Asymptotic Analysis - Volume 71, issue 4

Authors: Nishihara, Kenji

Article Type: Research Article

Abstract: We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term utt −Δu+b(t)ut +|u|ρ−1 u=0,  (t,x)∈R+ ×RN , (u,ut )(0,x)=(u0 ,u1 )(x),  x∈RN   (*). When b(t)=b0 (t+1)−β with −1<β<1 and b0 >0, we want to seek for the asymptotic profile as t→∞ of the solution u to (*) in the supercritical case ρ>ρF (N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation ϕt −1/b(t) Δϕ=0,  (t,x)∈R+ ×RN .  (**) When N=1, the …decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of (**) with source term −1/b(t) (utt +|u|ρ−1 u). Thus, we will show θ0 GB (t,x)  (θ0 : suitable constant) to be an asymptotic profile of u, where GB (t,x) is the fundamental solution of (**). Show more

Keywords: wave equation, time-dependent damping, supercritical exponent, asymptotic profile

DOI: 10.3233/ASY-2010-1018

Citation: Asymptotic Analysis, vol. 71, no. 4, pp. 185-205, 2011

Authors: Cardoso, Fernando | Cuevas, Claudio | Vodev, Georgi

Article Type: Research Article

Abstract: We prove optimal dispersive estimates at high frequency for the Schrödinger group for a class of real-valued potentials V(x)=O(〈x〉−δ ), δ>n−1, and V∈Ck (Rn ), k>kn , where n≥4 and (n−3)/2≤kn <n/2. We also give a sufficient condition in terms of L1 →L∞ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.

Keywords: dispersive estimates, high frequency

DOI: 10.3233/ASY-2010-1020

Citation: Asymptotic Analysis, vol. 71, no. 4, pp. 207-225, 2011

Authors: Surnachev, Mikhail

Article Type: Research Article

Abstract: We study a special class of solutions to semilinear elliptic equations in conical domains. We prove that under some initial assumption on the solution it is asymptotically close to a linear combination of special solutions of the corresponding homogeneous equation. We also give the existence result for solutions with asymptotics of this type.

Keywords: asymptotic, cone, weighted spaces, Emden–Fowler, semilinear, absorption

DOI: 10.3233/ASY-2010-1021

Citation: Asymptotic Analysis, vol. 71, no. 4, pp. 227-243, 2011

Article Type: Other

Citation: Asymptotic Analysis, vol. 71, no. 4, pp. 245-245, 2011