Nondegeneracy and uniqueness for boundary blow-up elliptic problems

In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Δu=λa(x)u^p+g(x,u) in Ω, with u=+∞ on ∂Ω, has a unique positive solution for large enough λ, and determine its asymptotic behavior as λ→+∞. Here p>1, a(x) is a continuous function which can be singular near ∂Ω and g(x,u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u^p by e^u or a “logistic type” function f(u).     

A Discrete Phenomenon in the C and Analytic Regularity of the Blow-Up Curve of Solutions to the Liouville Equation in One Space Dimension

We give a complete discussion of the C^∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case.     

Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms

This paper deals with the blow-up properties of the solution to a semilin-ear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditions for that the classical solution blows up in the finite time, secondly give necessary conditions and a sufficient condition for that two components blow up simultaneously, and then obtain the uniform blow-up profiles in the interior. Finally we describe the asymptotic behavior of the blow-up solution in the boundary layer.     

On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation with initial condition in the energy space and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in a sharp and stable upper bound on the blow-up rate: .

In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is,

In this paper, we prove the sharp lower bound

by exhibiting the dispersive structure in the scaling invariant space for this log-log regime. In addition, we will extend to the pure energy space a dynamical characterization of the solitons among the zero energy solutions.

     

On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

Blow-up for Joseph–Egri equation: Theoretical approach and numerical analysis

This work develops the theory of the blow-up phenomena for Joseph–Egri equation. The existence of the nonextendable solution of two initial-boundary value problems (on a segment and a half-line) is demonstrated. Sufficient conditions of the finite-time blow-up of these solutions, as well as the analytical estimates of the blow-up time, are obtained. A numerical method that allows to precise the blow-up moment for specified initial data is proposed.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Blow-up of solutions for a system of nonlocal singular viscoelastic equations

The main propose of this paper is to study the blow-up of solutions of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. where the blow-up of solutions in finite time with nonpositive initial energy combined with a positive initial energy are shown.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity

This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation $u_t - \triangle u - \triangle u_t + u = f(u)$ with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.