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 work deals with a semilinear parabolic systemwhich is coupled both in the equations and in the boundary conditions. The blow-up phenomena of its positive solutions are studied using the scaling method, the Green function and Schauder estimates. The upper and lower bounds of blow-up rates are then obtained. Moreover we show the influences of the reaction terms and the boundary absorption te...

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...

In each dimension N ≥ 3 and for each real number λ ≥ 1, we construct a family of complete rotationally symmetric solutions to Ricci flow on R which encounter a global singularity at a finite time T . The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−(λ+1). Near the origin, blow-ups of such a solution converge uniformly to the Bryant soli...

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...

In this work, we investigate the following Kirchhoff-type equation with variable exponent nonlinearities u_{tt}-M(‖∇u‖²)△u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u. We proved the blow up of solutions in finite time by using modified energy functional method.