American Institute of Mathematical Sciences

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form \begin{document}$ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $\end{document}, like Fokker-Plank and \begin{document}$ q $\end{document}-Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense \begin{document}$ p $\end{document}-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

Let \begin{document}$ \Omega $\end{document} be the smooth bounded domian in \begin{document}$ \mathbb{R}^2 $\end{document}, \begin{document}$ W_0^{1, 2}(\Omega) $\end{document} be the standard Sobolev space. We concern a Trudinger-Moser inequality involving \begin{document}$ L^p $\end{document} norms. For any \begin{document}$ p>1 $\end{document}, denote

\begin{document}$ \lambda_p(\Omega) = \inf\limits_{u\in W_0^{1, 2}(\Omega), u\not\equiv0} \frac{\|\nabla u\|_2^2}{\|u\|_p^2}. $\end{document}

We prove that for any \begin{document}$ p>1 $\end{document} and any \begin{document}$ 0\leq\tau<\lambda_p $\end{document}, there exists a positive real number \begin{document}$ \tau^\ast $\end{document} such that if \begin{document}$ \tau^\ast <\tau<\lambda_p $\end{document}, the supremum

\begin{document}$ \begin{equation*} \sup\limits_{u\in W_0^{1, 2}(\Omega), \, \| \nabla u\|_{2}^2\leq4 \pi}\int_{\Omega}e^{ u^2 (1+\tau\|u\|_p^2)}dx, \end{equation*} $\end{document}

can not be achieved by any \begin{document}$ u\in W_0^{1, 2}(\Omega) $\end{document} with \begin{document}$ \| \nabla u\|_{2}^2\leq4 \pi $\end{document}. This is based on a method of energy estimate, which is developed by [14, 15, 16].

In this paper we prove the existence of infinitely many radial solutions of \begin{document}$ \Delta u + K(r)f(u) = 0 $\end{document} on the exterior of the ball of radius \begin{document}$ R>0 $\end{document}, \begin{document}$ B_{R} $\end{document}, centered at the origin in \begin{document}$ {\mathbb R}^{N} $\end{document} with \begin{document}$ u = 0 $\end{document} on \begin{document}$ \partial B_{R} $\end{document} and \begin{document}$ \lim_{r \to \infty} u(r) = 0 $\end{document} where \begin{document}$ N>2 $\end{document}, \begin{document}$ f $\end{document} is odd with \begin{document}$ f<0 $\end{document} on \begin{document}$ (0, \beta) $\end{document}, \begin{document}$ f>0 $\end{document} on \begin{document}$ (\beta, \infty), $\end{document} \begin{document}$ f $\end{document} superlinear for large \begin{document}$ u $\end{document} and \begin{document}$ 0< K(r) \leq \frac{K_{1}}{r^{\alpha}} $\end{document} with \begin{document}$ 2<\alpha <2(N-1) $\end{document} for large \begin{document}$ r $\end{document}.

In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual \begin{document}$ L^{2}( \mathbb R^2) $\end{document}-based Sobolev spaces \begin{document}$ H^s( \mathbb R^2) $\end{document} for \begin{document}$ s>\frac{5}{4} $\end{document} which coincides with the best available result in the literature proved employing more complicated tools.

Using bifurcation theory, we prove the existence of spiky steady states and investigate the stability of bifurcating solutions of the one-dimensional continuous neighbour based chemotaxis model, in which the one-step jumping probability rate of cells is determined only by the chemoattractant concentration at the destination. These spiky steady states are crucial when we model cell aggregation, the most important phenomenon in chemotaxis.

In this paper, we consider the problem

\begin{document}$ \begin{equation*} f^{q-1}(x) = \int_{\Omega}\frac{|x|^{\alpha}|y|^{\beta}f(y)}{|x-y|^{n-\gamma}}dy, \; f>0, \; x\in\overline{\Omega}, \end{equation*} $\end{document}

where \begin{document}$ \Omega $\end{document} is the unit ball in \begin{document}$ \mathbb{R}^n(n\geq3) $\end{document} centered at the origin, \begin{document}$ 1<\gamma<n $\end{document} and \begin{document}$ \alpha, \beta>0 $\end{document}, \begin{document}$ q_\gamma: = \frac{2n}{n+\gamma}<q<2 $\end{document}. We will investigate the asymptotic behavior of energy maximizing positive solution as \begin{document}$ q\rightarrow (\frac{2n}{n+\gamma})^{+} = (q_\gamma)^+ $\end{document}. We also show that the energy maximizing positive solution concentrate at a point, which is located at the boundary as \begin{document}$ q\rightarrow (q_\gamma)^{+} $\end{document}. In addition, the energy maximizing positive solution is non-radial provided that \begin{document}$ q $\end{document} closes to \begin{document}$ q_\gamma $\end{document}.

