Decay estimates of solutions to the IBq equation

Obstructions to Uniform Estimates for Solutions to the ∂-equation

We show that if, for every bounded ∂-closed (0, 1)-form f , a pseudoconvex domain Ω admits a solution to ∂u = f that is continuous up to the boundary and has uniform estimates in terms of ‖f‖∞, then each p ∈ ∂Ω must necessarily admit a peak function in the class A(Ω) := O(Ω) ∩ C(Ω). We use this fact to examine some geometrical obstructions to uniform estimates for the ∂-equation.

Decay Estimates and Smoothness for Solutions of the Dispersion Managed Non-linear Schrödinger Equation

We study the decay and smoothness of solutions of the dispersion managed non-linear Schrödinger equation in the case of zero residual dispersion. Using new x-space versions of bilinear Strichartz estimates, we show that the solutions are not only smooth, but also fast decaying.

Decay estimates of solutions to wave equations in conical sets

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.

Interior estimates for solutions of Abreu’s equation

Here (u) denotes the inverse of the Hessian matrix uij = ∂u ∂xi∂xj . We call this PDE Abreu’s equation since the expression S(u) appears in [1], in the study of the differential geometry of toric varieties. In this paper we will be primarily interested in the case when A is a constant. Solutions to Equation 1 then correspond to certain Kahler metrics of constant scalar curvature, as we will rec...

Decay Estimates for the Wave Equation with Potential*

Decay Mode Solutions to (2 + 1)-Dimensional Burgers Equation, Cylindrical Burgers Equation and Spherical Burgers Equation

Three (2 + 1)-dimensional equations—Burgers equation, cylindrical Burgers equation and spherical Burgers equation, have been reduced to the classical Burgers equation by different transformation of variables respectively. The decay mode solutions of the Burgers equation have been obtained by using the extended G G ′       -expansion method, substituting the solutions obtained into the cor...