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We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the dimension of the manifold is n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂M.

Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by Andre in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor, which measure irregularity along divisors on blowups of the original space; this generalizes another result of Andre on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties., 27 pages; v4: refereed version; some technical edits in 2.2

Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$.

The world of fashion has lived in a bubble long before the concept found its way into social networks. Well before the social networks themselves, in fact. The very emergence of fashion as an idea occurred within the bubble of the social life at Palace of Versailles, France, where the notorious Louis XIV sported great interest in the looks of his court in addition to those of his own. The article is an outsider’s attempt to have a sober look at the fashion industry that until recently seemed to have maintained the “structure of feeling” of the 17th century Palace of Versailles. Today’s social realities, however, put fashion in the state of a shock that probably equals that of the Storming of the Bastille in 1789, even though it is presumably much less sudden. Written in the manner of the most popular contemporary fashion media format – a bullet list, the text presents a conceptual analysis of the world’s second most wasteful yet poisonously attractive industry, critically reflecting on such characteristic values of the fashion field as concept and features, hierarchy, arrogance, resources and philosophy.

In this article, we are concerned with a problem for the p -Laplacian parabolic equation with logarithmic nonlinearity; the blow-up result of the solution is proven. This work is completed Boulaaras’ work in Math. Methods Appl. Sci., (2020), where the author did not study the blowup of the solution.

In this paper, we consider a pseudo-parabolic equation, which was studied extensively in recent years. We generalize and extend the existing results in the following three aspects. First, we consider the vacuum isolating phenomenon with the initial energy $$J(u_0)$$ satisfying $$J(u_0)\le 0$$ and $$0

We answer a question of Ciliberto’s about cylinders in rational surfaces obtained by blowing up the plane at points in general position. Bibliography: 13 titles.

We consider the focusing \begin{document}$ L^2 $\end{document} -supercritical Schrodinger equation in the exterior of a smooth, compact, strictly convex obstacle \begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document} . We construct a solution behaving asymptotically as a solitary wave on \begin{document}$ \mathbb{R}^3, $\end{document} for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

We show that there are no conformal metrics $g=e^{2u}g_{\mathbb {R}^{4}}$ on $\mathbb {R}^{4}$ induced by a smooth function u ≤ C with Δu(x) → 0 as $|x|\to \infty $ having finite volume and finite total Q-curvature, when Q(x) = 1 + A(x) with a negatively definite symmetric 4-linear form A(x) = A(x,x,x,x). Thus, in particular, for suitable smooth, non-constant $f_{0}\le {\max \limits } f_{0}=0$ on a four-dimensional torus any “bubbles” arising in the limit λ ↓ 0 from solutions to the problem of prescribed Q-curvature Q = f0 + λ blowing up at a point p0 with dkf0(p0) = 0 for $k=0,\dots ,3$ and with d4f0(p0)

A quenching phenomenon of a degenerate parabolic equation with nonlinear source and singular boundary condition in one-dimensional space is investigated. We establish the results that quenching will occur in a finite time on the boundary x = 0 or x = 1, respectively. And the blowing up of ut at the quenching point and quenching rate estimates are discussed.

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank $3$ vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct a non-trivial part of the moduli space by blowing up along a variety in a special case., Comment: 23 pages

Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in \cite{MM1}, we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions., Comment: 27 pages

In this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.

The lightcone singularity at the origin is resolved by blowing up the singular point to $CP^1$. The Lorentz group acts on the resolved lightcone and has $CP^1$ as a special orbit. Using Wigner's method of associating unitary irreducible representations of the Poincar\'e group to particle states, we find that the special orbit gives rise to new vacuum states. These vacuum states are labelled by the principal series representations of $SL(2,C)$. Some remarks are included on the applications of these results to gauge theories and asymptotically flat spacetimes., Comment: 8 pages, new material added

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product., Comment: This is the definitive complete version; the previous one was an incomplete draft, uploaded by mistake

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazarsfeld in a uniform manner and study the blow-up invariance of some classical holomorphic invariants., Comment: Final version with several typos fixed, to appear in Journal de Math\'ematiques Pures et Appliqu\'ees. 36 pages