We deal with monotonicity with respect to \begin{document}$ p $\end{document} of the first positive eigenvalue of the \begin{document}$ p $\end{document}-Laplace operator on \begin{document}$ \Omega $\end{document} subject to the homogeneous Neumann boundary condition. For any fixed integer \begin{document}$ D>1 $\end{document} we show that there exists \begin{document}$ M\in[2 e^{-1}, 2] $\end{document} such that for any open, bounded, convex domain \begin{document}$ \Omega\subset{{\mathbb R}}^D $\end{document} with smooth boundary for which the diameter of \begin{document}$ \Omega $\end{document} is less than or equal to \begin{document}$ M $\end{document}, the first positive eigenvalue of the \begin{document}$ p $\end{document}-Laplace operator on \begin{document}$ \Omega $\end{document} subject to the homogeneous Neumann boundary condition is an increasing function of \begin{document}$ p $\end{document} on \begin{document}$ (1, \infty) $\end{document}. Moreover, for each real number \begin{document}$ s>M $\end{document} there exists a sequence of open, bounded, convex domains \begin{document}$ \{\Omega_n\}_n\subset{{\mathbb R}}^D $\end{document} with smooth boundaries for which the sequence of the diameters of \begin{document}$ \Omega_n $\end{document} converges to \begin{document}$ s $\end{document}, as \begin{document}$ n\rightarrow\infty $\end{document}, and for each \begin{document}$ n $\end{document} large enough the first positive eigenvalue of the \begin{document}$ p $\end{document}-Laplace operator on \begin{document}$ \Omega_n $\end{document} subject to the homogeneous Neumann boundary condition is not a monotone function of \begin{document}$ p $\end{document} on \begin{document}$ (1, \infty) $\end{document}.

Let \begin{document}$ u $\end{document} be a solution to the Cauchy problem for a nonlinear diffusion equation

\begin{document}$ \begin{equation*} \begin{cases} \partial_t u = \mathrm{div}\, (|\nabla u|^{p-2} \nabla u) + u^\alpha & \quad\mathrm{in}\quad{\bf R}^N\times(0, \infty), \\ u(x, 0) = \lambda+\varphi(x) & \quad\mathrm{in}\quad{\bf R}^N, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ N \ge 1 $\end{document}, \begin{document}$ 2N/(N+1)<p\neq2 $\end{document}, \begin{document}$ \alpha \in (-\infty, 1) $\end{document}, \begin{document}$ \lambda>0 $\end{document} and \begin{document}$ \varphi\in BC({\bf R}^N)\, \cap\, L^1({\bf R}^N) $\end{document} with \begin{document}$ \varphi\geq0 $\end{document} in \begin{document}$ {\bf R}^{N} $\end{document}. Then the solution \begin{document}$ u $\end{document} behaves like a positive solution to ODE \begin{document}$ \zeta' = \zeta^\alpha $\end{document} in \begin{document}$ (0, \infty) $\end{document}. In this paper we show that the large time behavior of the solution \begin{document}$ u $\end{document} is described by a rescaled Barenblatt solution.

In this paper, we study equations involving fully nonlinear nonlocal operators

\begin{document}$ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $\end{document}

We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by \begin{document}$ (\Phi_{1}, \Phi_{2}) $\end{document}-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.

In this work we study asymptotic properties of global solutions for an initial value problem of a second order fractional differential equation with structural damping. The evolution equation considered includes plate equation problems. We show asymptotic profiles depending on the exponents of the Laplace operators involved in the equation and optimality of the decay rates for the associated energy and the \begin{document}$ L^{2} $\end{document} norm of solutions.

In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [15], in which the magnetic vector field is bounded in critical Sobolev spaces.

We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel Hörmander-type condition which only allows for several noises in the magnetic direction.

This paper concerns with the motion of the interface for a damped hyperbolic Allen–Cahn equation, in a bounded domain of \begin{document}$ \mathbb{R}^n $\end{document}, for \begin{document}$ n = 2 $\end{document} or \begin{document}$ n = 3 $\end{document}. In particular, we focus the attention on radially symmetric solutions and extend to the hyperbolic framework some well-known results of the classic parabolic case: it is shown that, under appropriate assumptions on the initial data and on the boundary conditions, the interface moves by mean curvature as the diffusion coefficient goes to \begin{document}$ 0 $\end{document}.

We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate \begin{document}$ (1+t)^{-3 / 4} $\end{document} in \begin{document}$ L^{2} $\end{document} norm and the micro-rotational velocity tends to the equilibrium state with the faster rate \begin{document}$ (1+t)^{-5 / 4} $\end{document} in \begin{document}$ L^{2} $\end{document} norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.

In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents \begin{document}$ u $\end{document}, while no dispersal is assumed in the other equation for the infective humans \begin{document}$ v $\end{document}. The underlying spatial region \begin{document}$ [g(t), h(t)] $\end{document} (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [33], such a model was considered where the growth rate of \begin{document}$ u $\end{document} due to the contribution from \begin{document}$ v $\end{document} is given by \begin{document}$ cv $\end{document} for some positive constant \begin{document}$ c $\end{document}. Here this term is replaced by a nonlocal reaction function of \begin{document}$ v $\end{document} in the form \begin{document}$ c\int_{g(t)}^{h(t)}K(x-y)v(t,y)dy $\end{document} with a suitable kernel function \begin{document}$ K $\end{document}, to represent the nonlocal effect of \begin{document}$ v $\end{document} on the growth of \begin{document}$ u $\end{document}. We first show that this problem has a unique solution for all \begin{document}$ t>0 $\end{document}, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term \begin{document}$ cv $\end{document}. We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [33] where the term \begin{document}$ cv $\end{document} was used; in particular, small nonlocal dispersal rate of \begin{document}$ u $\end{document} alone no longer guarantees successful spreading of the disease as in the model of [33].

We study the emergent dynamics of the Cucker-Smale (C-S for brevity) ensemble under adaptive couplings. For the adaptive couplings, we basically consider two types of couplings: Hebbian vs. anti-Hebbian. When the Hebbian rule is employed, we present sufficient conditions leading to the mono-cluster flocking using the Lyapunov functional approach. On the other hand, for the anti-Hebbian rule, the possibility of mono-cluster flocking mainly depends on the integrability of the communication weight function and the regularity of the adaptive law at the origin. In addition, we perform numerical experiments and compare them with our analytic results.

We prove the existence of positive radial solutions to the problem

\begin{document}$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $\end{document}

where\begin{document}$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $\end{document} \begin{document}$ f:(0,\infty )\rightarrow \mathbb{R} $\end{document} is \begin{document}$ p $\end{document}-superlinear at \begin{document}$ \infty $\end{document} with possible singularity at \begin{document}$ 0, $\end{document} and \begin{document}$ \lambda $\end{document} is a small positive parameter. A nonexistence result is also established when \begin{document}$ f $\end{document} has semipositone structure at \begin{document}$ 0. $\end{document}

We study the existence of positive solutions of the following degenerate coercive quasilinear elliptic equations:

\begin{document}$ \!-\text{div}(g^2(u)\nabla u)\!+\!\lambda g(u)g'(u)|\nabla u|^2\!+\!V(x)u\! = \!\beta u^{(1-\gamma)(2^*-1)}\!+\!f(u), \ x\!\in\! \mathbb{R}^N, $\end{document}

where \begin{document}$ g(t)\in C(\mathbb{R}, \mathbb{R}) $\end{document}, \begin{document}$ V(x)\in C(\mathbb{R}^N, \mathbb{R}) $\end{document}, \begin{document}$ \lambda, \gamma \in \mathbb{R} $\end{document}, \begin{document}$ \beta\geq 0 $\end{document} and \begin{document}$ 2^* = \frac{2N}{N-2} $\end{document}, \begin{document}$ N\geq3 $\end{document}. The novelty of this paper is that \begin{document}$ g(t) $\end{document} is non-increasing with respect to \begin{document}$ |t| $\end{document} and \begin{document}$ \lim_{|t|\rightarrow +\infty} g(t) = 0. $\end{document} The main results of this paper can be regarded as a supplement to the case that \begin{document}$ g(t) $\end{document} is non-decreasing with respect to \begin{document}$ |t| $\end{document} which has been extensively studied recently.

We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.