Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions

Single Point Blow–up Solutions to the Heat Equation with Nonlinear Boundary Conditions

We study finite blow-up solutions of the heat equation with nonlinear boundary conditions. We provide a sufficient condition for the single point blow-up at the origin and a precise spacial singularity of the blow-up profile. Mathematics subject classification (2010): 35K20, 35B44.

Minimal Class of Blow-up Approximate Self-similar Solutions of the Heat Equation

On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions

We obtain some sufficient conditions under which solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions tend to zero or blow up in a finite time. We also give the asymptotic behavior of solutions which tend to zero as t→ ∞. Finally, we obtain the asymptotic behavior near the blow-up time of certain blow-up solutions and describe their blow-up set.

Blow-up of solutions to a nonlinear dispersive rod equation

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Mathematics Subject Classification(2000): 30C70, 37L05, 35Q58, 58E35

The Blow-up Rate Estimates for a System of Heat Equations with Nonlinear Boundary Conditions

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.