American Institute of Mathematical Sciences

For a discrete system \begin{document}$ (X,T) $\end{document}, the flip system \begin{document}$ (X,T,F) $\end{document} can be regarded as the action of infinite dihedral group \begin{document}$ D_\infty $\end{document} on the space \begin{document}$ X $\end{document}. Under this action, \begin{document}$ X $\end{document} is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the flip systems of shifts of finite type. The proof relies mostly on combinatorial calculations instead of the usual approach via zeta function. Here, we are able to obtain more precise asymptotic result for this \begin{document}$ D_\infty $\end{document}-action on shifts of finite type as compared to other group actions on systems available in the literature.

In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential \begin{document}$ u(\omega t) $\end{document}. We show that if the frequency vector \begin{document}$ \omega $\end{document} is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy \begin{document}$ E $\end{document}. In contrast with previous results, the conditions of small potential \begin{document}$ u $\end{document} or large energy \begin{document}$ E $\end{document} are no longer needed.

We consider the wave equation with a cubic convolution

\begin{document}$ \partial_{t}^2 u-\Delta u = (|x|^{- \gamma}*u^2)u $\end{document}

in three space dimensions. Here, \begin{document}$ 0< \gamma<3 $\end{document} and \begin{document}$ * $\end{document} stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then \begin{document}$ \gamma\ge2 $\end{document} assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when \begin{document}$ 2\le \gamma<3 $\end{document}. In this paper, we consider the Cauchy problem for \begin{document}$ 2\le \gamma<3 $\end{document} in the space-time weighted \begin{document}$ L^ \infty $\end{document} space in which functions have critical decay rate. When \begin{document}$ \gamma = 2 $\end{document}, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in [13]). When \begin{document}$ 2< \gamma<3 $\end{document}, we also prove unique global existence of solutions for small data.

We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

Recently, Lindenstrauss and Tsukamoto established a double variational principle between mean dimension theory and rate distortion theory. The main purpose of this paper is to develop some new variational relations for the metric mean dimension and the rate distortion dimension. Inspired by the dimension theory of topological entropy, we introduce and explore the Bowen metric mean dimension of subsets. Besides, we give some new characterizations for the rate distortion dimension. Finally, the relation between the Bowen metric mean dimension of the set of generic points and the rate distortion dimension is also investigated.

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.

In this paper we discuss some thermoelastic and thermoviscoelastic models obtained from the Gurtin theory, based on the invariance of the entropy under time reversal. We derive differential systems where the temperature and the velocity are ruled by generalized versions of the Moore-Gibson-Thompson equation. In the one-dimensional case, we provide a complete analysis of the evolution, establishing an existence and uniqueness result valid for any choice of the constitutive parameters. This result turns out to be new also for the MGT equation itself. Then, under suitable assumptions on the parameters, corresponding to the subcritical regime of the system, we prove the exponential stability of the related semigroup.

We consider a map \begin{document}$ F $\end{document} of class \begin{document}$ C^r $\end{document} with a fixed point of parabolic type whose differential is not diagonalizable, and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of \begin{document}$ F $\end{document}, there exist invariant curves of class \begin{document}$ C^r $\end{document} away from the fixed point, and that they are analytic when \begin{document}$ F $\end{document} is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of \begin{document}$ F $\end{document} on them.

The present paper deals with a class of Schrödinger-poisson system. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by using purely variational methods. Comparing to the previous works, we encounter some new challenges because of nonlocal term. By doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many positive solutions.

In this paper, we use a variational approach to study traveling wave solutions of a gradient system in an infinite strip. As the even-symmetric potential of the system has three local minima, we prove the existence of a traveling wave that propagates from one phase to the other two phases, where these phases corresponds to the three local minima of the potential. To control the asymptotic behavior of the wave at minus infinity, we successfully find a certain convexity condition on the potential, which guarantees the convergence of the wave to a constant state but not to a one-dimensional homoclinic solution or other equilibria. In addition, a non-trivial steady state in \begin{document}$ \mathbb R^2 $\end{document} is established by taking a limit of the traveling wave solutions in the strip as the width of the strip tends to infinity.

