Blow-up Rate Research Articles

Large harmonic functions for fully nonlinear fractional operators

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain Ω ⊂ R N . We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case ( − Δ ) s u = 0 in Ω , where ( − Δ ) s denotes the fractional Laplacian of order 2 s ∈ ( 0 , 2 ) . We use the viscosity solution’s theory and Perron’s method to construct harmonic functions with zero exterior condition in Ω c and a boundary blow-up profile lim x → x 0 , x ∈ Ω dist ( x , ∂ Ω ) 1 − s u ( x ) = h ( x 0 ) , for all x 0 ∈ ∂ Ω , for any given boundary data h ∈ C ( ∂ Ω ) . Our method allows us to provide a blow-up rate for the gradient of the solution.

Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping R2 to S2 with arbitrary given coefficients ρ1∈R,ρ2>0. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.

The Asymptotic Behavior and Blow-Up Rate of a Solution with a Lower Bound on the Highest Existence Duration for Semi-Linear Pseudo-Parabolic Equations

This note addresses the initial-boundary value problem for a class of semi-linear pseudo-parabolic equations defined on a smooth bounded domain, with an emphasis on determining the asymptotic behavior and blow-up rate of the solution. Our analysis considers both low-initial energy and critical-initial energy cases, with a specific focus on establishing a lower bound on the maximal existence time of the solutions to this problem.

Canonical surgeries in rotationally invariant Ricci flow

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of Kleiner and Lott [Acta Math. 219 (2017), pp. 65–134]. Results of Bamler-Kleiner [Acta Math. 228 (2022), pp. 1–215] and Haslhofer [Proc. Amer. Math. Soc. 150 (2022), pp. 5433–5437] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source

Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source

Concentration of blow-up solutions for the Gross-Pitaveskii equation

Abstract We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 {L}^{2} supercritical case. Finally, in the L 2 {L}^{2} critical case, a delicate limit of blow-up solutions is analyzed.

Optimal Lower Bound for the Blow-Up Rate of the Magnetic Zakharov System Without the Skin Effect

Optimal Lower Bound for the Blow-Up Rate of the Magnetic Zakharov System Without the Skin Effect

Finite point blowup for the critical generalized Korteweg-de Vries equation

In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation. In this article, we answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate. Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.

On Fila-King conjecture in dimension four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation{ut=Δu+u3 in R4×(0,∞),u(x,0)=u0(x) in R4. We construct a positive infinite time blow-up solution u(x,t) with the blow-up rate ‖u(⋅,t)‖L∞(R4)∼ln⁡t as t→∞ and establish the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King [15, Conjecture 1.1].

Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

We study the insulated conductivity problem with inclusions embedded in a bounded domain in \mathbb{R}^{n} . The gradient of solutions may blow up as \varepsilon , the distance between inclusions, approaches to 0 . It was known that the optimal blow-up rate in dimension n = 2 is of order \varepsilon^{-1/2} . It has recently been proved that in dimensions n \ge 3 , an upper bound of the gradient is of order \varepsilon^{-1/2 + \beta} for some \beta > 0 . On the other hand, optimal values of \beta have not been identified. In this paper, we prove that when the inclusions are balls, the optimal value of \beta is [-(n-1)+\sqrt{(n-1)^{2}+4(n-2)}]/4 \in (0,1/2) in dimensions n \ge 3 .

Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

For any manifold N p N^p admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products M = N p × S q + 1 M = N^p \times S^{q+1} with doubly warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any k > 1 k > 1 there exists a solution with curvature blow-up rate ‖ R m ‖ ∞ ( t ) ≳ ( T − t ) − k \| Rm \|_{\infty } (t) \gtrsim (T-t)^{-k} with singularity modeled on a Ricci-flat cone at parabolic scales.

Existence, Blow-Up, and Blow-Up Rate of Weak Solution to Fourth-Order Non-Newtonian Polytropic Variation-Inequality Arising from Consumption-Investment Models

This article obtains the conditions for the existence and nonexistence of weak solutions for a variation-inequality problem. This variational inequality is constructed by a fourth-order non-Newtonian polytropic operator which is receiving much attention recently. Under the proper condition of the parameter, the existence of a solution is proved by constructing the initial boundary value problem of an ellipse by time discretization and some elliptic equation theory. Under the opposite parameter condition, we analyze the nonexistence of the solution. The results show that the weak solution will blow up in finite time. Finally, we give the blow-up rate and the upper bound of the blow-up time.

Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow

In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].

Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

<abstract><p>In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.</p></abstract>

Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds

Abstract In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

Singular set and curvature blow-up rate of the level set flow

Abstract Under certain conditions such as the 2-convexity, a singularity of the level set flow is of type I (in the sense that the rate of curvature blow-up is constrained before and after the singular time) if and only if the flow shrinks to either a round point or a C 1 {C^{1}} curve near that singular point. Analytically speaking, the arrival time is C 2 {C^{2}} near a critical point if and only if it satisfies a Łojasiewicz inequality near the point.

Optimizing brewing beer production using Aspergillus oryzae solid-state fermentation of sorghum koji as an adjunct

ABSTRACT This study employed solid-state fermentation (SSF) to substitute 30% barley malt with Kinmen waxy sorghum and Australian non-waxy sorghum inoculated with Aspergillus oryzae for koji preparation in the production of sorghum beer. Optimal conditions for koji preparation were determined, and various beer formulations were developed based on Kinmen sorghum koji. The optimal soaking time for Kinmen waxy and Australian non-waxy sorghum was 2.5 h at room temperature with a water absorption rate of 35%. After steaming, Kinmen sorghum and Australian sorghum exhibited blowup rates of 75.33% and 85.67%, respectively. Five beer types were fermented for 12 days using different mashing methods. Glucose content ranged from 9.93 to 17.6 g/L on the first day, decreasing to 0.03 g/L by the third day. On the 12th day, the alcohol content peaked at 3.03–4.39%, with a pH range of 4.3–5.7. Aroma analysis identified higher alcohols, such as ethanol, 3-methyl-1-butanol, and 2-phenyl ethanol, as the main contributors, with 2-phenyl ethanol imparting a solid rose flavor and ethanol providing a hop aroma and mild taste. Given the global competitiveness of the beer industry, our research emphasizes the use of Kinmen sorghum-based malt, offering unique flavors and a rich aroma, presenting significant growth potential in the market.

Blow-up of solutions to a scalar conservation law with nonlocal source arising in radiative gas

In this paper we consider a scalar conservation law with nonlocal source arising in radiative gas. We give a sufficient condition to assure that the solution blows up in a finite time. Moreover, the estimates of life span and blow-up rate are derived.

Localized Quantitative Estimates and Potential Blow-Up Rates for the Navier–Stokes Equations

Localized Quantitative Estimates and Potential Blow-Up Rates for the Navier–Stokes Equations