American Institute of Mathematical Sciences

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions \begin{document}$ 0\leq u(x, t)\leq 1 $\end{document} of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [5]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries.

Finally, as a complement of [5], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker–Planck equations for probability measures \begin{document}$ (\mu_t)_{t \geq 0} $\end{document} on the path space \begin{document}$ {\scr {C}}: = C([-r_0, 0];\mathbb R^d), $\end{document} is analyzed:

\begin{document}$ \partial_t \mu(t) = L_{t, \mu_t}^*\mu_t, \ \ t\ge 0, $\end{document}

where \begin{document}$ \mu(t) $\end{document} is the image of \begin{document}$ \mu_t $\end{document} under the projection \begin{document}$ {\scr {C}}\ni\xi\mapsto \xi(0)\in\mathbb R^d $\end{document}, and

\begin{document}$ \begin{align*} L_{t, \mu}(\xi)&: = \frac 1 2\sum\limits_{i, j = 1}^d a_{ij}(t, \xi, \mu)\frac{\partial^2} {\partial_{\xi(0)_i} \partial_{\xi(0)_j }} \\\; &\quad +\sum\limits_{i = 1}^d b_i(t, \xi, \mu)\frac{\partial}{\partial_{\xi(0)_i}}, \ \ t\ge 0, \xi\in {\scr {C}}, \mu\in \scr P^{\scr {C}}. \end{align*} $\end{document}

Under reasonable conditions on the coefficients \begin{document}$ a_{ij} $\end{document} and \begin{document}$ b_i $\end{document}, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.

Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.

In this paper we consider the non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. Since the strong nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density, there are a few results about global existence of classical solution to this model. In the paper, we obtain a global classical solution to the equations with large initial data and vacuum. Moreover, we get the uniqueness of the solution to this system without vacuum. The analysis is based on the assumption \begin{document}$ \kappa(\theta) = O(1+\theta^q) $\end{document} where \begin{document}$ q\geq0 $\end{document} and delicate energy estimates.

We give a complete classification of the traveling waves of the following quadratic Szegő equation :

\begin{document}$i{\partial _t}u = 2J\Pi \left( {{{\left| u \right|}^2}} \right) + \bar J{u^2},\;\;\;\;u\left( {0, \cdot } \right) = {u_0},$ \end{document}

and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.

Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a sufficiently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

We consider the discrete nonlinear Schrödinger equations on a one dimensional lattice of mesh \begin{document}$ h $\end{document}, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.

We consider a non-local traffic model involving a convolution product. Unlike other studies, the considered kernel is discontinuous on \begin{document}$ \mathbb R $\end{document}. We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, \begin{document}$ {{\bf{C^{}}}}([0,T], {{\bf{L^2}}}( \mathbb R)) $\end{document}, and smooth, \begin{document}$ {\bf{W}}^{2,2N}([0,T]\times \mathbb R) $\end{document}, solutions for the non-local traffic model.

The inviscid multi-layer quasi-geostrophic equations are considered over an arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions are imposed to accommodate the free fluctuations of the top and layer interfaces. Using the barotropic and baroclinic modes in the vertical direction, the elliptic system governing the streamfunctions and the potential vorticity is decomposed into a sequence of scalar elliptic boundary value problems, where the regularity theories from the two-dimensional case can be applied. With the initial potential vorticity being essentially bounded, the multi-layer quasi-equations are then shown to be globally well-posed, and the initial and boundary conditions are satisfied in the classical sense.

