Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms

We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as , where Ω is a smooth bounded domain in R^N, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.     

Boundary behaviour of explosive solution to quasilinear elliptic problems with nonlinear gradient terms

Based on the Karamata regular variation theory and the method of explosive sub and supersolution, the boundary behaviour of explosive solutions to the quasilinear elliptic equation was obtained, where the singular weight function is non-negative and non-trivial, which may be unbounded on the boundary, the nonlinear term is a Γ-varying function, whose variation at infinity is not regular. The results of this article emphasize the central role played by the gradient term and singular weight function.     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

Second order estimates for large solutions of elliptic equations

This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in R^N, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory.     

Global existence and blowup of solutions for nonlinear coupled wave equations with viscoelastic terms

In this paper, we study a system of nonlinear coupled wave equations with damping, source, and nonlinear strain terms. We obtain several results concerning local existence, global existence, and finite time blow‐up property with positive initial energy by using Galerkin method and energy method, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

Global existence and blowup of solutions for a parabolic equation with a gradient term

The author discusses the semilinear parabolic equation with . Under suitable assumptions on and , he proves that, if with , then the solutions are global, while if with 1$">, then the solutions blow up in a finite time, where is a positive solution of , with .

     

Some elliptic problems with singular natural growth lower order terms

In this paper we deal with a nonlinear elliptic problem, whose model is

     

Asymptotic stability and blowup of solutions for a class of viscoelastic inverse problem with boundary feedback

In this paper, we consider a nonlinear viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function, and initial data, we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity. Furthermore, we show that there are solutions under some conditions on initial data that blow up in finite time. Copyright © 2015 John Wiley & Sons, Ltd.     

Sufficient conditions for the blowup of a solution to the Boussinesq equation subject to a nonlinear Neumann boundary condition

Sufficient blowup conditions are obtained for a solution to the generalized Boussinesq equation subject to a nonlinear Neumann boundary condition.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Boundary blow-up rates of large solutions for elliptic equations with convection terms

By Karamata regular variation theory, a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of large solutions to the semilinear elliptic equations with convection terms

     

The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation

By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem −Δu=k(x)g(u), u>0, x∈Ω, u_|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN; g∈C¹((0,∞),(0,∞)), , for each ξ>0, for some γ>0; and for some α∈(0,1), is nonnegative on Ω, which is also singular near the boundary.     

Positive solutions of second-order boundary value problems with sign-changing nonlinear terms

In this paper the existence results of positive solutions are obtained for second-order boundary value problem

−u″=f(t,u),t∈(0,1),u(0)=u(1)=0,     

Asymptotic integration of some evolution problems with large high-frequency terms

We present results concerning the justification of the averaging method, the construction of a complete justified asymptotics, and the time stability of solutions of semilinear parabolic equations and Navier-Stokes systems with polynomial nonlinearities and large rapidly oscillating terms.     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.     

Bounds of the blowup time in parabolic equations with weighted source under nonhomogeneous Neumann boundary condition

In this paper, we are concerned with the bounds for blowup time of the solution to parabolic equations with weighted nonlinear source subject to nonhomogeneous Neumann boundary condition. We obtain the lower and upper bounds for blowup time of the solution to the problem in . Copyright © 2013 John Wiley & Sons, Ltd.