Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

We present new results regarding the existence of density of the real-valued solution to a 3-dimensional stochastic wave equation. The noise is white in time and with a spatially homogeneous correlation whose spectral measure μ satisfies that , for some . Our approach is based on the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration; we use the tools of Malliavin calculus. Let S₃ be the fundamental solution to the 3-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral and upper bounds for in terms of powers of t. These estimates are crucial in the analysis of the Malliavin variance, which can be done by a comparison procedure with respect to smooth approximations of the distribution-valued function S₃(t) obtained by convolution with an approximation of the identity.     

Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: −i∑αkk_∂u+aβu+M(x)u=Ru(x,u). Here M(x) is a general potential and R(x,u) is a self-coupling which is super-quadratic in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct linking levels of the variational functional ΦM such that the minimax value cM based on the linking structure of ΦM satisfies , where is the least energy of the “limit equation”. Thus we can show the c(C)-condition holds true for all and consequently obtain one least energy solution to the Dirac equation.     

Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d?5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)_‖2<‖ΔW_‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.     

Solving the nonnegative solution for a (shifted) nonsymmetric algebraic Riccati equation in the critical case

We are interested in computing the nonnegative solution of a nonsymmetric algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters c and α. There have been some iterative methods presented by Lu in [13] and Bai et al. in [2] to solve the minimal positive solution for or . While the equation has a unique nonnegative solution when c=1 and α=0, all the methods presented by Lu and Bai cannot be used to find the nonnegative solution. To cope with this problem, a shifted technique is used in this paper to transform the original Riccati equation into a new one so that all the methods can be effectively employed to solve the nonnegative solution. Numerical experiments are given to illustrate the results.     

Well-posedness and persistence properties for the Novikov equation

Recently, Novikov found a new integrable equation (we call it the Novikov equation in this paper), which has nonlinear terms that are cubic, rather than quadratic, and admits peaked soliton solutions (peakons). Firstly, we prove that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces (which generalize the Sobolev spaces Hs) with the critical index . Then, well-posedness in Hs with , is also established by applying Kato's semigroup theory. Finally, we present two results on the persistence properties of the strong solution for the Novikov equation.     

On a nonlinear elliptic eigenvalue problem

Let N(λ) be the number of the solutions of the equation: , where Ω is a bounded domain in with smooth boundary. Under suitable conditions on f, we proved that N(λ)→+∞ as λ→+∞.     

Existence, regularity and local uniqueness of the solutions to the Maxwell-Landau-Lifshitz system in three dimensions

We study a model of ferromagnetism governed by Maxwell's equations coupled with the non-linear Landau-Lifshitz equation of micromagnetism. We are interested in the cases of space-periodic solutions for 3D domains. Obtaining the regularity of the solution m in space and of the solutions E, H in space we state the existence theorem. Finally, we prove the local uniqueness of the solutions.     

On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ₁p|u|u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ₁ is. Scattering will occur either when the perturbation is defocussing and or when the mass of the solution is small enough and . When the quintic term is focusing, we show the blow up for certain solutions.     

Convergence of discrete schemes for the Perona-Malik equation

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation ut=(?^′x(ux)), , when the initial datum is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when has a whole interval where is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing ?(⋅) with the truncated function min(?(⋅),1), and it turns out that u solves a free boundary problem. The free boundary consists of the points dividing the region where |ux|>1 from the region where |ux|?1. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals , i.e., the standing solution of the convexified problem.     

Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3?n?6, we derive lower bound for positive solution of

     

Large energy entire solutions for the Yamabe equation

We consider the Yamabe equation in Rn, n?3. Let k?1 and . For all large k we find a solution of the form , where , for n?4, for n=3 and o(1)→0 uniformly as k→+∞.     

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 .