Related papers: Well-Posedness of Initial-Boundary Value Problems …
We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical…
We consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well…
We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…
The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the…
We consider the Cauchy problem of the nonlinear Schr\"odinger equation with the modulated dispersion and power type nonlinearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K. Chouk and M,…
We address the well-posedness of subelliptic Fokker-Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible polynomial…
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are…
We consider the initial value problem (IVP) associated to the cubic nonlinear Schr\"odinger equation with third-order dispersion \begin{equation*} \partial_{t}u+i\alpha \partial^{2}_{x}u- \partial^{3}_{x}u+i\beta|u|^{2}u = 0, \quad x,t \in…
We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full…
In this work we prove a global well-posedness result for a tridimensional rescaled Boussinesq system, with positive full viscosity and diffusivity parameters in the framework of critical Fourier-Besov spaces. This rescaled approach permits…
The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the…
In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local…
In this paper, we study the well-posedness and boundary stabilization of the initial-boundary value problem for the complex Ginzburg-Landau (CGL) equation on a finite interval. First, we establish a local well-posedness theory for the open…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and…
We consider the focusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation \[ iu_t + \Delta u = -|x|^{-b}|u|^{\alpha}u \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $\alpha=(4-2b)/(n-2)$. We prove the global well-posedness and…
We study the reaction-fractional-diffusion equation $u_t+(-\Delta)^{s} u=f(u)$ with ignition and monostable reactions $f$, and $s\in(0,1)$. We obtain the first optimal bounds on the propagation of front-like solutions in the cases where no…
In the first part of this work we study the local well-posedness of dispersive equations in the weighted spaces $H^s(\mathbb{R})\cap L^2(|x|^{2b}dx)$. We then apply our results for several dispersive models such as the Hirota-Satsuma…
The present work provides well-posedness and exponential decay results for the Blackstock-Crighton-Kuznetsov equation arising in the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. First, we treat the…
We consider the defocusing inhomogeneous nonlinear Schr\"{o}dinger equation $i\partial_tu+\Delta u= |x|^{-b}|u|^{\alpha}u,$ where $0<b<1$ and $0<\alpha<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with…
We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<\alpha \leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array}…