We exhibit C∞ type II blow up solutions to the focusing energy critical wave equation in dimension N = 4. These solutions admit near blow up time a decomposiiton u(t, x) = 1 λ N−2 2 (t) (Q+ ε(t))( x λ(t) ) with ‖ε(t), ∂tε(t)‖Ḣ1×L2 ≪ 1 where Q is the extremizing profile of the Sobolev embedding Ḣ → L∗ , and a blow up speed λ(t) = (T − t)e− √ |log(T−t)|(1+o(1)) as t → T.

This paper deals with the blow-up properties of positive solutions to a system of two heat equations ut = ∆u, vt = ∆v in BR× (0, T ) with Neumann boundary conditions ∂u ∂η = e vp , ∂v ∂η = e uq on ∂BR × (0, T ), where p, q > 1, BR is a ball in Rn, η is the outward normal. The upper bounds of blow-up rate estimates were obtained. It is also proved that the blow-up occurs only on the boundary.

Abstract For the Schrödinger flow from R × R to the 2-sphere S, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of th...

We study the blow-up behaviour of two reaction-diiusion problems with a quasilinear degenerate diiusion and a superlinear reaction. We show that in each case the blow-up is self-similar, in contrast to the linear diiusion limit of each in which the diiusion is only approximately self-similar. We then investigate the limit of the self-similar behaviour and describe the transition from a stable m...

We study attractive particle systems with stationary product measures. We utilize the property of attractivity and its link to coupling to build a growth process that samples from the stationary measure of the zero-range process, on fixed and finite lattices, with computation times scaling linearly with the number of particles N . The zero-range process with constant jump-rates and a single def...

The blow-up rate estimate for the solution to a semilinear parabolic equation ut = ∆u+V (x)|u|p−1u in Ω×(0, T ) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x, 0) = Mφ(x) as M goes to infinity, which have been found in [5], are improved under some reason...

This paper concerns a superlinear parabolic equation, degenerate in the time derivative. It is shown that the solution may blow up in finite time. Moreover it is proved that for a large class of initial data blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic. Various estimates are obtained which determine the asymptotic behaviour near the blow-up. The ...

Abstract This paper deals with the blow-up properties of the solution to the degenerate and singular parabolic system with nonlocal sources, absorptions and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution to exist globally or blow up in finite time are obtained. Furthermore, under c...