We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.
We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the formwhere m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blow-up set. The proof is based on an application of the intersection comparison method with an explicit ‘separable’ solution which has the same blow-up time as u. The method has a natural generalisation to the case of more general nonlinearities in the equation. A description of different fine structures of blow-up patterns in the semilinear case m = 1 and in the quasilinear one m > 1 is also presented. A numerical study of the semilinear equation is also made using an adaptive collocation method. This is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour.
Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition