A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality

Critical exponent of the fractional Langevin equation.

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, wh...

Fast Diffusion Equation with Critical Sobolev Exponent in a Ball

Critical Exponent for a Nonlinear Wave Equation with Damping

It is well known that if the damping is missing, the critical exponent for the nonlinear wave equation gu=|u| p is the positive root p0(n) of the equation (n&1) p&(n+1) p&2=0, where n 2 is the space dimension (for p0(1)= , see Sideris [14]). The proof of this fact, known as Strauss' conjecture [17], took more than 20 years of effort, beginning with Glassey doi:10.1006 jdeq.2000.3933, available ...

Multiple Solutions to a Magnetic Nonlinear Choquard Equation

We consider the stationary nonlinear magnetic Choquard equation (−i∇+ A(x))u+ V (x)u = (

Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation as t approaches the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.