Inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Remarks on the energy decay for nonlinear wave equations with nonlinear localized dissipative terms

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation with a nonlinear localized dissipation. To overcome a difficulty related to derivative-loss mechanism we employ a ‘loan’ method.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.     

Existence of global decaying solutions to the exterior problem for the Klein–Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity

We prove the existence of global decaying solutions to the exterior problem for the Klein–Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity. To derive the required estimates of solutions we employ a ‘loan’ method.     

Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t→+∞.We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t→+∞, with the same rate of the solution of the limit problem of parabolic type.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Global regularity for the initial value problem of a 2-D Kazhikhov–Smagulov type model

We prove global-in-time existence of regular solution to the initial value problem of a 2-D Kazhikhov–Smagulov type model for incompressible nonhomogeneous fluids with mass diffusion.     

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

Multiple solutions of nonlinear elliptic systems

We proved a multiplicity result for a nonlinear elliptic system in R^N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.     

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 .     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities.     

Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.     

Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients

This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach, and the density of the continuous functions in the domain of the Γ-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three.     

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.     

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.     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.