Related papers: Continuous and discrete one dimensional autonomous…

We study the asymptotic behavior of compressible isentropic flow when the initial mass is finite and the friction varies with time, which is modeled by the compressible Euler equation with time-dependent damping. In this paper, we obtain…

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We…

We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0,…

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant,…

For $\gamma>0$, we are interested in blow up solutions $u\in C^+(B)$ of the fractional problem in the unit ball $B$ \begin{equation}\label{2nov} \left\{\begin{array} {rcll} \Delta^{\frac{\alpha}{2}} u &=& u^\gamma&\ \text{in }B\\ u &=& 0&\…

We study the following equation \begin{equation*} \frac{\partial u(t,\,x)}{\partial t}= \Delta u(t,\,x)+b(u(t,\,x))+\sigma \dot{W}(t,\,x),\quad t>0, \end{equation*} where $\sigma$ is a positive constant and $\dot{W}$ is a space-time white…

We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012},…

Osgood functions in the source term are used to produce results for non-existence of local solutions into the framework of non-Gaussian diffusion equations. The critical exponent for non-existence of local solutions is found to depend on…

Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents…

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general…

In this paper we identify, for small $t$ and a fixed $T>0,$ the order $\alpha>0$ in the abstract fractional differential equation $$\partial^\alpha u(t)=Au(t),$$ where the time-fractional derivative $\partial^\alpha$ is understood in the…

A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order $\delta \in (1,2)$ is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on…

It is well known that the Leibniz rule for the integer derivative of order one does not hold for the fractional derivative case when the fractional order lies between 0 and 1. Thus it poses a great difficulty in the calculation of…

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up…

Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $\gamma$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the…

We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…

When addressing ordinary differential equations in infinite dimensional Banach spaces, an interesting question that arises concerns the existence (or non existence) of blowing up solutions in finite time. In this manuscript we discuss this…

Let $u(s,t)$ be a continuous potential density of a symmetric L\'evy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $\lambda \wedge T_{0}$, where $\lambda$ is an independent exponential…

For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at…