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.     

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

     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

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.     

Boundary blow-up solutions to elliptic systems of competitive type

We consider the elliptic system Δu=u^pv^q, Δv=u^rv^s in Ω, where p,s>1, q,r>0, and Ω⊂R^N is a smooth bounded domain, subject to different types of Dirichlet boundary conditions: (F) u=λ, v=μ, (I) u=v=+∞ and (SF) u=+∞, v=μ on ∂Ω, where λ,μ>0. Under several hypotheses on the parameters p,q,r,s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near ∂Ω. Some more general related problems are also studied.     

含梯度项的椭圆方程组的边界爆破解

研究了含梯度项的椭圆方程组的边界爆破解的性质,其中权函数a(x),b(x)为正并且满足一定的条件.利用上下解的方法及比较原则证明了正解的存在性与唯一性,并得到了边界爆破速率的估计.     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Qualitative properties of blow-up solutions to some semilinear elliptic systems in non-convex domains

In this paper we study qualitative properties of boundary blow-up solutions to some semilinear elliptic cooperative systems in bounded non-convex domains. In particular, by a careful adaptation of the celebrated moving plane procedure of Alexandrov–Serrin, we deduce symmetry and monotonicity results for blow-up solutions for this class of systems.     

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.     

Boundary blow-up rate and uniqueness of the large solution for an elliptic cooperative system of logistic type

This paper ascertains the blow-up rates of each of the components of a singular boundary value problem related to a cooperative system of logistic type, in order to establish the uniqueness of the large solution. Astonishingly, the cooperative coupling does not change the blow-up rates of the uncoupled system provided these blow-up rates are sufficiently close, though it changes exactly one of them, keeping invariant the other, when they are bounded away. This seems to be the first time where this change of behavior has been documented in the literature.     

The blow-up rates for systems of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.     

On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions

We give conditions on the nonlinearities of a reaction-diffusion equation with nonlinear boundary conditions that guarantee that any solution starting at bounded initial data is bounded locally around a certain point of the boundary, uniformly for all positive time. The conditions imposed are of a local nature and need only to hold in a small neighborhood of the point .

     

Boundary blow-up elliptic problems with nonlinear gradient terms

By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δu±q|∇u|=b(x)g(u), u>0 in Ω, u_|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, q?0, g∈C¹[0,∞),g(0)=0, g^′ is regularly varying at infinity with index ρ with ρ>0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary.     

The exact blow-up rates of large solutions for semilinear elliptic equations

In this paper, combining the method of lower and upper solutions with the localization method, we establish the boundary blow-up rate of the large positive solutions to the singular boundary value problem

     

Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation

A blow-up criterion for classical solutions to the compressible Navier-Stokes equations

In this paper, we obtain a blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations just in terms of the gradient of the velocity, analogous to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, the initial vacuum is allowed in our case.