For the coupled Ginzburg-Landau system in \begin{document}$ {\mathbb R}^2 $\end{document}

\begin{document}$ \begin{align*} \begin{cases} -\Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+ = 0, \\ -\Delta w^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^- = 0, \end{cases} \end{align*} $\end{document}

with following constraints for the constant coefficients

\begin{document}$ A_+, A_->0,\ B^2<A_+A_-,\ t^+, t^->0, $\end{document}

the radially symmetric solution \begin{document}$ w(x) = (w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2 $\end{document} of degree pair \begin{document}$ (1, 1) $\end{document} was given by A. Alama and Q. Gao in J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator \begin{document}$ {\mathcal L} $\end{document} around \begin{document}$ w $\end{document} and prove the non-degeneracy result under one more assumption \begin{document}$ B<0 $\end{document}: the kernel of \begin{document}$ {\mathcal L} $\end{document} is spanned by the functions \begin{document}$ \frac{\partial w}{\partial{x_1}} $\end{document} and \begin{document}$ \frac{\partial w}{\partial{x_2}} $\end{document} in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator \begin{document}$ {\mathcal L} $\end{document} will be given.

If \begin{document}$ (M,g) $\end{document} is a smooth compact rank \begin{document}$ 1 $\end{document} Riemannian manifold without focal points, it is shown that the measure \begin{document}$ \mu_{\max} $\end{document} of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure \begin{document}$ \mu_{\max} $\end{document} is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle \begin{document}$ SM $\end{document} with respect to \begin{document}$ \mu_{\max} $\end{document} is Bernoulli is acquired provided \begin{document}$ M $\end{document} is a compact surface with genus greater than one and no focal points.

We prove the existence of infinitely many sign changing radial solutions for a \begin{document}$ p $\end{document}-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary.

In this paper, we present two constructions of forward self-similar solutions to the \begin{document}$ 3 $\end{document}D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems.

We are concerned with blow-up mechanisms in a semilinear heat equation:

\begin{document}$ u_t = \Delta u + |x|^{2a} u^p , \quad x \in \textbf{R}^N , \, t>0, $\end{document}

where \begin{document}$ p>1 $\end{document} and \begin{document}$ a>-1 $\end{document} are constants. As for the Fujita equation, which corresponds to \begin{document}$ a = 0 $\end{document}, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if \begin{document}$ N\geq 11 $\end{document} and \begin{document}$ p> 1 + 4/(N-4-2\sqrt{N-1}) $\end{document}, then there exist radial blow-up solutions \begin{document}$ u_{\ell, {\rm HV}}(x, t) $\end{document}, \begin{document}$ \ell \in \bf{N} $\end{document}, such that

\begin{document}$ \lim\limits_{t\to T} \left( T-t \right)^{1/(p-1)} \| u_{\ell, {\rm HV}}(\cdot, t) \|_{L^{\infty}(\textbf{R}^N )} = \infty, $\end{document}

where \begin{document}$ T $\end{document} is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case \begin{document}$ a\not = 0 $\end{document}. As a consequence, we obtain an example of solutions that blow up at \begin{document}$ x = 0 $\end{document}, the zero point of potential \begin{document}$ |x|^{2a} $\end{document} with \begin{document}$ a>0 $\end{document}, and behave in non-self-similar manner for \begin{document}$ N > 10 + 8a $\end{document}. This last result is in contrast to backward self-similar solutions previously obtained for \begin{document}$ N < 10 + 8a $\end{document}, which blow up at \begin{document}$ x = 0 $\end{document}.

In this paper we consider the problem

\begin{document}$ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $\end{document}

where \begin{document}$ V_{\lambda} = \lambda+V_{0} $\end{document} with \begin{document}$ \lambda \geq 0 $\end{document}, \begin{document}$ V_0\in L^{N/2}({\mathbb{R}}^N) $\end{document}, \begin{document}$ I_{\mu} = \frac{1}{|x|^\mu} $\end{document} is the Riesz potential with \begin{document}$ 0<\mu<\min\{N, 4\} $\end{document} and \begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $\end{document} with \begin{document}$ N\geq 3 $\end{document}. Under some smallness assumption on \begin{document}$ V_0 $\end{document} and \begin{document}$ \lambda $\end{document} we prove the existence of two positive solutions of \begin{document}$ (P_\lambda) $\end{document}. In order to prove the main results, we used variational methods combined with degree theory.

In this paper, we consider the compressible Euler equations with time-dependent damping \begin{document}$ \frac{{\alpha}}{(1+t)^\lambda}u $\end{document} in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case \begin{document}$ \lambda\neq1 $\end{document} and \begin{document}$ \lambda = 1 $\end{document} respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for \begin{document}$ 1<\gamma<3 $\end{document} we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a \begin{document}$ C^1 $\end{document}-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

We investigate spreading properties of solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interactions. In this work the mutation process is described using a non-local convolution operator in the phenotype space. Initially equipped with a localized amount of infection, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed of the travelling wave solutions discussed in [1]. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable and the phenotype one are separated.

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.