Solution profiles beyond quenching for a radially symmetric multi-dimensional parabolic problem

Let T≤∞T≤∞, R>0R>0, Ω=(0,R)×(0,T)Ω=(0,R)×(0,T), χ{S}χ{S} be the characteristic function of the set SS. This article studies the following parabolic problem: rN−1ut−(rN−1ur)r=rN−1f(u)χ({u0f(0)>0, f′>0f′>0, f″≥0f″≥0, and limu→c−f(u)=∞limu→c−f(u)=∞. It is shown that under some additional conditions on ff, the problem has a weak solution, and all weak solutions of the problem tend to a unique steady-state (nonclassical) solution U(r)U(r) as tt tends to infinity. Furthermore, increasing the length RR increases the interval where U(r)≡cU(r)≡c by the same amount.