Blow-up Points Research Articles

Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent

In this paper, we deal with a time-weighted parabolic system coupled via nonlocal sources. Firstly, we give the critical Fujita exponent of solutions. Secondly, we propose the time-weighted blow-up profiles of solutions. Thirdly, the simultaneous and non-simultaneous blow-up of solutions are discussed under some conditions on the initial data. Fourthly, besides blow-up rates, we classify total blow-up and single point blow-up of solutions. At last, upper and lower bounds of blow-up time are studied according to precise blow-up phenomena.

Extremely low order time-fractional differential equation and application in combustion process

Fractional blow-up model, especially which is of very low order of fractional derivative, plays a significant role in combustion process. The order of time-fractional derivative in diffusion model essentially distinguishes the super-diffusion and sub-diffusion processes when it is relatively high or low accordingly. In this paper, the blow-up phenomenon and condition of its appearance are theoretically proved. The blow-up moment is estimated by using differential inequalities. To numerically study the behavior around blow-up point, a mixed numerical method based on adaptive finite difference on temporal direction and highly effective discontinuous Galerkin method on spatial direction is proposed. The time of blow-up is calculated accurately. In simulation, we analyze the dynamics of fractional blow-up model under different orders of fractional derivative. It is found that the lower the order, the earlier the blow-up comes, by fixing the other parameters in the model. Our results confirm the physical truth that a combustor for explosion cannot be too small.

Blow-up analysis for Boltzmann–Poisson equation in Onsager's theory for point vortices with multi-intensities

In this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one-sided Borel probability measure. If such a sequence blows up, we derive some estimate which is related to the behavior of solution near the blow-up point. Moreover, we study the two-intensities case to consider the sufficient condition for this estimate. Our main results are new for the standard mean field equation as well.

Quantization of the Blow-Up Value for the Liouville Equation with Exponential Neumann Boundary Condition

In this paper, we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition. In particular, we will obtain a sharp mass quantization result for the solution sequences at a blow-up point.

On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(\sigma_1,\sigma_2)$ at each blowup point has the expression $$\sigma_i=2(N_{i1}\mu_1+N_{i2}\mu_2+N_{i3}),$$ where $N_{ij}\in\mathbb{Z},~i=1,2,~j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $(\alpha_1^t,\alpha_2^t)$ are integers and $\rho_i\notin 4\pi\mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\log|x-x^k|\leq C,$$ where $x^k$ is the local maximum point of $u^k$ near $q$. (iv) If the blow up point $q$ is a vortex point $p_t$ and $\alpha_t^1,\alpha_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\log|x-p_t|\leq C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.

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.

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

We construct a solution for the Complex Ginzburg-Landau equation in some critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. 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.

The nonequivariant coherent-constructible correspondence for toric surfaces

We prove the nonequivariant coherent-constructible correspondence conjectured by Fang–Liu–Treumann–Zaslow in the case of toric surfaces. Our proof is based on describing a semi-orthogonal decomposition of the constructible side under toric point blow-up and comparing it with Orlov’s theorem.

Sharp estimates for solutions of mean field equations with collapsing singularity

ABSTRACTThe pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26], and Bartolucci-Tarantello [3] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.

Refined estimates for simple blow-ups of the scalar curvature equation on $S^n$

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

Blow-up solutions for L2 supercritical gKdV equations with exactly k blow-up points

In this paper we consider the slightly L2-supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2-supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.

Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

We consider the following exponential reaction–diffusion equation involving a nonlinear gradient term:∂tU=ΔU+α|∇U|2+eU,(x,t)∈RN×[0,T),α>−1. We construct for this equation a solution which blows up in finite time T>0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some sense, resulting in the change of the final blowup profile in comparison with the case α=0. The proof of the construction is inspired by the method of Merle and Zaag in 1997. It relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations in the initial data.

Boundary bubbling solutions for a supercritical Neumann problem with mixed nonlinearities

In this paper, we investigate the existence of boundary bubbling solutions for a supercritical Neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary ∂Ω, the behavior of the solutions is as a tower of k bubbles when the exponent goes to Sobolev critical exponent, and the blow-up point is a critical point of the mean curvature.

Fano 4-folds, flips, and blow-ups of points

We study smooth, complex Fano 4-folds X with large Picard number ρX, with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then ρX≤12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case ρX>6, in which we give some general results of independent interest.

A remark on a result of Ding-Jost-Li-Wang

Let ( M , g ) (M,g) be a compact Riemannian surface without boundary, W 1 , 2 ( M ) W^{1,2}(M) be the usual Sobolev space, and J : W 1 , 2 ( M ) → R J: W^{1,2}(M)\rightarrow \mathbb {R} be the functional defined by \[ J ( u ) = 1 2 ∫ M | ∇ u | 2 d v g + 8 π ∫ M u d v g − 8 π log ⁡ ∫ M h e u d v g , J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^udv_g, \] where h h is a positive smooth function on M M . In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997), Ding, Jost, Li and Wang obtained a sufficient condition under which J J achieves its minimum. In this note, we prove that if the smooth function h h satisfies h ≥ 0 h\geq 0 and h ≢ 0 h\not \equiv 0 , then the above result still holds. Our method is to exclude blow-up points on the zero set of h h .

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities: $$ \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitz\'eica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result.

On the profile of energy concentration at blow-up points for subconformal focusing nonlinear waves

We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the H 1 H^1 -norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this H 1 H^1 -energy can be distributed inside such cones. Our proof relies on new localized estimates—obtained using Carleman-type inequalities—for such nonlinear waves. These bound the L p + 1 L^{p+1} -norm in the interior of timelike cones by their H 1 H^1 -norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones.

Refined Regularity of the Blow-Up Set Linked to Refined Asymptotic Behavior for the Semilinear Heat Equation.

We consider , a solution of which blows up at some time , where , and . Define to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an -dimensional continuum for some , then S is in fact a manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable and reach significant small terms in the polynomial order for some . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.