We solve the prescribed-time mean-square stabilization problem, providing the first feedback solution to a stochastic null-controllability problem for strict-feedback nonlinear systems with stochastic disturbances. Our non-scaling backstepping design scheme’s key novel design ingredient is that, rather than employing “blowing up” time-varying scaling of the backstepping coordinate transformation, we introduce, instead, a damping in the backstepping target systems which grows unbounded as time approaches the terminal time. With this approach, even for deterministic systems, a simpler controller results and the control effort is reduced compared to previous designs. We achieve prescribed-time stabilization in the mean-square sense.

Inspired by the recent works of S. Rao--S. Yang--X.-D. Yang and L. Meng on the blow-up formulae for de Rham and Morse--Novikov cohomology groups, we give a new simple proof of the blow-up formula for Morse--Novikov cohomology by introducing the relative Morse--Novikov cohomology group via sheaf cohomology theory and presenting the explicit isomorphism therein.

A new method for blowing up the power of a singular cardinal is presented. It allows to blow up the power of a singular in the core model cardinal of uncountable cofinality. The method makes use of overlapping extenders.

In this paper, we study the blow-up, and global existence of solutions to the following time–space fractional diffusion problem 0 C D t α u ( x , t ) + ( − Δ ) β ∕ 2 u ( x , t ) = 0 J t γ | u | p − 1 u ( x , t ) , x ∈ R N , t > 0 , u ( x , 0 ) = u 0 ( x ) ∈ C 0 ( R N ) , where 0 α 1 − γ , 0 γ 1 , 0 β ≤ 2 , p > 1 , 0 J t γ denotes the left Riemann–Liouville fractional integral of order γ , 0 C D t α is the Caputo fractional derivative of order α and ( − Δ ) β ∕ 2 stands for the fractional Laplacian operator of order β ∕ 2 . We show that if p 1 + β ( α + γ ) ∕ α N , then every nonnegative solution blows up in finite time, and if p ≥ 1 + β ( α + γ ) ∕ α N and ‖ u 0 ‖ L q c ( R N ) is sufficiently small, where q c = α N ( p − 1 ) ∕ β ( α + γ ) , then the problem has global solutions. Finally, we give an upper bound estimate of the life span of blowing-up solutions.

In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u_t} +{M([u]^2_s){\mathcal {L}}_Ku}={|u|^{p-2}u\log |u|},\ \ \ &{}\hbox { in } \Omega \times (0,+\infty ),\\ \displaystyle u(x,t)=0,&{}\hbox { in }({\mathbb {R}}^N\setminus \Omega )\times (0,+\infty ),\\ \displaystyle u(x,0)=u_0(x),&{}\hbox { in }\Omega , \end{array}\right. \end{aligned}$$ where $$[u]_s$$ is the Gagliardo seminorm of u, $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary, $$00$$ be the mountain-pass level given in (2.4), and $${\widetilde{M}}\in (0,d]$$ be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for $$J(u_0)\le d$$ . Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for $$J(u_0)\le {\widetilde{M}}$$ , the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. In addition, for $$J(u_0)\le d$$ , we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1] , where the weak solution with low initial energy ( I ( 0 ) d , d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I ( 0 ) ≤ 0 and 0 I ( 0 ) d , respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.

We consider a thin-film equation modeling the epitaxial growth of nanoscale thin films. By exploiting the boundary conditions and the variational structure of the equation, we look for conditions on initial data that ensure the solution exists globally or blows up in finite time. Moreover, for global solution, we establish the exponential decays of solutions and energy functional, and give the concrete decay rate. As for blow-up solution, we prove that the solution grows exponentially and obtain the behavior of energy functional as t tends to the maximal existence time. Under the low initial energy, we get further two necessary and sufficient conditions for the solution existing globally and blowing up in finite time, respectively. A new sufficient condition such that the solution exists globally is obtained; we point out that this initial condition is independent to initial energy. Finally, we discuss the vacuum isolating phenomena of the solution.

