相关论文: Noisy integrate-and-fire equation: continuation af…

One of the most important challenges in mathematical neuroscience is to properly illustrate the stochastic nature of neurons. Among different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most…

The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network. It is used when slow conductance receptors are activated…

In this paper, we consider the initial-boundary value problems with several fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary condition) for the multi-component system of semi-linear classical damped wave equations…

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and…

We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-\Delta u =|v|^{p}+t^\sigma w_1(x),\,\, v_t-\Delta v…

We propose another integrate-and-fire model as a single neuron model. We study a globally coupled noisy integrate-and-fire model with inhibitory interaction using the Fokker-Planck equation and the Langevin equation, and find a reentrant…

In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning…

This paper deals with the blow-up properties of positive solutions to a system of two heat equations.

A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial…

We consider the Yangs-Mills equations in 4+1 dimensions. This is the energy critical case and we show that it admits a family of solutions which blow up in finite time. They are obtained by the spherically symmetric ansatz in the SO(4)…

Nonlinear dispersive partial differential equations such as the nonlinear Schr\"odinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing…

This work provides a description of the asymptotic behavior of sequences of solutions to an elliptic equation with a nonlocal exponential nonlinearity of Choquard type. The equation under consideration is a nonlocal analog of the classical…

In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the…

In this paper, we study weakly coupled systems for semilinear wave equations with distinct nonlinear memory terms in general forms, and the corresponding single semilinear equation with general nonlinear memory terms. Thanks to Banach's…

We study the possibility of non-simultaneous blow-up for positive solutions of a coupled system of two semilinear equations, $u_t = J*u-u+ u^\alpha v^p$, $v_t =\Delta v^+u^qv^\beta$, $p, q, \alpha, \beta>0$ with homogeneous Dirichlet…

We study a nonlinear system coupling the Darcy-Forchheimer-Brinkman equations with a convection-diffusion-reaction equation, arising in reactive transport through porous media. The model features a nonlinear viscosity coupling, Forchheimer…

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for…

In this work, we study the finite time blow-up phenomenon of three types of semilinear wave systems with multiple speeds, posed on asymptotically Euclidean manifolds. We establish the upper bound estimates for the lifespan of solutions when…

In this paper, we consider the long time behavior for the solution of a class of variable coefficient wave equation with nonlinear damping and logarithmic source. The existence and uniqueness of local weak solution can be obtained by using…

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis,…