Blow-up Points Research Articles

Feedback controllability for blowup points of semilinear heat equations

This paper studies a controllability problem for blowup points of two classes of semilinear heat equations.Our goal to act controls on the systems we studied is to make the corresponding solutions blow upat given points. This differs with the controllability problem of equations with the property of blowup in the references, where the purpose of using controls is to prevent blowupby controls. We obtain the feedback controls for our controllability problem of blowup points.

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem $$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}} \text { in } (M,g) . $$ We prove that for any k ∈ ℕ, there exists ek > 0 such that for all e ∈ (0, ek) the problem (P𝜖) has a symmetric solution ue, which looks like the superposition of k positive bubbles centered at the point ξ0 as e → 0. In particular, ξ0 is a towering blow-up point.

Nodal Blow-Up Solutions to Slightly Subcritical Elliptic Problems with Hardy-Critical Term

Abstract The paper is concerned with the slightly subcritical elliptic problem with Hardy-critical term { - Δ ⁢ u - μ ⁢ u | x | 2 = | u | 2 ∗ - 2 - ε ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω $\left\{\begin{aligned} \displaystyle-\Delta u-\mu\frac{u}{|x|^{2}}&% \displaystyle=|u|^{2^{\ast}-2-\varepsilon}u&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega\end{% aligned}\right.$ in a bounded domain Ω ⊂ ℝ N ${\Omega\subset\mathbb{R}^{N}}$ with 0 ∈ Ω ${0\in\Omega}$ , in dimensions N ≥ 7 ${N\geq 7}$ . We investigate the possible blow-up behavior of solutions as μ , ε → 0 ${\mu,\varepsilon\to 0}$ . In particular, we prove the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ , ε → 0 + ${\mu,\varepsilon\to 0^{+}}$ . The location of the negative blow-up point is determined by the geometry of Ω. Moreover, the asymptotic shape of the solutions is described in detail. An interesting new consequence of our results is that the type of blow-up solutions considered here exists for μ = O ⁢ ( ε α ) ${\mu=O(\varepsilon^{\alpha})}$ with α > N - 4 N - 2 ${\alpha>\frac{N-4}{N-2}}$ . The bound N - 4 N - 2 ${\frac{N-4}{N-2}}$ is sharp.

Effective cones of cycles on blowups of projective space

In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$, $n>4$, at $r$ very general points, the cone of divisors is not finitely generated as soon as $r> n+3$, whereas the cone of curves is generated by the classes of lines if $r \leq 2^n$. In fact, if $X_r^n$ is a Mori Dream Space then all the effective cones of cycles on $X_r^n$ are finitely generated.

Blow-up behavior for a degenerate elliptic $$\sinh $$ sinh -Poisson equation with variable intensities

In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure \(\mathcal P\) defined on the interval \([-1,1]\). Under Dirichlet boundary conditions we establish the exclusion of boundary blow-up points, we show that the concentration mass does not have residual \(L^1\)-terms (“residual vanishing”) and we determine the location of blow-up points in terms of Kirchhoff’s Hamiltonian. We allow \(\mathcal P\) to be a general Borel measure, which could be “degenerate” in the sense that \(\mathcal P(\{\alpha _-^*\})=0=\mathcal P(\{\alpha _+^*\})\), where \(\alpha _-^*=\min \mathrm {supp}\mathcal P\) and \(\alpha _+^*=\max \mathrm {supp}\mathcal P\). Our main results are new for the standard sinh-Poisson equation as well.

On condensate of solutions for the Chern–Simons–Higgs equation

This is the first part of our comprehensive study on the structure of doubly periodic solutions for the Chern–Simons–Higgs equation with a small coupling constant. We first classify the bubbling type of the blow-up point according to the limit equations. Assuming that all the blow-up points are away from the vortex points, we prove the non-coexistence of different bubbling types in a sequence of bubbling solutions. Secondly, for the CS type bubbling solutions, we obtain an existence result without the condition on the blow-up set as in [4]. This seems to be the first general existence result of the multi-bubbling CS type solutions which is obtained under nearly necessary conditions. Necessary and sufficient conditions are also discussed for the existence of bubbling solutions blowing up at vortex points.

On the existence and blow-up of solutions for a mean field equation with variable intensities

We study an elliptic problem with exponential nonlinearities describing the statistical mechanics equilibrium of point vortices with variable intensities. For suitable values of the physical parameters we exclude the existence of blow-up points on the boundary, we prove a mass quantization property and we apply our analysis to the construction of minimax solutions.

Estimates of the mean field equations with integer singular sources: Non-Simple blowup

Let $M$ be a compact Riemann surface, $\alpha j \gt -1$, and $h(x)$ a positive $C^2$ function of $M$. In this paper, we consider the following mean field equation: \[ \Delta u (x) + \rho \left( \frac{h(x) e^{u(x)}}{\int_{M} h (x) e^{u(x)}} - \frac{1}{\lvert M \rvert} \right) = 4\pi \sum^{d}_{j=1} \alpha_j \left( \delta_{q_j} - \frac{1}{\lvert M \rvert} \right) \textrm{ in } M \textrm{.} \] We prove that for $\alpha_j \in \mathbb{N}$ and any $\rho \gt \rho_0$, the equation has one solution at least if the Euler characteristic $\chi (M) \leq 0$, where $\rho_0 = \underset{M}{\max} ( 2K - \Delta \mathrm{ln} h + N^*)$, $K$ is the Gaussian curvature, and $N^* = 4\pi \sum^{d}_{j=1} \alpha_j$. This result was proved in [10] when $\alpha_j = 0$. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex $q_j$. In the case where $\alpha_j \notin \mathbb{N}$, the sharp estimate of solutions near $q_j$ has been obtained in [11]. However, if $\alpha_j \in \mathbb{N}$, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.

Some results of algebraic geometry over Henselian rank one valued fields

We develop geometry of affine algebraic varieties in K^{n} over Henselian rank one valued fields K of equicharacteristic zero. Several results are provided including: the projection K^{n} times {mathbb {P}}^{m}(K) rightarrow K^{n} and blowups of the K-rational points of smooth K-varieties are definably closed maps; a descent property for blowups; curve selection for definable sets; a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the K-topology; and extending continuous hereditarily rational functions, established for the real and p-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field {mathbb {R}} was developed by Fichou–Huisman–Mangolte–Monnier. The main results here are regulous versions of Nullstellensatz and Cartan’s theorems A and B.

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.

Continuous functions on the plane regular after one blowing-up

We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, we prove a Positivstellensatz without denominator in this ring.

Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result

The pioneering work by Brézis–Merle [3] applied to mean-field equations of Liouville type (1) (see below) implies that any unbounded sequence of solutions (i.e. a sequence of blow-up solutions) must exhibit only finitely many points (blow-up points) around which their “mass” concentrate. In this note, we describe some examples of blow-up solutions that violate such conclusion, in the sense that their mass may spread, as soon as we consider situations which mildly depart from Brézis–Merle's assumptions. The presence of a “residual” mass in blow-up phenomena was pointed out by Ohtsuka–Suzuki in [12], although such possibility was not substantiated by any explicit examples. We mention that for systems of Toda-type, this new phenomenon occurs rather naturally and it makes the calculation of the Leray Schauder degree much harder than the resolution of the single mean-field equation.

Existence and nonexistence of sign-changing solutions to elliptic critical equations

We consider the nonlinear equation $ -\Delta u = u ^{p-1}u -\varepsilon u \quad \mbox{in } \Omega , u =0 \quad \mbox{on } \partial \Omega ,$ where $\Omega $ is a smooth bounded domain in $\mathbb{R}^n$, $n \geq 4$, $ \varepsilon$ is a small positive parameter, and $p=(n+2)/(n-2)$. We study the existence of sign-changing solutions that concentrate at some points of the domain. We prove that this problem has no solutions with one positive and one negative bubble. Furthermore, for a family of solutions with exactly two positive bubbles and one negative bubble, we prove that the limits of the blow-up points satisfy a certain condition.

Blowup analysis for a nonlinear equation with negative exponent

In this paper we study the blowup behavior of the minimizing sequence to certain nonlinear elliptic equation with negative critical exponents. We analyze the location for blowup point and the asymptotic behavior of solutions near the blowup point.

On the Blow-Up of Solutions to Liouville-Type Equations

Abstract We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.

Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile

We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, 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. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.

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.