Instability of solitary wave solutions for the generalized BO-ZK equation

In this paper we study the generalized BO-ZK equation in two space dimensions

ut+upux+αHuxx+εuxyy=0.     

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.     

Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

Extending previous results of Oh-Zumbrun and Johnson-Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions. Key to our analysis is a nonlinear cancellation estimate observed by Johnson and Zumbrun, along with a detailed understanding of the Whitham averaged system. The latter motivates a careful analysis of the Bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave, allowing us to derive the sharpened low-frequency estimates needed to close the nonlinear iteration arguments.     

Spatial problem of Darboux type for one model equation of third order

For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ₀ such that for > ₀ the problem under consideration is uniquelly solvable in the Frechet space. In the case where the coefficients are constants, Bochner's method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.     

A PDE PROBLEM ARISING FROM CALCULATION OF THE MODEL FOR CONTINUOUS CASTING OF STEEL

This paper deals with the relationship between a certain three-dimensional elliptic problem and a two-dimensional parabolic problem, which arises from numerical calculation in the process of the continuous casting of steel.     

Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1/(ln(t+2))^2−δ,∀δ>0, when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1/(ln(ln(t+e²)))^2−δ,∀δ>0.     

On strong solutions of the multi-layer quasi-geostrophic equations of the ocean

In this article, we study the multi-layer quasi-geostrophic equations of the ocean. The existence of strong solutions is proved. We also prove the existence of a maximal attractor in L²(Ω)$L^2\left(\Omega \right)$ and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our estimates rely on a new formulation that we introduce for the multi-layer quasi-geostrophic equation of the ocean, which replaces the nonhomogeneous boundary conditions (and the nonlocal constraint) on the stream-function by a simple homogeneous Dirichlet boundary condition. This work improves the results given in [C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier–Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl. 4 (2) (1994) 465–489].     

On the correct formulation of a multidimensional problem for strictly hyperbolic equations of higher order

A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higherorder strictly hyperbolic systems in the conic domain with special orientation.     

The equations of potential flow of compressible viscous fluid at low Reynolds number

The existence, uniqueness, and stabilization of solutions are investigated for two approximate models of viscous compressible fluid.     

Global well-posedness of the Cauchy problem of the fifth-order shallow water equation

By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation is globally well-posed in the Sobolev space Hs(R) provided .     

The existence and decay of solutions of a damped Kirchhoff-Carrier equation in Banach spaces

This paper is concerned with the study of the existence and decay of solutions of the following initial value problem:

(∗)     

Stabilization of flows through porous media

In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope of b in c : The equilibrium is unstable if b′(c) < 0, whereas b′(c) > 0 implies exponential stability. Zhaoyong Feng: He is grateful to the DFG for financial support through the Graduiertenkolleg 615 “Interaction of Modeling, Computation Methods and Software Concepts for Scientific-Technological Problems”.     

Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δ_ut$-\mathrm{\Delta }{u}_t$ and the linear dissipative term _ut$u_t$ by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.     

Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order

In this paper, the asymptotic behavior of the solution to the initial–boundary value problem for a nonlinear evolution equation of fourth order

equation(1)

_utt−_a1uxx−_a2uxxt−_a3uxxtt=φ_(ux)x$u_{tt}-a_1{u}_{xx}-a_2{u}_{xxt}-a_3{u}_{xxtt}=\phi {\left(u_x\right)}_x$

is studied. The sufficient conditions for blow-up of the solutions to the initial–boundary value problems for Eq. (1) are given.     

Remarks on the method of modulus of continuity and the modified dissipative Porous Media Equation

We employ Besov space techniques and the method of modulus of continuity to obtain the global well-posedness of the modified Porous Media Equation.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

On the singular set of the parabolic obstacle problem

This paper is devoted to regularity results and geometric properties of the singular set of the parabolic obstacle problem with variable right-hand side. Making use of a monotonicity formula for singular points, we prove the uniqueness of blow-up limits at singular points. These results apply to parabolic obstacle problem with variable coefficients.     

Approximate radial symmetry for solutions of a class of boundary value problems in ring-shaped domains

This paper deals with some classical boundary value problems defined in the ring-shaped region $\Omega \backslash\overline{B(R_0)} \subset \Bbb{R}^N$, where B(R ₀) is the N-ball of radius R ₀ centered at the origin and contained in . The authors investigate how the geometry of is affected by some overdertermined data. In particular they show that must be nearly spherical if the overdetermined data are nearly constants.     

Boundedness of global solutions of a porous medium equation with a localized source

In this paper a localized porous medium equation _ut=u^r(Δu+af(u(_x0,t)))$u_t=u^r(\mathrm{\Delta }u+\mathit{af}(u(x_0,t)))$ is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.