Approaching Critical Decay in a Strongly Degenerate Parabolic Equation

The Cauchy problem in R-n, n >= 1, for the parabolic equation u(t) = u(p) Delta u (star) is considered in the strongly degenerate regime p >= 1. The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies t(1/p)vertical bar vertical bar u(center dot, t)vertical bar vertical bar(L infinity(Rn)) -> infinity as t -> infinity. (0.1) The first result of this study complements this by asserting that given any positive f is an element of C-0([0,infinity)) fulfilling f (t) -> +infinity as t -> infinity one can find a positive nondecreasing function phi is an element of C-0([0,infinity)) such that whenever u(0) is an element of C-0(R-n) is radially symmetric with 0 < u(0) < phi(|center dot|), the corresponding minimal solution u satisfies t(1/p)vertical bar vertical bar u(center dot, t)vertical bar vertical bar L-infinity(R-n)/f (t) -> 0 as t -> infinity. Secondly, (star) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data u0. It is shown that if the connected components of {u(0) > 0} comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to (star) satisfies 0 < lim inf (t -> infinity) {t(1/p vertical bar)vertical bar vertical bar u(center dot, t)vertical bar vertical bar(L infinity(Rn)) <= lim sup (t -> infinity) {t(1/p)vertical bar vertical bar u(center dot, t)vertical bar vertical bar(L infinity(Rn))} < infinity Under a somewhat complementary hypothesis, particularly fulfilled if {u(0) > 0} contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.