Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.     

Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: The constraint equations

This paper deals with the construction of initial data for the coupled Einstein-Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order.The common Lichnerowicz-York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Application of the G class of functions in the parabolic class

In this paper,the application of the G class of functions in the parabolic class is considered. The regularity of the solution for the first boundary value problem of parabolic equation in divergence form is proved.     

On the uniqueness and structure of solutions to a coupled elliptic system

In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.     

Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand–Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.     

Viscosity solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace equation with nonlinear source

In the present paper we consider the Dirichlet problem in a convex domain for the multidimensional p-Laplace equation with nonlinear source. We prove the existence of the unique continuous viscosity solution under quite general assumptions on the structure of the source. Received: 18 January 2006     

On the existence and regularity of solutions of a quasilinear mixed equation of Leray-Lions type

Our aim in this article is to study the existence and regularity of solutions of a quasilinear elliptic-hyperbolic equation. This equation appears in the design of blade cascade profiles. This leads to an inverse problem for designing two-dimensional channels with prescribed velocity distributions along channel walls. The governing equation is obtained by transformation of the physical domain to the plane defined by the streamlines and the potential lines of fluid. We establish an existence and regularity result of solutions for a more general framework which includes our physical problem as a specific example.     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group

We propose a direct method to control the first-order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T-direction and then second-order weak differentiability in the horizontal directions.     

On a nonclassical fractional boundary-value problem for the Laplace operator

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.     

The Neumann problem for the Hénon equation,trace inequalities and Steklov eigenvalues

We consider the Neumann problem for the Hénon equation. We obtain existence results and we analyze the symmetry properties of the ground state solutions. We prove that some symmetry and variational properties can be expressed in terms of eigenvalues of a Steklov problem. Applications are also given to extremals of certain trace inequalities.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

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.     

Estimates on fractional power dissipative equations in function spaces

We study the fractional power dissipative equations, whose fundamental semigroup is given by e^{−t(−Δ)α} with α>0. By using an argument of duality and interpolation, we extend space-time estimates of the fractional power dissipative equations in Lebesgue spaces to the Hardy spaces and the modulation spaces. These results are substantial extensions of some known results. As applications, we study both local and global well-posedness of the Cauchy problem for the nonlinear fractional power dissipative equation u_t+(−Δ)^αu=|u|^mu for initial data in the modulation spaces.     

On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator

In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic-hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation.     

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R³. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs₀×Hs₀−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.     

Averaging principle and hyperbolic evolution equations

An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry’s results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

The Third Problem for the Laplace Equation with a Boundary Condition from L p

The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in L_p and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C¹ boundary the explicit solution of the problem is calculated.