In this paper, we investigate Strichartz estimates for discrete linear Schrödinger and discrete linear Klein-Gordon equations on a lattice \begin{document}$ h\mathbb{Z}^d $\end{document} with \begin{document}$ h>0 $\end{document}, where \begin{document}$ h $\end{document} is the distance between two adjacent lattice points. As for fixed \begin{document}$ h>0 $\end{document}, Strichartz estimates for discrete Schrödinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis [21]. Our main result shows that such inequalities hold uniformly in \begin{document}$ h\in(0,1] $\end{document} with additional fractional derivatives on the right hand side. As an application, we obtain local well-posedness of a discrete nonlinear Schrödinger equation with a priori bounds independent of \begin{document}$ h $\end{document}. The theorems and the harmonic analysis tools developed in this paper would be useful in the study of the continuum limit \begin{document}$ h\to 0 $\end{document} for discrete models, including our forthcoming work [7] where strong convergence for a discrete nonlinear Schrödinger equation is addressed.

We study the weakly coupled critical elliptic system

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u = \mu_{1}|u|^{2^{*}-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u & \text{in }\Omega,\\ -\Delta v = \mu_{2}|v|^{2^{*}-2}v+\lambda\beta |u|^{\alpha}|v|^{\beta-2}v & \text{in }\Omega,\\ u = v = 0 & \text{on }\partial\Omega, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded smooth domain in \begin{document}$ \mathbb{R}^{N} $\end{document}, \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ 2^{*}: = \frac{2N}{N-2} $\end{document} is the critical Sobolev exponent, \begin{document}$ \mu_{1},\mu_{2}>0 $\end{document}, \begin{document}$ \alpha, \beta>1 $\end{document}, \begin{document}$ \alpha+\beta = 2^{*} $\end{document} and \begin{document}$ \lambda\in\mathbb{R} $\end{document}.

We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on \begin{document}$ \Omega $\end{document}, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as \begin{document}$ \lambda\to -\infty $\end{document}.

We also obtain existence of infinitely many solutions to this system in \begin{document}$ \Omega = \mathbb{R}^N $\end{document}.

We study a nonlinear coupled fluid–structure system modelling the blood flow through arteries. The fluid is described by the incompressible Navier–Stokes equations in a 2D rectangular domain where the upper part depends on a structure satisfying a damped Euler–Bernoulli beam equation. The system is driven by time-periodic source terms on the inflow and outflow boundaries. We prove the existence of time-periodic strong solutions for this problem under smallness assumptions for the source terms.

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.

This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.

This paper concerns the following nonlinear Choquard equation:

\begin{document}$ \begin{equation} -\varepsilon^{2}\Delta w+V(x)w = \varepsilon^{-\theta}W(x)(I_\theta*(W|w|^p))|w|^{p-2}w,\quad x\in\mathbb{R}^N, ~~~~~~~~~~~~(*)\end{equation} $\end{document}

where \begin{document}$ \varepsilon>0,\ N>2,\ I_\theta $\end{document} is the Riesz potential with order \begin{document}$ \theta\in(0,N),\ p\in\big[2,\frac{N+\theta}{N-2}\big),\ \min V>0 $\end{document} and \begin{document}$ \inf W>0 $\end{document}. Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for \begin{document}$ (\ast) $\end{document}. The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set.

In this paper we study bifurcation solutions of a free boundary problem modeling the growth of necrotic multilayered tumors. The tumor model consists of two elliptic differential equations for nutrient concentration and pressure, with discontinuous terms and two free boundaries. The novelty is that different types of boundary conditions are imposed on two free boundaries. By bifurcation analysis, we show that there exist infinitely many branches of non-flat stationary solutions bifurcating from the unique flat stationary solution.

In this paper, we study Cauchy problem of the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Taking advantage of a coupling structure of the equations and using the damping effect of the growth term \begin{document}$ g(n) $\end{document}, we obtain the necessary estimates of solution \begin{document}$ (n,c,u) $\end{document} without the diffusion term \begin{document}$ \Delta n $\end{document}. These uniform estimates enable us to establish the global-in-time existence of almost weak solutions for the system.

