Related papers: Blowup solutions for the nonlinear Schr\"odinger e…
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…
We study the focusing mass-critical nonlinear Schr\"odinger equation, and construct certain solutions which blow up at exactly $m$ points according to the log-log law.
We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$ for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale. This is…
We consider the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS), an $L^2$-critical nonlinear Schr\"odinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality,…
In this paper we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \begin{align}\label{inls} i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the…
Proceeding from the hydrodynamic approach, we construct exact solutions to nonlinear Schr\"odinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They…
This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and…
We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial…
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker spacetimes. For the case of accelerated expansion, we show that blow-up in a finite time…
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…
In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schr\"odinger equation for general initial data in $L^2(\mathbb R^d)$, extending previous knowledge for mass near the ground-state threshold…
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 the nonlinear heat equations with Neumann boundary conditions $$ \begin{cases} u_{t}=\Delta u & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\…
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up…
We investigate the long time dynamics of the nonlinear Schr\"odinger equation (NLS) with combined powers on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$. Different from the previously studied NLS-models with single power on the…
In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new…
We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schr\"odinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole…
The study of blow-up solution of time-fractional heat equations is of significant and wide-ranging interest for its multitude of applications. These types of equations are used to model several real problems in science and engineering. This…
Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of $u_t = u_{xx} + e^{u^m}$ with the…
The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions…