Related papers: Non-homogeneous initial boundary value problems fo…
Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schr\"odinger equation on the finite interval, $0 < x < L$, with $q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$ and…
A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion-reaction problem with a reaction parameter tending…
Mixed boundary value problems for the Navier-Stokes system in a polyhedral domain are considered. Different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are prescribed on the faces of a polyhedron. The authors…
We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate…
In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schr\"odinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like $\left|x\right|^{-b}$ for $0<b<\min \left\{\frac{N}{2},4\right\} $. We…
We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the…
We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy…
In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on…
We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary…
We consider the inhomogeneous biharmonic nonlinear Schr\"odinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +\Delta^2 u -|x|^{-b} |u|^{2\sigma}u = 0,$$ where $\sigma>0$ and $b>0$. We first study the local well-posedness in $\dot…
This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and…
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat…
We study the $\bar\partial$ equation subject to various boundary value conditions on bounded simply connected Lipschitz domains $D\subset\mathbb C$: for the Dirichlet problem with datum in $L^p(bD, \sigma)$, this is simply a restatement of…
This paper is devoted to the analysis of a focusing nonlinear biharmonic Schr\"odinger equation in the presence of an unbounded growing up inhomogeneous term. The first main contribution of this work is the derivation of an inhomogeneous…
The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity…
We prove the local Hadamard well-posedness of the ``good'' Boussinesq equation formulated on the half-line with nonzero Robin boundary conditions. These boundary data involve the Dirichlet and Neumann boundary values as well as the second…
We prove the local well-posedness of the initial boundary value problem for the nonlinear quadratic Schr\"odinger equation under low initial-boundary regularity assumption via the boundary integral operator method introduced by…
It was shown by the second author in (CPAA, 2019) for the biharmonic Schr\"odinger equation and most recently by Himonas and Mantzavinos (Nonlinearity, 2020) for 2D Schr\"odinger equation that Fokas method based formulas are capable of…
In this paper, we consider the initial-boundary value problem to the nonhomogeneous incompressible Navier-Stokes equations. Local strong solutions are established, for any initial data $(\rho_0, u_0)\in (W^{1,\gamma} \cap L^\infty)\times…
We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…