Related papers: Blowing-up Nonautonomous Vector Fields: Infinite D…
For a class of $(N+1)$-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding…
We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in…
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of…
We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from…
In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-\Delta…
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…
Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfu{\ss} conjecture…
In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery…
A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential…
The paper describes a novel method for studying the stability of nonautonomous dynamical systems. This method based on the flow and divergence of the vector field with coupling to the method of Lyapunov functions. The necessary and…
We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the…
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…
We deal with existence and non-existence of non-negative entire solutions that blow-up at infinity for a quasilinear problem depending on a non-negative real parameter. Our main objectives in this paper are to provide far more general…
We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and…
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with…
We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference…
The article is devoted to investigating the initial boundary value problem for the damped wave equation in the scale-invariant case with time-dependent speed of propagation on the exterior domain. By presenting suitable multipliers and…
The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u=0 &\text{on $(0,\infty)\times \Gamma_0$,}\\…
An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed,…
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper…