Asymptotics for a nonlinear integral equation with a generalized heat kernel

Abstract

This paper is concerned with a nonlinear integral equation) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P ) behaving like a multiple of the integral kernel G as t → ∞.

Global solutions for a nonlinear integral equation with a generalized heat kernel

“…In addition, it follows from a similar argument as in the proof of [16, Lemma 2.1] (see also [18]) that:…”

Large time behavior of ODE type solutions to nonlinear diffusion equations

“…and established a method to obtain the higher order expansions of the solution behaving like a suitable multiple of the Gauss kernel as t → ∞. (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.…”

“…(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…”

“…Furthermore, using nice properties of H(τ, w, ∇w) (see e.g. (4.1) and (4.11)), we apply the arguments in [10] and [11] to obtain the precise description of the large time behavior of w.…”

Asymptotic Expansions of Solutions of Fractional Diffusion Equations

“…This paper is concerned with the Cauchy problem for a nonlinear fractional diffusion equation 1) where N ≥ 1, ∂ t := ∂/∂t, (−∆) θ/2 is the fractional power of the Laplace operator with 0 < θ < 2, p > 1 + θ/N and ϕ ∈ L ∞ (R N ) ∩ L 1 (R N ). We say that a continuous function u in R N × (0, ∞) is a solution of (1.1) if u satisfies u(x, t) = Problem (1.1) appears in the study of nonlinear problems with anomalous diffusion and the Laplace equation with a dynamical boundary condition and it has been studied extensively by many mathematicians (see [1], [6], [7], [10], [11], [14], [17], [18], [24] and references therein). Among others, Sugitani [24] showed that, if 1 < p ≤ 1 + θ/N , then problem ( (See [17] and [18].)…”