Asymptotic Expansions of Solutions of Fractional Diffusion Equations

Abstract

In this paper we obtain the precise description of the asymptotic behavior of the solution u ofwe develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equationwhere 0 < θ < 2 and p > 1 + θ/N .

Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms

“…In this section we shall review the results from [15], which dealt with the fractional heat equation. Remark 7.1 In the case when γ = 0 and β = 0 in (7.2), the inequality has already been introduced in [16, Lemma 3.2] (see also [3] for a more general result).…”

“…Furthermore, we can also prove Theorem 4.1 by using (7.3) directly after obtaining (5.5). As seen in [15], sophisticated techniques are needed to prove (7.1) and (7.2). In this paper, however, we need only estimates for the L 2 framework and so there is room for giving simpler proofs.…”

“…In this paper, however, we need only estimates for the L 2 framework and so there is room for giving simpler proofs. It should be emphasized that independently from the expanding theory established in [15], one can actually give alternative proofs based on Lemma 5.1 which is a generalized result in [6], and Lemma 5.3. This is one of our novelties.…”

“…They have additionally obtained the higher order (up to the first order) asymptotic profiles of the solution to (1.1) after identifying the above leading term (the zero-th order expansion) only in the one dimensional case. However, since the solution to (1.1) is very close to the one of the heat equation as t → ∞, one can expect further asymptotic expansions of the solution to (1.1) under more heavy moment condition on the initial data in all space dimensional cases (see e.g., [15]). When we want to observe the difference between them, we have to get the asymptotic expansions of the solution to (1.1) higher than the second order.…”

Large time behavior of ODE type solutions to nonlinear diffusion equations

“…(See also [10] and [12].) However, in the case of m = 1, due to the nonlinear term div H(τ, w, ∇w), we can not apply the arguments in [7]- [10] and [12] to problem (1.10) directly. Indeed, it is difficult to apply their arguments to the Cauchy problem for nonlinear diffusion equations of the form…”