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In this paper we obtain the precise description of the asymptotic behavior of the solution u of the fractional diffusion equation ∂tu + (-Δ) θ/ 2 u = 0 in RN × (0,∞) with the initial data φ ϵ LK := L1(RN, (1 + |x|)K dx), where 0 < θ < 2 and K ≥ 0. This enables us to obtain the asymptotic behavior of the hot spots of the solution u. Furthermore, we develop the arguments in [K. Ishige and T. Kawakami, Math. Ann., 353 (2012), pp. 161-192] and [K. Ishige, T. Kawakami, and K. Kobayashi, J. Evol. Equ., 14 (2014), pp. 749-777] and establish a method to obtain the asymptotic expansions of the solutions to inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations ut = Δu + f(u) on ℝN. Here f ∈ C¹, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x→∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasi-convergent in the sense that all their limit profiles as t→∞are steady states. If N = 1, then all positive bounded, localized solutions are quasi-convergent. We show that such a general conclusion is not valid if N ≥ 3, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose ω-limit set is infinite and contains only one equilibrium. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of classes of global and blow-up solutions of a semilinear parabolic equation of the "limit" Cahn-Hilliard type ut = -Δ(Δu + |u|p-1 u) in RN x R+, p > 1, with bounded integrable initial data. We show that in some {p,N}-parameter ranges it admits a countable set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exponent p = p0 = 1 + 2/N, where the first simplest profile is shown to be stable. Unlike the blow-up case, we show that, for p = p0, the set of global decaying source-type similarity solutions is continuous and determine the stable mass-branch. We prove that there exists a countable spectrum of critical exponents {p = pl = 1 + 2/N+l, l = 0, 1, 2, ...} creating bifurcation branches, which play a key role in general description of solutions globally decaying as t → ∞. [ABSTRACT FROM AUTHOR]