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In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or $0$, the solution exists globally in time or blows up in finite…
We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, \[ u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T). \] Building on the work of Ghoul, Liu, and Masmoudi…
This paper investigates the Cauchy problem for the semilinear damped wave equation $u_{tt}+\mathcal{L}_{a,b}u+u_t=|u|^p$ with the mixed local-nonlocal operator $\mathcal{L}_{a,b}:=-a\Delta+b(-\Delta)^{\sigma}$, where $a,b\in\mathbb{R}_+$…
We consider the focusing nonlinear Schr\"odinger equations $i\partial_t u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$.…
A degenerate Keller-Segel system with diffusion exponent $2n/(n+2)<m<2-\frac{2}{n}$ in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on $L^{2n/(n+2)}$ norm of the…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
In this paper, we consider the following equation: \[ i\frac{\partial u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in…
We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the Smoluchowski coagulation equations with diffusion, valid in any dimension. If the collision propensities \alpha(n,m) of mass n and mass m particles grow more slowly than…
The motion of rarefied gases for uniform shear flow at the kinetic level is governed by the spatially homogeneous Boltzmann equation with a deformation force. In the paper we study the corresponding Cauchy problem with initial data of…
This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schr\"odinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for…
This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed…
We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0$$ where $\delta=\delta(x)$ is the delta function supported at the origin.…
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that…
We study the focusing NLS equation in $\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical) regime, with $H^1$ data at the mass-energy threshold $ \mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where $Q$ is the ground…
We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions…
At time zero, there are $N$ identical point particles in the line (1D) which are characterized by their positions and velocities. Both values are given randomly and independently from each other, with arbitrary probability densities. Each…
We extend the thermodynamics of quantum BTZ black holes by treating the quantum backreaction strength parameter $\nu$ as a thermodynamic variable. We find various novel features. The critical point appears at $\nu_c=1$ and a first order…
The L^1-critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena. In this paper we study this critical mass phenomenon in the…
We consider the modified Korteweg-de Vries equation. Given a self-similar solution, and a subcritical perturbation of any size, we prove that there exists a unique solution to the equation which behaves at blow-up time as the self-similar…
This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+…