Blow-up, beyond quenching, and multi-dimensional quenching due to local and nonlocal sources

In this study, the following three topics are investigated: a blow-up criterion for a degenerate parabolic problem, beyond quenching, and multi-dimensional quenching due to local and nonlocal nonlinear sources.;The exact position b* ∈ (0,1/2) of the concentrated source such that the solution it of a degenerate parabolic problem never blows up for b ∈ (0, b*] ∪ [b*, 1), and u always blows up in a finite time for b ∈ (b*,1 -- b*) is found This also implies that u does not blow up in infinite time.;For beyond quenching, existence of a weak solution is given. It is proved that as t tends to infinity, all weak solutions (not necessarily obtained via regularization) tend to a unique steady-state (nonclassical) solution. A computational method to find the solution profiles beyond quenching is devised. For illustration, some examples are given.;A multi-dimensional quenching problem due to a nonlinear source having local and nonlocal features on the boundary ∂B of a ball B in a domain D is studied. This problem has a unique nonnegative continuous solution u, which is a strictly increasing function of t in D, and which attains its absolute maximum on ∂B. If tq is finite, then u quenches everywhere on ∂B only It is shown that there exists a unique critical size |Dˆ|* such that u exists globally for |Dˆ| ≤ |Dˆ|* and quenches in a finite time for |Dˆ| > |Dˆ|*. This implies that quenching does not occur in infinite time. A formula for |Dˆ|* is given. For radially symmetric solutions in spatially spherical domains Dˆ, the formula for |Dˆ|* is used to deduce the critical radii of the nonlinear source for the solution to exist globally or quench in a finite time.