Blow-up Points Research Articles

Asymptotic Analysis for Sacks-Uhlenbeck 𝛼-Harmonic Maps From Degenerating Riemann Surfaces

In this paper, we investigate the asymptotic analysis and qualitative behaviour for a general sequence of Sacks-Uhlenbeck α \alpha -harmonic maps from degenerating Riemann surfaces. This answers an open problem proposed by J. D. Moore, aiming at developing a partial Morse theory for closed parametrized minimal surfaces with arbitrary codimensions in compact Riemannian manifolds. In technical terms, we shall deal with three types of necks which give rise to new difficulties. We establish a generalized energy identity involving some new quantities, such as Pohozaev type constants which measure the extent to which the Pohozaev type identity fails, and some locating constants which characterize the limit position of the blow-up points in long hyperbolic cylinders. Furthermore, we show that the limit of necks are all geodesics in the target manifold where some length formulas are given. Finally, we exploit some geometric and topological conditions to ensure that the limiting neck geodesics appeared in a blow-up sequence of α \alpha -harmonic maps with finite Morse index are all of finite length, which implies that energy identity hold.

Nirenberg problem on high dimensional spheres: blow up with residual mass phenomenon

In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in Malchiodi and Mayer(J Differ Equ 268(5):2089–2124, 2020; Int Math Res Not 18:14123–14203, 2021). Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a non-zero weak limit, which necessarily constitutes a solution of the Nirenberg problem itself. Our focus lies in providing a comprehensive description of such blowing up solutions, including precise determinations of blow-up points and blow-up rates. Additionally, we compute the topological contribution of these solutions to the difference in topology between the level sets of the associated Euler-Lagrange functional. Such an analysis is intricate due to the potential degeneracy of the solutions involved. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate.

Singularity formation for the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime

AbstractWe study the formation of singularities of smooth solutions to the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime. The system is in the spherically symmetric form, and its coefficients and nonhomogeneous terms contain a parameter reflecting the mass of the black hole, which makes it highly nonlinear and complicated. To overcome the influence of the mass parameter of black hole, we introduce a pair of suitable auxiliary variables related to it and derive their characteristic decompositions to establish the estimates of the smooth solution. We show that, for a kind of initial data, the smooth solution develops singularity in finite time and the mass‐energy density itself approaches infinity at the blowup point.

Dynamic analysis of a fast slow modified Leslie–Gower predator–prey model with constant harvest and stochastic factor

In this paper, the dynamic properties of a fast slow modified Leslie–Gower predator–prey model with constant harvest are discussed. The results show that the system appears different Hopf bifurcations and limit cycles as parameters change, and the stability of the internal equilibriums changes accordingly. The system also appears Bogdanov–Takens bifurcation phenomenon. For the fast–slow system of this model, the second order approximate perturbation manifolds, and an approximate manifold passing the fold point are constructed. Then using the criterion of inflection point curve and blow-up method, the existence of canard cycles is analyzed. By comparing with the model with nonlinear harvest, the different effects of constant harvest on the model are highlighted. In addition, the effect of stochastic factors on the fast–slow system is considered.

Singularity formation for the cylindrically symmetric rotating relativistic Euler equations of Chaplygin gases

This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

Varieties covered by affine spaces, uniformly rational varieties and their cones

It was shown in Kaliman snd Zaidenberg (2023) [26] that the affine cones over flag manifolds and rational smooth projective surfaces are elliptic in the sense of Gromov. The latter remains true after successive blowups of points on these varieties. In the present article we extend this to smooth projective spherical varieties (in particular, toric varieties) successively blown up along smooth subvarieties. The same holds, more generally, for uniformly rational projective varieties, in particular, for projective varieties covered by affine spaces. It occurs also that stably uniformly rational complete varieties are elliptic.

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−2 and ε>0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.In general, the solutions of (⁎) may blow-up at multiple points a1,⋯,ak of Ω as ε→0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k=1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation -Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u =\\displaystyle {\\Big (\\int _{\\Omega }\\frac{u^{2_{\\mu }^*} (\\xi )}{|x-\\xi |^{\\mu }}d\\xi \\Big )u^{2_{\\mu }^*-1}}+\\varepsilon u,~~~\ ext {in}~\\Omega ,\\\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ ext {on}~\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Nge 4, 0<mu le 4, varepsilon >0 is a small parameter, Omega is a bounded domain in mathbb {R}^N, and 2_{mu }^*=frac{2N-mu }{N-2} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for varepsilon small.

The Blow-Up Analysis on B2(1) Affine Toda System: Local Mass and Affine Weyl Group

Abstract It has been established that the local mass of blow-up solutions to Toda systems associated with the simple Lie algebras $\textbf{A}_{n}$, $\textbf{B}_{n}$, $\textbf{C}_{n}$, and $\textbf{G}_{2}$ can be represented by a finite Weyl group. In particular, at each blow-up point, after a sequence of bubbling steps (via scaling) is performed, the transformation of the local mass at each step corresponds to the action of an element in the Weyl group. In this article, we present the results in the same spirit for the affine $\textbf{B}_{2}^{(1)}$ Toda system with singularities. Compared with the Toda system with simple Lie algebras, the computation of local masses is more challenging due to the infinite number of elements of the affine Weyl group of type $\textbf{B}_{2}^{(1)}$. In order to give an explicit expression for the local mass formula, we introduce two free integers and write down all the possibilities into $8$ types. This shows a striking difference to previous results on Toda systems with simple Lie algebras. The main result of this article seems to provide the first major advance in understanding the relation between the blow-up analysis of affine Toda system and the affine Weyl group of the associated Lie algebras.

Construction of Blowup Solutions for the Complex Ginzburg-Landau Equation with Critical Parameters

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T T only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.

The $$C^0$$-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li (Commun Math Phys 200(2):421–444, 1999) gave a uniform asymptotic estimate. Later, Zhang (Commun Math Phys 268(1):105–133, 2006) and Gluck (Nonlinear Anal 75(15):5787–5796, 2012) improved Li’s estimate in the sense of $$C^0$$ -convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao–Wang–Zhou (J Math Anal Appl 418:142–162, 2014) proved a similar asymptotic estimate of Li (1999). In this paper, we will prove a $$C^0$$ -convergence result in this boundary blow-up process. Our method is different from Gluck (2013), Zhang (2006).

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation

This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation with zero-flux boundary condition for u and zero-Neumann boundary condition for v, where Ω is a bounded domain in with smooth boundary , is a rotation matrix with . We show that:Let be a general smooth bounded domain.If , then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω.If and contains a line segment, then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies on the line segment of .Let be a disc in with radius R > 0 centered at origin. Although there is a rotation effect in system (*), solutions still preserve radial symmetry of initial data. If nonnegative radially symmetric initial data u 0 satisfies , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.

The Nirenberg Problem on Half Spheres: A Bubbling-off Analysis

Abstract In this paper, we perform a refined blow-up analysis of finite energy approximated solutions to a Nirenberg-type problem on half spheres. The latter consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular, we give a precise location of blow-up points and blow-up rates. Such an analysis shows that the blow-up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed, besides the combination of interior and boundary blow ups, there are nonsimple blow-up points for subcritical solutions having zero or nonzero weak limit. The formation of such nonsimple blowups is governed by a vortex problem, unveiling an unexpected connection with Euler equations in fluid dynamic and mean fields type equations in mathematical physics.

Feedback controllability for blowup points of the heat equation

This paper is concerned with a controllability problem of blowup points for the heat equation. It can be described as follows: In the absence of control, the solution to the linear heat equation globally exists in a bounded domain Ω. We wonder whether for a given time T>0 and a point a in this domain, we can find a feedback control, acting on a given internal subset ω of this domain, such that the corresponding solution to the heat equation blows up at time T and holds a as a unique blowup point. In this paper, we positively answer the question, when a∈ω. On the contrary, when a∈Ω∖ω‾, we show this is not possible, for any open-loop or feedback control which guarantees that the solution is L∞(Ω) whenever it exists.

GLOBAL ATTRACTIVITY, ASYMPTOTIC STABILITY AND BLOW-UP POINTS FOR NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS’ SOLUTIONS AND APPLICATIONS IN BANACH SPACE BC(R+) WITH COMPUTATIONAL COMPLEXITY

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space [Formula: see text] using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space [Formula: see text] are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to [Formula: see text]. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with [Formula: see text] dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

Polar exploration of complex surface germs

We prove that the topological type of a normal surface singularity ( X , 0 ) (X,0) provides finite bounds for the multiplicity and polar multiplicity of ( X , 0 ) (X,0) , as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of ( X , 0 ) (X,0) . A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of ( X , 0 ) (X,0) , which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of ( X , 0 ) (X,0) through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted.