Related papers: Sharp Sobolev estimates for concentration of solut…
In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the…
In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution…
Due to extreme difficulties in numerical simulations of Euler-Maxwell equations, which are caused by the highly complicated structures of the equations, this paper concerns the simplification of Euler-Maxwell system through the…
We study the nonlinear fractional equation $(-\Delta)^s u = f(u)$ in $\mathbb{R}^n$, for all fractions $0<s<1$ and all nonlinearities $f$. For every fractional power $s \in (0,1)$, we obtain sharp energy estimates for bounded global…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also…
In this article, we first prove that for general dispersive equations on Riemannian symmetric spaces of compact type $\mathbb{X}=U/K$, of rank $1$ and $2$, the Sobolev regularity threshold $\alpha >1/2$ for the initial data, is sufficient…
This paper studies a nonlinear diffusion equation with memory: $$u_t=\nabla\cdot \big( D(x)\cdot\int_0^t K(t-s) \nabla\cdot\Phi(u(x,s))ds \big)+f(x,t)$$ Where $K$ is memory Kernel and $D(x)$ is bounded. Under monotonicity and growth…
We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial…
We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity we obtain a fractional Sobolev type time regularity of…
The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean…
We consider diffraction at random point scatterers on general discrete point sets in $\R^\nu$, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence…
We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the…
In the context of some bidimensionnal Navier-Stokes model, we exhibit a family of exact oscillating solutions $\{u_{\epsilon}\}_{\epsilon}$ defined on some strip $[0,T]\times\R^2$ which does not depend on $\epsilon\in]0,1]$. The exact…
In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v…
In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and…
We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the…
We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-\Delta u+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to…
We consider the equation $$-\Delta u+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the…
We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we…
We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…