\begin{document}$ \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {} = \lambda \frac{f(u)}{u^\gamma}, \ u>0 && \quad\mbox{ in } \, \Omega, \\ u & {} = 0 && \quad\mbox{ on } \partial\Omega, \end{alignedat} \right. $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document} with \begin{document}$ C^2 $\end{document} boundary, \begin{document}$ N \geq 1 $\end{document}, \begin{document}$ \lambda >0 $\end{document} is a real parameter,

\begin{document}$ \mathcal{L}_{p,q} u : = {\rm{div}}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u), $\end{document}

\begin{document}$ 1<p<q< \infty $\end{document}, \begin{document}$ \gamma \in (0,1) $\end{document}, and \begin{document}$ f $\end{document} is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [1], we prove existence of three positive solutions in the positive cone of \begin{document}$ C_\delta(\overline{\Omega}) $\end{document} and in a certain range of \begin{document}$ \lambda $\end{document}.