Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

We consider a nonlinear reaction-diffusion equation on the whole space Rd. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L² only. Then we adapt the short trajectory method to establish the existence of the global attractor and, if d?3, we find an upper bound of its Kolmogorov's ε-entropy.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H²(M)?L^2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.     

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation −ε²Δu+u=up in Ω⊆RN, where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N?3 and for k∈{1,…,N−2}. We impose Neumann boundary conditions, assuming 1<p<(N−k+2)/(N−k−2) and ε→⁰⁺. This result settles in full generality a phenomenon previously considered only in the particular case N=3 and k=1.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lε$L_{\epsilon }$ (where ε is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line  , with in addition a simple eigenvalue in 0. Moreover, for ε<0$\epsilon <0$, there is a pair of imaginary eigenvalues which meet in 0 when ε=0$\epsilon =0$ and which disappear in the essential spectrum for ε>0$\epsilon >0$. For ε>0$\epsilon >0$ small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T₀,∞)$(T_0,\infty )$ possibly except a discrete set of isolated points.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

Pullback attractors for a class of non-autonomous nonclassical diffusion equations

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation u_t−εΔu_t−Δu+f(u)=g(t), ε∈[0,1], in a bounded domain in R^N. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0.     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

Total boundedness and bornologies

A set A in a metric space is called totally bounded if for each ε>0 the set can be ε-approximated by a finite set. If this can be done, the finite set can always be chosen inside A. If the finite sets are replaced by an arbitrary approximating family of sets, this coincidence may disappear. We present necessary and sufficient conditions for the coincidence assuming only that the family is closed under finite unions. A complete analysis of the structure of totally bounded sets is presented in the case that the approximating family is a bornology, where approximation in either sense amounts to approximation in Hausdorff distance by members of the bornology.     

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low (n?10) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n?10.Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction-diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.     

Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system

An elliptic system is considered in a smooth bounded domain, subject to Dirichlet boundary conditions of three different types. Based on the construction of certain upper and sub-solutions, we obtain some conditions on the parameters ai,bi,ci (i=1,2) and the exponents m,n,p,q to ensure the existence of positive solutions. Furthermore, uniqueness and boundary behavior of positive solutions is also discussed.     

Random modulation of solitons for the stochastic Korteweg–de Vries equation

We study the asymptotic behavior of the solution of a Korteweg–de Vries equation with an additive noise whose amplitude ε   tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg–de Vries equation. We prove that up to times of the order of 1/ε²$1/{\epsilon }^2$, the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as ε tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation.     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ?. The latter can be of hyperbolic type if the Cattaneo-Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the ?ojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.