Related papers: A new gap in the critical exponent for semi-linear…
We prove new time decay estimates for the linearized $\beta$-plane equation near the Couette flow on the plane that combine inviscid damping and the dispersion of Rossby waves. Specifically, we show that the profiles of the velocity field…
We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - \Delta u = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous…
We propose that nonequilibrium quantum criticality in open systems at both zero and finite temperatures can be described by a master equation of the Lindblad form. We derive this equation from a system coupling microscopic to a heat bath.…
A quantitatively reliable theoretical description of the dynamics of fluctuations in non-equilibrium is indispensable in the experimental search for the QCD critical point by means of ultra-relativistic heavy-ion collisions. In this work we…
Motivated by recent experimental progress to read out quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing…
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d<p<1+4/(d-2)$. More…
In this paper, we study the finite-time blow up of solutions to the following semilinear wave equation with time-dependent damping \[ \partial_t^2u-\Delta u+\frac{\mu}{1+t}\partial_tu=|u|^p \] in $\mathbb{R}_{+}\times\mathbb{R}^n$. More…
We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive…
Lifespan estimates for semilinear damped wave equations of the form $\partial_t^2u-\Delta u+\partial_tu=|u|^p$ in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the…
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…
In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem…
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-\Delta u + \partial_tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\…
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 /…
We investigate critical transport and the dynamical exponent through the spreading of an initially localized particle in quadratic Hamiltonians with short-range hopping in lattice dimension $d_l$. We consider critical dynamics that emerges…
We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step…
This paper deals with a class of semilinear wave equation with nonlinear damping term $|u_{t}|^{m-2}u_t $ and nonlinear source term $g(x)|u|^{p-2}u$ on the manifolds with conical singularities. Firstly, we prove the local existence and…
In this note two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case, when the damping and the mass terms…
In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…
We study a time-fractional semilinear heat equation $$\partial^{\alpha}_t u -\Delta u = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^{\alpha}$ denotes the…
We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale…