Asymptotic Analysis - Volume 43, issue 1-2

Authors: Sadek, A.I.

Article Type: Research Article

Abstract: The paper gives sufficient conditions for the asymptotic stability of the zero solution of system (1.1). By constructing Lyapunov functionals, we obtained the results in a simple form.

Keywords: stability, Lyapunov functional, non‐autonomous differential equations of third order with delay

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 1-7, 2005

Authors: Andreu, F. | Mazón, J.M. | Moll, J.S.

Article Type: Research Article

Abstract: We study existence, uniqueness and the asymptotic behaviour of the entropy solutions for the Total Variation Flow with nonlinear boundary conditions. To prove the existence we use the nonlinear semigroup theory and for the uniqueness we apply Kruzhkov's method of doubling variables both in space and in time. We show that when the initial data are in L2 , the entropy solutions are strong solutions. Respect to the asymptotic behaviour, we show that entropy solutions stabilize as t→∞ by converging to a constant function.

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 9-46, 2005

Authors: Agarwal, Ravi P. | O'Regan, Donal | Wong, Patricia J.Y.

Article Type: Research Article

Abstract: We consider the following system of integral equations ui (t)=μ∫0 1 gi (t,s)f(s,u1 (s),u2 (s),…,un (s)) ds, t∈[0,1], 1≤i≤n, where μ>0, the function f may take negative values and f(·,u1 ,u2 ,…,un ) may be singular at uj =0, j∈{1,2,…,n}. Our aim is to establish criteria such that the above system has a constant‐sign solution. To illustrate the generality of the results obtained, application to a well known boundary value problem is included. We also extend the above problem to that on the half‐line [0,∞) ui (t)=μ∫0 ∞ gi (t,s)f(s,u1 (s),u2 (s),…,un (s)) ds, t∈[0,∞), 1≤i≤n.

Keywords: semipositone, system of singular integral equations, constant‐sign solutions, boundary value problems

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 47-74, 2005

Authors: Wei, Juncheng | Winter, Matthias | Yeung, Wai‐Kong

Article Type: Research Article

Abstract: Of concern is the following singularly perturbed semilinear elliptic problem \[\left\{\begin{array}{@{}l}{\varepsilon }^{2}\Delta u-u+u^{p}=0\quad \hbox{in}\varOmega ,\\u>0\quad \hbox{in}\varOmega \quad \hbox{and}\quad \dfrac{ \curpartial u}{\curpartial \nu}=0\quad \hbox{on}\curpartial \varOmega ,\end{array}\right.\] where Ω is a bounded domain in RN with smooth boundary \[$\curpartial \varOmega $ , ε>0 is a small constant and \[$1<p<(\frac{N+2}{N-2})_{+}$ . Associated with the above problem is the energy functional Jε defined by \[J_{\varepsilon }[u]:=\int_{\varOmega }\bigl(\frac{\varepsilon ^{2}}{2}{|\nabla u|}^{2}+\frac{1}{2}u^{2}-F(u)\bigr)\,\mathrm{d}x\] for u∈H1 (Ω), where F(u)=∫0 u sp  ds. Ni and Takagi [Comm. Pure Appl. Math. 44 (1991), 819–851; Duke Math. J. 70 (1993), 247–281] proved …that for a single boundary spike solution uε , the following asymptotic expansion holds: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[w]-c_{1}\varepsilon H(P_{\varepsilon })+\mathrm{o}(\varepsilon )\bigr],\end{equation} where I[w] is the energy of the ground state, c1 >0 is a generic constant, Pε is the unique local maximum point of uε and H(Pε ) is the boundary mean curvature function at \[$P_{\varepsilon }\in \curpartial \varOmega $ . Later, Wei and Winter [C. R. Acad. Sci. Paris, Ser. I 337 (2003), 37–42; Calc. Var. Partial Differential Equations 20 (2004), 403–430] improved the result and obtained a higher‐order expansion of Jε [uε ]: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[\omega]-c_{1}\varepsilon H(P_{\varepsilon })+\varepsilon ^{2}\left[c_{2}\left(H(P_{\varepsilon })\right)^{2}+c_{3}R(P_{\varepsilon })\right]+\mathrm{o}(\varepsilon ^{2})\bigr],\end{equation} where c2 and c3 >0 are generic constants and R(Pε ) is the scalar curvature at Pε . However, if N=2, the scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature. In this paper, we consider this case and assume that 2≤p<+∞. Without loss of generality, we may assume that the boundary near \[$P\in \curpartial \varOmega $ is represented by the graph {x2 =ρP (x1 )}. Then we have the following higher order expansion of Jε [uε ]: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[w]-c_{1}\varepsilon H({P_{\varepsilon }})+c_{2}\varepsilon ^{2}\left(H({P_{\varepsilon }})\right)^{2}+\varepsilon ^{3}\left[P\left(H({P_{\varepsilon }})\right)+c_{3}S({P_{\varepsilon }})\right]+\mathrm{o}(\varepsilon ^{3})\bigr],\end{equation} where H(Pε )=ρ″Pε (0) is the curvature, P(t)=A1 t+A2 t2 +A3 t3 is a polynomial, c1 , c2 , c3 and A1 , A2 , A3 are generic real constants and S(Pε )=ρPε (4) (0). In particular c3 <0. Some applications of this expansion are given. Show more

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 75-110, 2005

Authors: Malusa, Annalisa

Article Type: Research Article

Abstract: One of the recent advances in the investigation on nonlinear elliptic equations with a measure as forcing term is a paper by G. Dal Maso, F. Murat, L. Orsina and A. Prignet in which it has been introduced the notion of renormalized solution to the problem \[-\mathrm{div}(a(x,\nabla u))=\mu\quad\mbox{in }\varOmega,\qquad u=0\quad\mbox{on }\curpartial\varOmega.\] Here Ω is a bounded open set of RN , N≥2, the operator is modelled on the p‐Laplacian, and μ is a Radon measure with bounded variation in Ω. The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. …We provide a new proof of this stability result, based on the properties of the truncations of the renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one. Show more

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 111-129, 2005

Authors: Stoyanov, Luchezar

Article Type: Research Article

Abstract: A comprehensive version of the Ruelle–Perron–Frobenius theorem is considered with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator.

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 131-150, 2005

Authors: Stolk, Christiaan C.

Article Type: Research Article

Abstract: We construct parametrices for initial value problems of the form \[(*)\quad (\curpartial _{z}-\mathrm {i}A(z,x,D_{x})+B(z,x,D_{x}))u=0,\quad z>z_{0},\quad u(z_{0},\cdot)=u_{0},\] where $(z,x)\in \mathbb{R} \times \mathbb{R} ^{n}$ , A(z,x,Dx ) is a family of order 1 pseudodifferential operators with homogeneous real principal symbol a(z,x,ξ), and B(z,x,Dx ) is a family of order γ>0 pseudodifferential operators with non‐negative homogeneous real principal symbol b(z,x,ξ). The parametrix is a family of pseudodifferential operators when A=0, and a Fourier integral operator with real phase function if A≠0. A priori this leads to symbols of type $(\rho,\delta)=(1-\frac{\gamma}{2},\frac{\gamma}{2})$ , which limits our construction to γ<1, and …leads to operators with a complicated symbol calculus in the case γ=1. With an additional assumption on B we obtain symbols of type $(\rho,\delta)=(1-\frac{\gamma}{L},\frac{\gamma}{L})$ , for some L≥2. The assumption implies in particular that the first L−1 derivatives of b vanish where b=0. Parametrices for (*) are constructed for the case when 2γ<L. Show more

Keywords: Fourier integral operators, pseudodifferential initial value problem

Citation: Asymptotic Analysis, vol. 43, no. 1-2, pp. 151-169, 2005