In this paper, we use the Green’s function method to get the pointwise convergence rate of the semilinear pseudo-parabolic equations. By using this precise pointwise structure and introducing negative index Sobolev space condition on the initial data, we release the critical index of the nonlinearity for blowing up. Our result shows that the global existence does not only depend on the nonlinearity but also the initial condition.

We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension $\le 9$. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compactness in $C^4$. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension $\le 7$, when the Yamabe invariant is positive. We also prove that, for dimension $\ge 8$, the Weyl tensor has to vanish at possible blow up points of a sequence of blowing up solutions. This implies the compactness result in dimension $\ge 8$ when the Weyl tensor does not vanish anywhere. To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations., 40 pages, published version

In this paper, we investigate the qualitative structures of a Ricker competition model near a degenerate fixed point with eigenvalues ±1. Firstly, by the Takens's theorem, we asymptotically embed a...

Let u be a map from a Riemann surface M to a Riemannian manifold N and α > 1 , the α energy functional is defined as E α ( u ) = 1 2 ∫ M [ ( 1 + | ▽ u | 2 ) α − 1 ] d V . We call u α a sequence of Sacks–Uhlenbeck maps if u α are critical points of E α and sup α > 1 ⁡ E α ( u α ) ∞ . In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere S K − 1 . The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3] , [4] ).

International audience; We consider the Schr\"odinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.

We are concerned with the existence of blowing-up solutions to the following boundary value problem \begin{document}$ -\Delta u = \lambda V(x) e^u-4\pi N {\mathit{\boldsymbol{\delta}}}_0\;\hbox{ in } \Omega, \quad u = 0 \;\hbox{ on }\partial \Omega, $\end{document} where \begin{document}$ \Omega $\end{document} is a smooth and bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} such that \begin{document}$ 0\in\Omega $\end{document} , \begin{document}$ V $\end{document} is a positive smooth potential, \begin{document}$ N $\end{document} is a positive integer and \begin{document}$ \lambda>0 $\end{document} is a small parameter. Here \begin{document}$ {\mathit{\boldsymbol{\delta}}}_0 $\end{document} defines the Dirac measure with pole at \begin{document}$ 0 $\end{document} . We assume that \begin{document}$ \Omega $\end{document} is \begin{document}$ (N+1) $\end{document} -symmetric and we find conditions on the potential \begin{document}$ V $\end{document} and the domain \begin{document}$ \Omega $\end{document} under which there exists a solution blowing up at \begin{document}$ N+1 $\end{document} points located at the vertices of a regular polygon with center \begin{document}$ 0 $\end{document} .

In this study, we consider a special case of the family of birational maps f:C² → C² , which were dynamically classified by [13]. We identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations. In particular, we highlight all of the integrable and periodic mappings.

Using the normal forms introduced in Chap. 10 the transitions near the singularities, showing up after blowing up the contact points, are described in terms of the functions that are introduced in Chap. 11. Together with Chap. 8 on the blow-up technique this chapter forms the heart of the book.

We are concerned with the existence of blowing-up solutions to the following boundary value problem − Δ u = λ a ( x ) e u − 4 π N δ 0 in Ω , u = 0 on ∂ Ω , where Ω is a smooth and bounded domain in R 2 such that 0 ∈ Ω , a ( x ) is a positive smooth function, N is a positive integer and λ > 0 is a small parameter. Here δ 0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution u λ blowing up at 0 and satisfying λ ∫ Ω a ( x ) e u λ → 8 π ( N + 1 ) as λ → 0 + .

In this article, we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them.

We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative $Q$-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infinitely many complete metrics with constant $Q$-curvature conformal to $\mathbb S^m\times\mathbb R^d$, $m\geq4$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d\leq m-3$; which give infinitely many solutions to the singular constant $Q$-curvature problem on round spheres $\mathbb S^n$ blowing up along a round subsphere $\mathbb S^k$, for all $0\leq k, Comment: LaTeX2e, 19 pages, final (revised) version. To appear in Int. Math. Res. Not. IMRN