相关论文: On Singularity formation for the L^2-critical Boso…

We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schr\"odinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly…

For the 3d cubic nonlinear Schr\"odinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time…

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler…

We consider ground state solutions $u \geq 0$ for the $L^2$-critical boson star equation $$ \sqrt{-\Delta} \, u - \big (|x|^{-1} \ast |u|^2 \big) u = -u \quad {in $\R^3$}. $$ We prove analyticity and radial symmetry of $u$. In a previous…

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$…

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up…

We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the…

We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical…

We consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schr\"odinger equations \[ i\partial_t u - \Delta^2 u + \mu \Delta u = - |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq…

Consider the energy critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to…

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require,…

We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $\alpha_\star\leq…

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave…

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…

We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…

In this paper we consider nodal radial solutions $u_\epsilon$ to the problem \[ \begin{cases} -\Delta u=\lambda ue^{u^2+|u|^{1+\epsilon}}&\text{ in }B,\\ u=0&\text{ on }\partial B. \end{cases} \] and we study their asymptotic behaviour as…

In this paper, we consider the following equation: \[ i\frac{\partial u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in…

We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schr\"odinger equation with inverse-square potential, \[ i\partial_t u + \Delta u + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and…