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In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of…
It is shown that self-similar blow-up for a fourth-order reaction-diffusion equation is incomplete in the sense that, in general, there exists a self-similar extension of solutions after blow-up. Other types of complete blow-up of non…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schr\"odinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for…
We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ \begin{align}\label{inls} i \partial_t u +\Delta u +V(x)|u|^{\frac{4-2b}{N}}u = 0, \end{align} where $V(x) = k(x)|x|^{-b}$, with $b>0$. Under suitable…
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…
We establish the optimal regularity for solutions to the multiple membranes problem, and perform a blow-up analysis at points on the free boundary with the highest multiplicity. This leads to a complete classification of blow-up profiles in…
It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The…
We establish several bounds for solutions to elliptic/parabolic cross-diffusion systems of $m$ equations ($m\ge2$) on 2d/3d domains $\Og$. We settle the existence and global existence problems in these cases and also provide new…
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…
Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type $$ u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren}…
In this paper, we consider the mass-critical focusing nonlinear Schr\"odinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data…
Finding four-dimensional de Sitter spacetime solutions in string theory has been a vexing quest ever since the discovery of the accelerating expansion of the universe. Building on a recent analysis of bubble-nucleation in the decay of…
In this report we discuss and propose a correction to a convergence and stability issue occurring in the work of Da et al.[2015], in which they proposed a numerical model to simulate soap bubbles.
Bubbles are point-like regular solutions of the higher-dimensional Kaluza-Klein equations that appear as naked singularities in four dimensions. We analyze all such possible solutions in 5D Kaluza-Klein theory that are static and…
We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer $k$, we construct a set of codimension $2k$ in…
In this paper, we consider existence of positive solutions for the Schr\"odinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} \Delta_pu+\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\…
I construct solutions to Einstein's equations in 6 dimensions with bulk cosmological constant and intersecting 4-branes. Solutions exist for a continuous range of 4-brane tension, with long distance gravity localized to a 3+1 dimensional…
We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is…
The prospect of self-consistent propagation of more than two pulses contemporaneously through multi-resonant media raises open questions: whether soliton solutions exist, and whether a useful generalization of two-level pulse Area can be…