The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations

This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front.     

On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.     

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.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

A fourth order parabolic equation with nonlinear principal part

In this paper, we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions. On the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions.     

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.     

Pullback attractors for closed cocycles

In this article, using a result of Pata and Zelik (2007) [45], we derive a general result on the existence of pullback attractors for closed cocycles acting on a Banach space, where the strong continuity is replaced by a much weaker requirement that the cocycle be a closed map. As application, we prove the existence of the pullback attractor of a cocycle associated with the z-weak solutions of a non-autonomous two-dimensional primitive equations of the ocean.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Indirect stability of the wave equation with a dynamic boundary control

In this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in of the energy. In a second step, under appropriated conditions on the boundary, called the multiplier control conditions, we establish a polynomial decay in of the energy. Later, we show in a particular case that such a polynomial decay is available even if the previous conditions are not satisfied. For this aim, we consider our system on the unit square of the plane. Using a method based on a Fourier analysis and a specific analysis of the obtained 1‐d problems combining Ingham's inequality and an interpolation method, we establish a polynomial decay in of the energy for sufficiently smooth initial data. Finally, in the case of the unit disk, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained decay is optimal in the domain of the operator.     

Bifurcation for a free boundary problem modeling the growth of multi-layer tumors

In this paper we study bifurcations for a free boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. An important feature of this problem is that the surface tension effect of the free boundary is taken into account. By reducing this problem into an abstract bifurcation equation in a Banach space, overcoming some technical difficulties and finally using the Crandall–Rabinowitz bifurcation theorem, we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ?, we prove the existence of the uniform global attractor A^?. Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in Chepyzhov and Vishik (2007) [22].     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

Orbital stability of peakons for the Degasperis-Procesi equation with strong dispersion

In this paper, we study the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line. Using the method in [Z. Lin, Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2009) 125-146], we prove that the shapes of these peakons are stable under small perturbations. Some previous results are extended.     

Spherically symmetric internal layers for activator-inhibitor systems—I. Existence by a Lyapunov-Schmidt reduction

Reaction-diffusion systems of activator-inhibitor type are studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. Under the condition that the activator diffuses slowly, reacts rapidly and the inhibitor diffuses rapidly, reacts moderately, we show that the system admits a family of spherically symmetric internal transition layer equilibria. The method of proof consists of rigorous asymptotic expansions and a Lyapunov-Schmidt reduction.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Stability of transonic shocks in supersonic flow past a wedge

In this paper we study the stability of transonic shocks in steady supersonic flow past a wedge. We take the potential flow equation as the mathematical model to describe the compressible flow. It is known that in generic case such a problem admits two possible location of shock, connecting the flow ahead it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine-Hugoniot conditions and entropy condition. In this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled.     

Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential

The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.     

Existence and uniqueness analysis of a detached shock problem for the potential flow

We study a problem for two-dimensional steady potential and isentropic Euler equations in a bounded domain, where an artificial detached shock interacts with a wedge. Using the stream function, we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted Hölder spaces. The proof is based on various Hölder and Schauder estimates for second-order elliptic equations and fixed point theorems. Moreover, we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional.