Blow-up Sets Research Articles

Prescribed blow-up sets for sequences of solutions to a non-local Q-curvature equation in ℝ3

Let [Formula: see text] be an open connected domain and [Formula: see text] stand for the Laplacian in [Formula: see text]. For a uniformly bounded sequence [Formula: see text] having a fixed sign and non-positive bi-harmonic functions [Formula: see text], we prove that there exists a sequence of solutions [Formula: see text] to [Formula: see text] with a total curvature [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is the zero set given by [Formula: see text] In particular, when [Formula: see text] is smoothly bounded and [Formula: see text], we obtain that [Formula: see text] blows up on [Formula: see text] for any [Formula: see text]. This extends the work of Hyder et al. who treated the analogue in even dimensions and complements a previous result in dimension three. We also consider a radial case with no blowing up phenomena and large total curvature.

Limiting behavior of sequences of properly embedded minimal disks

We develop a theory of “minimal $\theta$-graphs” and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of embedded minimal disks, even when there is no curvature blow-up. Our methods work in a more general Riemannian setting, including hyperbolic space. This allows us to establish the existence of a complete, simply connected, minimal surface in hyperbolic space that is not properly embedded.

Blow-up analyses in reaction–diffusion equations with nonlinear nonlocal boundary flux

In this paper, we study a reaction–diffusion system involving coupled sources and nonlinear nonlocal boundary flux. By using the comparison principle, we give the criteria on blow-up and global solutions. There exist no nontrivial global solutions under some conditions on the coefficients. After classifying simultaneous and non-simultaneous blow-up of the components of solutions, we obtain blow-up rates and sets. Lower and upper bounds of blow-up time are determined quantificationally for all dimensions of the space domains.

Blow-up analyses in reaction–diffusion equations with Fujita exponents

In this paper, we study the reaction–diffusion equations with variable coefficients in some bounded domains. At least one of the components of solutions blows up for every initial data in some exponent regions, where the Fujita exponents are determined by the exponents of the sources and the coefficients and the dimension of the domain. We also show the classifications of simultaneous and nonsimultaneous blow-up of the components of solutions. The asymptotic properties are discussed including blow-up rates and sets. Moreover, the upper and the lower bounds of blow-up time are given for all dimensions of domains, respectively.

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Let [Formula: see text] be an integer. For any open domain [Formula: see text], non-positive function [Formula: see text] such that [Formula: see text], and bounded sequence [Formula: see text] we prove the existence of a sequence of functions [Formula: see text] solving the Liouville equation of order [Formula: see text] [Formula: see text] and blowing up exactly on the set [Formula: see text], i.e. [Formula: see text] thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of [Formula: see text] and to the case [Formula: see text]. Several related problems remain open.

Blow-up time and boundary layer for solutions in parabolic equations with different diffusion

This paper deals with parabolic equations with different diffusion coefficients and coupled nonlinear sources, subject to homogeneous Dirichlet boundary conditions. We give many results about blow-up solutions, including blow-up time estimates for all of the spatial dimensions, the critical non-simultaneous blow-up exponents, uniform blow-up profiles, blow-up sets, and boundary layer with or without standard conditions on nonlocal sources. The conditions are much weaker than the ones for the corresponding results in the previous papers.

Blow-up sets for a complex-valued semilinear heat equation

This paper is concerned with finite-time blow-up solutions of a one-dimensional complex-valued semilinear heat equation. We characterize the location and the number of blow-up points from the viewpoint of zeros of the solution.

Controlling area blow-up in minimal or bounded mean curvature varieties

Consider a sequence of minimal varieties $M_i$ in a Riemannian manifold $N$ such that the measures of the boundaries are uniformly bounded on compact sets. Let $Z$ be the set of points at which the areas of the $M_i$ blow up. We prove that $Z$ behaves in some ways like a minimal variety without boundary. In particular, it satisfies the same maximum and barrier principles that a smooth minimal submanifold satisfies. For suitable open subsets $W$ of $N$, this allows one to show that if the areas of the $M_i$ are uniformly bounded on compact subsets of $W$, then the areas are in fact uniformly bounded on all compact subsets of $N$. Similar results are proved for varieties with bounded mean curvature. The results about area blow-up sets are used to show that the Allard Regularity Theorems can be applied in some situations where key hypotheses appear to be missing. In particular, we prove a version of the Allard Boundary Regularity Theorem that does not require any area bounds. For example, we prove that if a sequence of smooth minimal submanifolds converge as sets to a subset of a smooth, connected, properly embedded manifold with nonempty boundary, and if the convergence of the boundaries is smooth, then the convergence is smooth everywhere.

Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions

In this paper, we apply positivity-preserving local discontinuous Galerkin (LDG) methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustion problems. However, previous numerical methods are mainly based on a second order finite difference method. This is because the positivity-preserving property can hardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets. Recently, we have applied discontinuous Galerkin methods to linear hyperbolic equations involving δ-singularities and obtained good approximations. For nonlinear problems, some special limiters are constructed to capture the singularities precisely. We will continue this approach and study parabolic equations with blow-up solutions. We will construct special limiters to keep the positivity of the numerical approximations. Due to the Dirichlet boundary conditions, we have to modify the numerical fluxes and the limiters used in the schemes. Numerical experiments demonstrate that our schemes can capture the blow-up sets, and high-order approximations yield better numerical blow-up time.

Note on nonsimultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with the nonsimultaneous blow-up problems for with coupled nonlinear boundary flux . The main results extend the existence of two components blowing up simultaneously in the paper J. Math. Anal. Appl. 2009;356:215–231 to the case of components blowing up simultaneously. We verify that blow-up phenomena are independent of the choosing of the initial data in some exponent regions, while, in some others, the blow-up phenomena depend sensitively on the choosing of initial data. Moreover, the blow-up rates and sets are obtained for blow-up components.

Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source , , , , , , and , , which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.

On the finite difference approximation for blow-up solutions of the porous medium equation with a source

We consider the porous medium equation with a nonlinear source term, ut=(um)xx+uβ, x∈(0,L), t>0, whose solution blows up in finite time. Here β⩾m>1, L>0 are parameters. To approximate the solution near blow-up time and to estimate the blow-up time numerically, the concept of adaptive time meshes is introduced so as to construct a finite difference scheme whose solution also blows up in finite time. For such schemes, we show not only the convergence of the numerical solution but also the convergence of the numerical blow-up time. Moreover, the numerical blow-up sets are classified. It is interesting that although the convergence of the numerical solution is guaranteed, the numerical blow-up sets are sometimes different from that of the PDE. However, the blow-up shapes are reproduced numerically by our schemes.

Blow-up for the non-local [formula omitted]-Laplacian equation with a reaction term

In this paper we study the blow-up phenomenon for the non-local p-laplacian equation with a reaction term, ut(x,t)=∫AJ(x−y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy+uq(x,t),x∈Ω,t∈[0,T], with Dirichlet conditions (A=RN,u≡0 in RN∖Ω) or Neumann conditions (A=Ω). Those problems are the non-local analogous to the equation vt=Δpv+vq with the corresponding conditions. We determine in both cases which are the global existence exponents, that coincide with the exponents of the corresponding local problem for Neumann boundary conditions. However, we observe differences with respect to the global existence exponents of the local Dirichlet problem. Moreover, we show that the blow-up rate is the same as the one that holds for the ODE ut=uq, that is, limt↗T(T−t)1q−1‖u(⋅,t)‖∞=(1q−1)1q−1. We also find some differences between the local and the non-local models concerning the blow-up sets. Precisely we show that regional blow-up is not possible for non-local problems. Finally, we include some numerical experiments which illustrate our results.

Non-simultaneous blow-up and blow-up rates for reaction–diffusion equations

This paper considers blow-up solutions for reaction–diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of n components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of k ( ∈ { 1 , 2 , … , n } ) components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of n components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215–231].

Non-simultaneous blow-up in localized reaction–diffusion equations

This article deals with blow-up solutions in reaction–diffusion equations coupled via localized exponential sources, subject to null Dirichlet conditions. The optimal and complete classification is obtained for simultaneous and non-simultaneous blow-up solutions. Moreover, blow-up rates and blow-up sets are also discussed. It is interesting that, in some exponent regions, blow-up phenomena depend sensitively on the choosing of initial data, and the localized nonlinearities play important roles in the blow-up properties of solutions.

Non-simultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution ( u 1 , u 2 , … , u n ) to heat equations coupled via nonlinear boundary ∂ u i ∂ η = u i p i u i + 1 q i + 1 ( i = 1 , 2 , … , n ) . It is proved that there exist suitable initial data such that u i ( i ∈ { 1 , 2 , … , n } ) blows up alone if and only if q i + 1 < p i . All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k , i , n ) of u i − k and u i leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that u i − k , u i − k + 1 , … , u i ( i ∈ { 1 , 2 , … , n } , k ∈ { 0 , 1 , 2 , … , n − 1 } ) blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained.

Non-simultaneous blow-up for [formula omitted]-componential parabolic systems

This paper deals with the non-simultaneous and simultaneous blow-up for some parabolic systems ( u i ) t = Δ u i + u i p i coupled via nonlinear boundary flux ∂ u i ∂ η = u i + 1 q i + 1 ( i = 1 , 2 , … , n ) . For radially symmetric solutions, we obtain that one component can blow up by itself and may provide sufficient help to the blow-up of the other k ( ∈ { 0 , 1 , … , n − 2 } ) ones under suitable initial data. In particular, such phenomena happen for every initial data in some exponent regions. It is interesting that there exist initial data such that any two components blow up simultaneously, either of which blows up depending on itself and also can give sufficient help to the other components blowing up simultaneously. A necessary and sufficient condition is obtained on the simultaneous blow-up of at least two components for all initial data. Moreover, the non-simultaneous and simultaneous blow-up rates and sets are determined.

Asymptotic analysis for a localized nonlinear diffusion equation

In this paper we study a localized nonlinear diffusion equation u t = Δ u m + λ 1 u p + λ 2 u q ( 0 , t ) subject to null Dirichlet boundary condition with p , q ≥ 0 , max { p , q } > m > 1 , and λ 1 , λ 2 > 0 . We investigate interactions among the localized and local sources, nonlinear diffusion with the zero boundary value condition to establish blow-up rates and uniform blow-up profiles of solutions under different dominations. In addition, as results of the interactions of multiple nonlinearities, the blow-up sets of solutions, namely, total versus single point blow-up of solutions are also determined.

Non-simultaneous blow-up in a parabolic system with three components

This paper deals with heat equations coupled via nonlinear boundary flux ∂ u 1 ∂ η = u 1 p 11 + u 2 p 12 , ∂ u 2 ∂ η = u 2 p 22 + u 3 p 23 , ∂ u 3 ∂ η = u 3 p 33 + u 1 p 31 . A necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions. Three kinds of exponent regions are obtained as follows, (i) only one component blows up for every initial data; (ii) the existence of two components blowing up simultaneously while the third one remains bounded, also with two kinds of blow-up rates ( 1 2 ( p 22 − 1 ) , 1 2 ( p 33 − 1 ) ) and ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) ; (iii) e.g., u 1 remains bounded and u 2 , u 3 blow up simultaneously with blow-up rate ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) for every initial data. Moreover, the eight kinds of simultaneous blow-up rates and blow-up sets are discussed.

Capacity and blow-up for the 3 + 1 dimensional wave operator

The size (in terms of Hausdorff content) of the blow-up sets, for weak solutions to the non-homogeneous wave equation in three space dimensions, are studied when the right hand side is a function of the space and time variables, in a mixed L p -norm space. The methods rely on non-linear elliptic retarded potentials. 2000 Mathematics Subject Classification: 35L05, 35L67, 31C15.