Abstract
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.
Highlights
We are concerned with the global existence and blow-up phenomenon for a quasilinear parabolic equation with nonlinear inner absorption term ut = [(|∇u|p + 1) u,i],i − f (u), (x, t) ∈ Ω × (0, t∗), (1)subjected to the nonlinear Neumann boundary and initial conditions (|∇u|p + 1) ∂u ∂] = g (u),(x, t) ∈ ∂Ω × (0, t∗), (2)u (x, 0) = u0 (0) ≥ 0, x ∈ Ω, (3)
For the problems with nonlinear Neumann boundary conditions, Payne et al [20] studied the semilinear heat equation with inner absorption term ut = Δu − f (u), (x, t) ∈ Ω × (0, t∗). They established conditions on nonlinearity to guarantee that the solution u(x, t) exists for all time t > 0 or blows up at some finite time t∗
An upper bound for t∗ was derived
Summary
They derived conditions on the data which guarantee the blow-up or the global existence of the solution. They established conditions on nonlinearity to guarantee that the solution u(x, t) exists for all time t > 0 or blows up at some finite time t∗. They showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data; upper and lower bounds for the blow-up time were derived when blow-up occurs; see [21].
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