For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates ...

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...

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...

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution...

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.

We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer k, we construct a set of codimension 2k in the space of initial data giving rise to solutions that blow-up according to the given profile.

We prove the finite time blow-up for solutions of the 3D incompressible Euler equations, which happens along the fluid particle trajectories starting from a set of points. This set is specified by the relation between the deformation tensor and the Hessian of pressure both coupled with the vorticity directions, associated with the initial data. As a corollary of this result we prove the finite ...

We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation ut uxx − a x, t f u , 0 < x < 1, t ∈ 0, T , with boundary conditions ux 0, t 0, ux 1, t b t g u 1, t , blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-u...

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampère equation with power source, with the following basic 2D model 0.1 (0.1) ut = −|Du|+ |u|u in R × R+, where in two-dimensions |D2u| = uxxuyy − (uxy) and p > 1 is a fixed exponent. For a class of “dominated concave” and compactly supported radial initial data u0(x) ≥ 0, th...