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem

\begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\left( {{u}^{\prime }}/\sqrt{1-{{u}^{\prime }}^{2}} \right)}^{\prime }}=\lambda f(u),\ \text{in }\left( -L,L \right), \\ & u(-L)=u(L)=0, \\ \end{align} \\\end{array} \right. $\end{document}

where \begin{document}$ \lambda >0 $\end{document} is a bifurcation parameter, \begin{document}$ L>0 $\end{document} is an evolution parameter, \begin{document}$ f\in C[0, \infty )\cap C^{2}(0, \infty ) $\end{document} and there exists \begin{document}$ \beta >0 $\end{document} such that \begin{document}$ \left( \beta -z\right) f(z)>0 $\end{document} for \begin{document}$ z\neq \beta $\end{document}. In particular, we find that the bifurcation curve \begin{document}$ S_{L} $\end{document} is monotone increasing for all \begin{document}$ L>0 $\end{document} when \begin{document}$ f(u)/u $\end{document} is of Logistic type, and is either \begin{document}$ \subset $\end{document}-shaped or S-shaped for large \begin{document}$ L>0 $\end{document} when \begin{document}$ f(u)/u $\end{document} is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.

We study the regularity of the isometric embedding \begin{document}$ X: $\end{document} \begin{document}$ (B(O, r), g) $\end{document} \begin{document}$ \rightarrow $\end{document} \begin{document}$ (\mathbb{R}^3, g_{can}) $\end{document} of a 2-ball with nonnegatively curved \begin{document}$ C^4 $\end{document} metric into \begin{document}$ \mathbb{R}^3 $\end{document}. Under the assumption that \begin{document}$ X $\end{document} can be expressed in the graph form, we show \begin{document}$ X \in C^{2,1} $\end{document} near \begin{document}$ P $\end{document}, which is optimal by Iaia's example.

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document}, \begin{document}$ d = 5,6 $\end{document}, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document}, \begin{document}$ d = 5, 6 $\end{document}, in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on \begin{document}$ \mathbb T^2 $\end{document}. We prove the local well-posedness for given data in \begin{document}$ H^s( \mathbb T^2) $\end{document} whenever \begin{document}$ s> 5/3 $\end{document}. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the \begin{document}$ \mathbb R^2 $\end{document} and \begin{document}$ \mathbb R\times \mathbb T $\end{document} settings.

We are concerned with the vortex sheet solutions for the inviscid two-phase flow in two dimensions. In particular, the nonlinear stability and existence of compressible vortex sheet solutions under small perturbations are established by using a modification of the Nash-Moser iteration technique, where a priori estimates for the linearized equations have a loss of derivatives. Due to the jump of the normal derivatives of densities of liquid and gas, we obtain the normal estimates in the anisotropic Sobolev space, instead of the usual Sobolev space. New ideas and techniques are developed to close the energy estimates and derive the tame estimates for the two-phase flows.

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax–Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

When \begin{document}$ P$\end{document} is the fractional Laplacian \begin{document}$ (-\Delta )^a$\end{document}, \begin{document}$ 0<a<1$\end{document}, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set \begin{document}$ \Omega \subset{\Bbb R}^n$\end{document}: \begin{document}$ r^+Pu(x, t)+\partial_tu(x, t) = f(x, t)$\end{document} on \begin{document}$ \Omega \times \, ]0, T[\, $\end{document}, \begin{document}$ u(x, t) = 0$\end{document} for \begin{document}$ x\notin\Omega$\end{document}, \begin{document}$ u(x, 0) = 0$\end{document}, is known to be solvable in relatively low-order Sobolev or Hölder spaces. We now show that in contrast with differential operator cases, the regularity of \begin{document}$ u$\end{document} in \begin{document}$ x$\end{document} at \begin{document}$ \partial\Omega$\end{document} when \begin{document}$ f$\end{document} is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schrödinger Dirichlet problem \begin{document}$ r^+Pv(x)+Vv(x) = g(x)$\end{document} on \begin{document}$ \Omega$\end{document}, \begin{document}$ \text{supp } v\subset\overline\Omega$\end{document}, with \begin{document}$ V(x)\in C^\infty$\end{document}. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a \begin{document}$ \text{dist}(x, \partial\Omega )^a$\end{document} singularity.