The equation ut = ∆u + u with homegeneous Dirichlet boundary conditions has solutions with blow-up if p > 1. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.

Abstract Let D be an open set in $$\mathbb {R}^d$$ ℝ d and $$u:D\to \mathbb {R}$$ u : D → a (non-negative) local minimizer of $$\mathcal F_\Lambda $$ ℱ...

In the present work, we establish an optimal estimate for the electric potential difference between closely adjacent spherical perfect conductors in n dimensional space (n ≥ 2). This result indicates that electric fields blow up as a pair of spherical perfect conductors approach each other, and provides the lower bound with the optimal blowup rate which was recently established by Bao, Li and Y...

This paper deals with a degenerate parabolic equation vt = ∆v + av1 ∥v∥1 α1 subject to homogeneous Dirichlet condition. The local existence of a nonnegative weak solution is given. The blow-up and global existence conditions of nonnegative solutions are obtained. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)∇u)+ f (x,u, |∇u|2, t) under nonlinear boundary condition ∂u/∂n + g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-...