Blow-up Estimates Research Articles

Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordström Spacetime

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, |Q|=M,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ |Q|=M, $$\\end{document} Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}. In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component α̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\underline{\\alpha }} $$\\end{document} not decaying asymptotically along the event horizon H+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+, $$\\end{document} a result previously unknown in the literature.

A Priori Estimates for Finite-Energy Sign-Changing Blowing-Up Solutions of Critical Elliptic Equations

Abstract We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger–Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^{0}$-theory for positive solutions of Schrödinger–Yamabe-type equations. To deal with the sign-changing case, we develop a method of proof that combines an a priori bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp a priori blow-up estimates for a linear problem.

Blow-up of positive solutions for the semilinear heat equation with a potential

We study nonnegative solutions of the equation in , t>0, under the assumption that satisfies . We establish a new blow-up criterion that depends only on and p. In addition, as the supreme of decreases, we find an interesting phenomenon that the Fujita exponent goes to ∞, in the sense that every nonnegative solution blows up in finite time whenever p>1. Furthermore, we obtain the blow-up rate estimate in the subcritical case. In the end, under the assumption of , we give the refined blow-up estimate of the blow-up solutions near the blow-up time.

L2-blowup estimates of the wave equation and its application to local energy decay

We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].

Blow-up estimates for a higher-order reaction–diffusion equation with a special diffusion process

Blow-up estimates for a higher-order reaction–diffusion equation with a special diffusion process

Estimates and asymptotics for the stress concentration between closely spaced stiff C1,γ inclusions in linear elasticity

This paper is concerned with the stress concentration phenomenon in elastic composite materials when the inclusions are very closely spaced. We investigate the gradient blow-up estimates for the Lamé system of linear elasticity with partially infinite coefficients to show the dependence of the degree of stress enhancement on the distance between inclusions in a composite containing densely placed stiff inclusions. In this paper, we assume that the inclusions to be of C1,γ, weaker than the previous C2,γ assumption. To overcome this new difficulty, we make use of W1,p estimates for elliptic system with right hand side in divergence form, instead of a direct W2,p argument for C2,γ inclusion case, and combine with the Campanato's approach to establish the optimal gradient estimates, including upper and lower bounds. Moreover, an asymptotic formula of the gradient near the blow-up point is obtained for some symmetric C1,γ inclusions.

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-\Delta)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).

Blow-up estimate in a reaction–diffusion equation with nonlinear nonlocal flux and source

This paper deals with a reaction–diffusion equation with nonlocal inner source and nonlocal boundary flux. Besides the influence of exponents, we find out that the weighted functions and the measure of the domain could lead to the existence of global solutions for any initial data. Larger weighted functions require smaller initial data in guaranteeing the existence of global solutions to the problem. Blow-up rates are obtained under different dominations of nonlinearities in one dimensional space. The upper and lower bounds of blow-up time are determined for all spatial dimensions, respectively. Moreover, the solutions blow up only on the boundary when the nonlocal flux dominates the blow-up.

A sharp blow-up estimate for the Lebesgue norm

We prove that if $$p>1$$ and $$\psi :]0,p-1[\rightarrow ]0,\infty [$$ is nondecreasing, then $$\begin{aligned} \sup _{0<\varepsilon<p-1} \psi (\varepsilon ) \Vert f\Vert _{L^{p-\varepsilon }(0,1)}\approx & {} \sup _{0<t<1} \psi \left( \frac{p-1}{1-\log t}\right) \Vert f^*\Vert _{L^{p}(t,1)} \\&\mathop {\Updownarrow }\limits _{{\begin{array}{c} \psi \,\in \,\Delta _2\,\cap \, L^\infty . \end{array}}} \end{aligned}$$ Here f is a Lebesgue measurable function on (0, 1) and $$f^*$$ denotes the decreasing rearrangement of f. The proof generalizes and makes sharp an equivalence previously known only in the particular case when $$\psi $$ is a power; such case had a relevant role for the study of grand Lebesgue spaces. A number of consequences are discussed, among which: the behavior of the fundamental function of generalized grand Lebesgue spaces, an analogous equivalence in the case the assumption on the monotonicity of $$\psi $$ is dropped, and an optimal estimate of the blow-up of the Lebesgue norms for functions in Orlicz–Zygmund spaces.

Correction to: Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

Correction to: Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4,5] from the whole space R3 to the case of bounded smooth domain Ω⊂R3. Lower global blow-up estimate on ‖n‖L∞(Ω) is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.

Asymptotic boundary estimates to infinity Laplace equations with Γ-varying nonlinearity

In a previous paper of the authors (Wang et al. (2014) [40]), the asymptotic estimates of boundary blow-up solutions were established to the infinity Laplace equation Δ∞u=b(x)f(u) in Ω⊂RN, with the nonlinearity 0≤f∈C[0,∞) regularly varying at ∞, and the weighted function b∈C(Ω¯) positive in Ω and vanishing on the boundary. The present paper gives a further investigation on the asymptotic behavior of boundary blow-up solutions to the same equation by assuming f to be Γ-varying. Note that a Γ-varying function grows faster than any regularly varying function. We first quantitatively determine the boundary blow-up estimates with the first expansion, relying on the decay rate of b near the boundary and the growth rate of f at infinity, and further characterize these results via examples possessing various decay rates for b and growth rates for f. In particular, we pay attention to the second-order estimates of boundary blow-up solutions. It was observed in our previous work that the second expansion of solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the classical Laplacian. The second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow-up solutions between the infinity Laplacian and the classical Laplacian.

Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.

Blowup estimates for a family of semilinear SPDEs with time-dependent coefficients

Abstract We investigate the blowup and stability of semilinear stochastic partial differential equations with time-dependent coefficients using stopping times of exponential functionals of Brownian martingales and a non-homogeneous heat semigroup. In particular we derive lower bounds for the probability of blowup in finite time, and we provide sufficient conditions for the existence of global positive solutions.

Some remarks on the paper “On the blow up criterion of 3D Navier–Stokes equations” by J. Benameur

We indicate some important simplifications and extensions of the analysis recently given by J. Benameur to derive blow-up estimates for strong solutions to 3D incompressible Navier–Stokes equations in homogeneous Sobolev spaces H˙s(R3), s>1/2, in case of finite-time existence.

On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.

Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations

In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation Δ∞u=b(x)f(u) in Ω, where Δ∞ is the ∞-Laplacian, the nonlinearity f is a positive, increasing function in (0,∞), and the weighted function b∈C(Ω¯) is positive in Ω and may vanish on the boundary. We first establish the exact boundary blow-up estimates with the first expansion when f is regularly varying at infinity with index p>3 and the weighted function b is controlled on the boundary in some manner. Furthermore, for the case of f(s)=sp(1+c˜g(s)), with the function g normalized regularly varying with index −q<0, we obtain the second expansion of solutions near the boundary. It is interesting that the second term in the asymptotic expansion of boundary blow-up solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the boundary blow-up problems involving the classical Laplacian.

On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation

We show a new lower bound on the $\dot{H}^{\frac{3}{2}}\left(\mathbb {T}^3\right)$Ḣ32T3 norm of a possible blow-up solution to the Navier-Stokes equation, and also comment on the extension of this result to the whole space. This estimate can be seen as a natural limiting result for Leray's blow-up estimates in $L^p\left(\mathbb {R}^3\right)$LpR3, 3 &amp;lt; p &amp;lt; ∞. We also show a lower bound on the blow-up rate of a possible blow-up solution of the Navier-Stokes equation in $\dot{H}^{\frac{5}{2}}\left(\mathbb {T}^3\right)$Ḣ52T3, and give the corresponding extension to the case of the whole space.

Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition

In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.

Nondegeneracy of blow-up points for the parabolic Keller–Segel system

This paper is concerned with the parabolic Keller–Segel system{ut=∇⋅(∇u−um∇v)in Ω×(0,T),Γvt=Δv−λv+uin Ω×(0,T), in a domain Ω of RN with N⩾1, where m,Γ>0, λ⩾0 are constants and T>0. When Ω≠RN, we impose the Neumann boundary conditions on the boundary. Under suitable assumptions, we prove the local nondegeneracy of blow-up points. This seems new even for the classical Keller–Segel system (m=1). Lower global blow-up estimates are also obtained. In the singular case 0<m<1, as a prerequisite, local existence and regularity properties are established.