We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. 2009 , which considered the spec...

For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms L and Ḣ, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up solutions, and prove a result on the concentration of the L norm at the origin. Two disp...

In this paper we consider the asymptotic behavior in time of solutions to the heat equation with certain nonlinear Neumann boundary conditions, ∂u/∂n = F (u). Here F is a function which grows superlinearly. In general solutions exist for only a finite time before “blowing up”, or they decay to zero as time approaches infinity. In both one and two space-dimensions we establish some conditions on...

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,    ut = uxx − λ(u + 1) log(u + 1) (x, t) ∈ R+ × (0, T ), −ux(0, t) = (u + 1) log(u + 1)(0, t) t ∈ (0, T ), u(x, 0) = u0(x) x ∈ R+, with p, q, λ > 0. We describe in terms of p, q and λ when the solution is global in time and when it blows up in f...

In this paper, we study the blowup rate estimate for a system of semilinear parabolic equations. The blowup rate depends on whether the two components of the solution of this system blow up simultaneously or not.