Asymptotic Analysis - Volume 91, issue 2

Authors: Zhou, Shuangshuang | Zheng, Sining

Article Type: Research Article

Abstract: In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut =up Δu+uq |∇u|2 −ur , subject to null Dirichlet boundary condition. We study the existence of global solutions and the large time behavior for them. The main effort is paid to obtain uniform asymptotic profiles for decay solutions, under various dominations of the nonlinear diffusion or absorption. It is shown that the large time property of the solution u behaves just like (1+(r−1)t)(−1/r−1) if the decay is governed by the nonlinear absorption with 1<r<p+1. Otherwise, the asymptotic profiles would possess the form …of (1+pt)−1/p W, with W solving various homogeneous Dirichlet elliptic equations: (i) −ΔW=W1−p −W if r=p+1<q+2; (ii) −ΔW=W1−p −W+W−1 |∇W|2 if r>p+1=q+2; and (iii) −ΔW=W1−p if r,q+2>p+1. Show more

Keywords: degenerate parabolic equation, gradient term, global solution, large time behavior

DOI: 10.3233/ASY-141254

Citation: Asymptotic Analysis, vol. 91, no. 2, pp. 91-102, 2015

Authors: Cappiello, Marco | Nicola, Fabio

Article Type: Research Article

Abstract: We consider the Euler equations on Td with analytic data and prove lower bounds for the radius of spatial analyticity ε(t) of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of ε(t) on the radius of analyticity of the initial datum.

Keywords: Euler equations, radius of analyticity

DOI: 10.3233/ASY-141260

Citation: Asymptotic Analysis, vol. 91, no. 2, pp. 103-110, 2015

Authors: Hu, Weiwei | Kukavica, Igor | Ziane, Mohammed

Article Type: Research Article

Abstract: We address the global regularity for the 2D Boussinesq equations with positive viscosity and zero diffusivity. We prove that for data (u0 ,ρ0 ) in Hs ×Hs−1 , where 1<s<2, the persistence of regularity holds, i.e., the solution (u(t),ρ(t)) exists and belongs to Hs ×Hs−1 for all positive t. Given the existing results, this provides the persistence of regularity for all s≥0. In addition, we address the Hs ×Hs persistence and establish it for all s>1.

Keywords: Boussinesq equations, global existence, regularity, zero diffusivity, Navier–Stokes equations

DOI: 10.3233/ASY-141261

Citation: Asymptotic Analysis, vol. 91, no. 2, pp. 111-124, 2015

Authors: Ma, Manjun | Ou, Chunhua

Article Type: Research Article

Abstract: We study the existence, uniqueness and asymptotic expansions to perturbed Poisson–Boltzmann equations on an unbounded domain in R2 or R3 . First, a shooting method is applied to prove the existence and uniqueness of the exact solution. For the approximation to the regularly perturbed Poisson–Boltzmann equation, the solution via the classical method fails. We develop a novel approximate solution in terms of generalized asymptotic expansions. For the singularly perturbed problem, we show that a formula of asymptotic expansions with a boundary layer near the left end point provides a valid approximation. All our results are proved rigorously.

Keywords: asymptotic analysis, singular perturbations, Poisson–Boltzmann equation

DOI: 10.3233/ASY-141262

Citation: Asymptotic Analysis, vol. 91, no. 2, pp. 125-146, 2015

Authors: de Bonis, Ida | Giachetti, Daniela

Article Type: Research Article

Abstract: We deal with the existence of nonnegative solutions to parabolic problems which are singular in the u variable whose model is ut −Δp u=f(x,t)(1/uθ +1) in Ω×(0,T), u(x,t)=0 on ∂Ω×(0,T), u(x,0)=u0 (x) in Ω. Here Ω is a bounded open subset of RN , N≥2, 0<T<+∞, θ>0, Δp u=div (|∇u|p−2 ∇u) with p>1. As far as the data are concerned, we assume f(x,t)∈Lr (0,T;Lm (Ω)), with 1/r+N/pm<1, f(x,t)≥0 a.e. in Ω×(0,T) and u0 (x)≥0 a.e. in Ω. We consider also the case where the right-hand side depends on the gradient of the solution. In this last case the …model of the right-hand side is F(x,t,u,∇u)=(f(x,t)+D|∇u|q )/uθ , with θ>0, D>0, 1<q<p and f(x,t) as before. Show more

Keywords: nonlinear parabolic equations, singular lower order terms, nonnegative solutions, existence

DOI: 10.3233/ASY-141257

Citation: Asymptotic Analysis, vol. 91, no. 2, pp. 147-183, 2015