N. Mizoguchi et E. Yanagida, Critical exponents for the decay rate of solutions in a semilinear parabolic equation, ARCH R MECH, 145(4), 1998, pp. 331-342
This paper is concerned with the Cauchy problem
u(t) = u(xx) - \u\p(-1)u in R x (0, infinity),
u(x, 0) = u(0)(x) in R.
A solution It is said to decay fast if t(1/(p-1))u --> 0 as t --> infinity
uniformly in R, and is said to decay slowly otherwise. For each nonnegative
integer k, let Lambda(k) be the set of uniformly bounded functions on R wh
ich change sign k times, and let p(k) > 1 be defined by p(k) = 1 + 2/(k + 1
). It is shown that any nontrivial bounded solution with u(0) is an element
of Lambda(k) decays slowly if 1 < p < p(k), whereas there exists a nontriv
ial fast decaying solution with u(0) is an element of Lambda(k) if p greate
r than or equal to p(k).