Related papers: One-dimensional aggregation equation after blow up…

In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear…

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…

The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions after blow-up. Here we study an approach based on random…

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of…

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a selfconsistant…

We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…

Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified…

A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions,…

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…

We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by…

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…

In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…

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…

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. Optimal conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of…

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…

This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics…

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy…

We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of…

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective…

We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…