Blow-up Points Research Articles

Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps

We study the dynamics of the family of rational maps of the form $$\begin{aligned} f_{d,{\uplambda }}(z)={\uplambda }\left( z+\frac{1}{z^{d-1}}\right) , \quad d \ge 3, \quad {\uplambda }\in \mathbb C{\setminus } \{0\}. \end{aligned}$$ Among other things, we show that the parameter planes for these maps contain infinitely many copies of the Mandelbrot set as well as infinitely many “blowup points”, i.e., parameters for which the critical orbits map to \(\infty \), so the Julia set is the entire sphere. Our efforts are aided by the useful observation that for fixed \(d \ge 3\), this family is conformally conjugate on the entire Riemann sphere to the family of relaxed Newton maps for \(p_d(z) = z^d-1\). The conjugacy allows us to move from one family to the other in order to find simpler proofs of our results, as well as establishing a dictionary of results from one family to the other.

A priori estimates and blow-up behavior for solutions of −Q N u = Ve u in bounded domain in ℝ N

Let $Q_{N}$ be $N$-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, $N$-Laplacian operator and anisotropic Laplacian operator. In this paper, we firstly obtain the properties for $Q_{N}$, which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of $-Q_{N}u=Ve^{u}$ in bounded domain in $\mathbb{R}^{N}$, $N\geq 2$ are established. Finally, the behavior of sole blow-up point is further considered.

Almost collapse mass quantization in 2D Smoluchowski–Poisson equation

We study Smoluchowski–Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation, several profiles of blowup solution have been noticed: blowup threshold on L1 norm of the initial value, finiteness of blowup points, formation of delta singularities called collapses, occurrence of type II blowup, exclusion of the boundary blowup point, and so forth. Here, we show the collapse mass quantization with possible residual terms. Copyright © 2015 John Wiley & Sons, Ltd.

Convergence rate, location and [formula omitted] condition for fully bubbling solutions to [formula omitted] Toda systems

It is well known that the study of SU(n+1) Toda systems is important not only to Chern–Simons models in Physics, but also to the understanding of holomorphic curves, harmonic sequences or harmonic maps from Riemann surfaces to CPn. One major goal in the study of SU(n+1) Toda system on Riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions. In this article we use a unified approach to study fully bubbling solutions to general SU(n+1) Toda systems and we prove three major sharp estimates important for constructing bubbling solutions: the closeness of blowup solutions to entire solutions, the location of blowup points and a ∂z2 condition.

Single blow-up point and critical speed for a parabolic problem with a moving nonlinear source on a semi-infinite interval

Let $v$ and $T$ be positive numbers, $D=\left ( 0,\infty \right )$, $\Omega =D\times \left ( 0,T\right ]$, and $\bar {D}$ be the closure of $D$. This article studies the first initial-boundary value problem, \[ \begin {array}{l} u_{t}-u_{xx}=\delta (x-vt)f\left ( u(x,t)\right ) \text { in }\Omega , \\ u(x,0)=\psi (x)\text { on }\bar {D}, \\ u(0,t)=0,u(x,t)\rightarrow 0\text { as }x\rightarrow \infty \text { for } 0<t\leq T, \end {array} \] where $\delta \left ( x\right )$ is the Dirac delta function, and $f$ and $\psi$ are given functions. It is shown that if the solution $u$ blows up in a finite time $t_{b}$, then it blows up only at the point $x=vt_{b}$. A criterion for $u$ to exist globally and a criterion for $u$ to blow up in a finite time are given. Furthermore, the problem is shown to have a critical speed $v^{\ast }$ of the moving nonlinear source such that no blowup occurs for $v\geq v^{\ast }$ and blowup occurs in a finite time for $v<v^{\ast }$. The formula for computing $v^{\ast }$ is also derived.

Boundary regularity for the supercritical Lane-Emden heat flow

We establish a Pacard-type monotonicity formula and Morrey bounds up to the boundary for smooth solutions of the Lane-Emden heat flow $$u_t-\Delta u = |u|^{p-2}u$$ on a general, smoothly bounded domain $$\Omega \subset \mathbb {R}^n$$ , $$n\ge 3$$ , for exponents $$p>2^*=2n/(n-2)$$ , extending our previous work on the problem. As a consequence we obtain partially regular, self-similar tangent maps at any first blow-up point of the flow, and partial regularity at the blow-up time if the energy is uniformly bounded from below.

Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

We study the long-time dynamics of complex-valued modified Korteweg-de Vries (mKdV) solitons, which are recognized because they blow-up in finite time. We establish stability properties at the H^1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H^1 stable, improving our previous result, where we only proved H^2 stability. The main new ingredient of the proof is the use of a B\"acklund transformation which links the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We also prove that negative energy breathers are asymptotically stable. Since we do not use any method relying on the Inverse Scattering Transformation, our proof works even under rough perturbations, provided a corresponding local well-posedness theory is available.

Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type

In the present paper, we deal with the degenerate parabolic system of porous medium type with non-linear diffusion m and interaction q in \({\rm{I\!R}}^N\) in the critical case of \({q =m+\frac{2}{N}}\). We establish the \({\varepsilon}\) -regularity theorem for weak solutions. As an application, the structure of asymptotics of blow-up solution is clarified. In particular, we show that the solution behaves like the δ-function at the blow-up points. Moreover, we prove that the number of blow-up points is finite, which can be controlled in terms of the mass of initial data. We also give a sharp constant for the \({\varepsilon}\) -regularity theorem.

Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation

We construct a solution to a complex nonlinear heat equation which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. We note that the real and imaginary parts of the constructed solution blow up simultaneously, with the imaginary part dominated by the real.

Sign-changing blow-up solutions for Hénon type elliptic equations with exponential nonlinearity

We study the existence of sign-changing solutions with multiple concentration to the following boundary value problem−Δu=ε2|x|2α(eu−e−u) in Ω,u=0 on ∂Ω, where α>0, Ω is a smooth bounded domain in R2 containing the origin, ε>0 is a small parameter. In particular we prove that if α≠1 then a nodal solution exists with a number of mixed positive and negative blow-up points up to 4.

Multiple blow-up solutions for an exponential nonlinearity with potential in [formula omitted

We study the following boundary value problem (0.1){Δu+λa(x)up−1eup=0,u>0inΩ;u=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, λ>0 is a small parameter, the function a(x)≥0 is a smooth potential, and the exponent p satisfies 0<p<2. We construct a family of solutions to problem (0.1) which blows up, as λ→0, at some points of Ω which stay outside the zero set of a(x). We relate the number of possible blow-up points with the zero set of a(x).

Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non-locally conformally flat manifolds

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.

Formation of singularities in one-dimensional Chaplygin gas

We investigate the formation and propagation of singularities for the system of one-dimensional Chaplygin gas. Under suitable assumptions we construct a physically meaningful solution containing a new type of singularities called "delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By a careful analysis, we study the behavior of the solution in a neighborhood of a blow-up point. The formation of this new kind of singularities is related to the envelop of different characteristic families, instead of characteristics of the same family in the standard situation. This shows that the blow-up phenomenon for systems with linearly degenerate characteristics is quite different from the problem of shock formation for the system with genuinely nonlinear characteristic fields. Different initial data can lead to different delta-like singularities: the delta-like singularity with point-shape and the delta-like singularity with line-shape.

Morse Indices of Multiple Blow-Up Solutions to the Two-Dimensional Gel'fand Problem

Blow-up solutions to the two-dimensional Gel'fand problem are studied. It is known that the location of the blow-up points of these solutions is related to a Hamiltonian function involving the Green function of the domain. We show that this implies an equivalence between the Morse indices of the solutions and the associated critical points of the Hamiltonian.

On the location of two blowup points on an annulus for the mean field equation

We consider the mean field equation on two-dimensional annular domains, and prove that if P1 and P2 are two blowup points of a blowing-up solution sequence of the equation, then we must have P1=−P2.

A local estimate for super-Liouville equations on closed Riemann surfaces

Continuing our work on the super-Liouville equations, a variational problem motivated by the supersymmetric extension of the Liouville functional in quantum field theory, we study the profile of blow-up solutions near the blow-up point and establish a local estimate for the bubbling sequences.

Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$

In this paper, we construct multivortex solutions of the elliptic governing equationfor the self-dual Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$when the Chern-Simons coupling parameter is sufficiently small, and thelocation of singular points satisfy suitable conditions.Our solutions show concentration phenomena at some points of the singular pointsas the coupling parameter tends to zero, and they are locally asymptotically radialnear each blow-up point.

Logarithmic scaling of the collapse in the critical Keller–Segel equation

A reduced Keller–Segel equation (RKSE) is a parabolic–elliptic system of partial differential equations which describes bacterial aggregation and the collapse of a self-gravitating gas of Brownian particles. We consider RKSE in two dimensions, where solution has a critical collapse (blow-up) if the total number of bacteria exceeds a critical value. We study the self-similar solutions of RKSE near the blow-up point. Near the collapse time, t = tc, the critical collapse is characterized by the L ∝ (tc − t)1/2 scaling law with logarithmic modification, where L is the spatial width of the collapsing solution. We develop an asymptotic perturbation theory for these modifications and show that the resulting scaling agrees well with numerical simulations. The quantitative comparison of the theory and simulations requires several terms of the perturbation series to be taken into account.

Single Point Blow-Up and Regional Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

Let. This article considers the following degenerate semilinear parabolic initial-boundary value problem, where be constants. We obtained the blow up set and find the conditions of single point blow-up.

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.