Abstract
We investigate a quasi-linear parabolicproblem with nonlocal absorption, for which the comparison principle is not always available. Thesufficient conditions are established via energy method to guaranteesolution to blow up or not, and the long time behavior is alsocharacterized for global solutions.
Highlights
In this paper, a quasilinear parabolic problem with a nonlocal term are considered: ut= up(uxx + u − u(t)), x ∈ (0, a), t > 0, ux(0, t) = ux(a, t) = 0, t > 0, (1)u(x, 0) = u0(x), x ∈ [0, a], where p > 1, a 0, u(t) =a −0 udx 1 a a 0 udx.The initial datum u0(x) ∈C2+β([0, a]) (0 < β < 1) satisfies u0x(0) = u0x(a) = 0
We always assume that u0(x) > 0 on [0, a]
When p > 2, analogously as above, we find that d dt a u2−pdx = (p − 2)E(t) ≤ (p − 2)E(0) < 0, 0 < t < Tmax
Summary
We say that a nonnegative solution u of problem (1) blows up at a time T ≤ ∞ if it satisfies lim sup max u(x, t) = ∞. Property of solution to the following problem of the form ut = up Assume u is a positive solution of problem (1) and Tmax is its maximal existence time.
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