Blow-up Phenomena Research Articles

Cauchy problem for the Lane-Emden heat flow with sign-changing initial data

This article concerns the blow-up phenomenon of sign-changing solutions to the Lane-Emden heat flow. We construct sign-changing weak sub-solutions and localization of the positive and negative parts of the sign-changing solutions. We also extend the results to a nonlinear and finite diffusion equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/66/abstr.html

Modeling urbanization dynamics by labor force migration.

Individual participants in human society collectively exhibit aggregation behavior. In this study, we present a simple microscopic model of labor force migration based on the active Brownian particles framework. In particular, agent-based simulations show that the model produces clusters of agents from a random initial distribution. Furthermore, two empirical regularities called Zipf's and Okun's laws were observed. To reveal the mechanism underlying the reproduced aggregation phenomena, we use our microscopic model to derive an extended Keller-Segel system, which is a classic model describing the aggregation behavior of biological organisms called taxis. The obtained macroscopic system indicates that the concentration of the workforce in the real world can be explained through a new type of taxis central to human behavior, highlighting the relevance of urbanization to blow-up phenomena in the derived PDE system. We then characterize the transition between the aggregation and diffusion regimes both analytically and computationally. The predicted long-term dynamics of urbanization-originating in the asymmetric natures of employed and unemployed agents-are compared with global empirical data, particularly in the realms of labor statistics and urban indicators.

Movable Singularity of Some Hamiltonian Systems Associated with Blowup Phenomena

The motivation for this paper comes from a blowup problem for a semi-linear wave equation and a heat equation. Following an idea of J. Leray, we study a radially symmetric self-similar solution which has singularities on the characteristic cone. This naturally leads to the study of a Hamiltonian system called a profile equation. The novelty of this paper is that we focus on the movable singularity of the Hamiltonian system and we use Borel summability in constructing a singular solution. By “movable singularity” we mean that the singularity does not appear in the coefficients of the equation and depends on the respective solution. In the proof of our theorem we reduce the Hamiltonian system to a simpler form by a method similar to the so-called Birkhoff reduction. We obtain the parametrization of a singular solution by an elementary function. We also give applications to a semi-linear wave equation and a heat equation.

Suppression of blowup by slightly superlinear degradation in a parabolic–elliptic Keller–Segel system with signal-dependent motility

In this paper, we consider an initial–Neumann boundary value problem for a parabolic–elliptic Keller–Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension N≥2. In the current work, when N≤3, we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order slogs, unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works (Fujie and Jiang, 2020; Lyu and Wang, 2023) which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively.

Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs

Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs

Blow-Up Phenomena for a Non-Newton Filtration Equation with Local Linear Boundary Dissipation

In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of the blow-up solution with arbitrary high initial energy. Meanwhile, the upper and lower estimates of the blow-up time are discussed.

On the global existence of solutions to a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source

This paper is devoted to a chemotaxis system with signal-dependent motility under homogeneous Neumann boundary conditions in a bounded domain. We prove that this problem possesses a global classical solution which is uniformly bounded under weaker conditions than that obtained by Lv and Wang (2020). The findings of our study demonstrate the presence of a consistent decay rate, effectively ruling out the occurrence of blow-up phenomena in the system across all spatial dimensions.

Effects of multiplicative noise on the fractional Hartree equation

This paper is dedicated to radial solutions to the Cauchy problem for the fractional Hartree equation with multiplicative noise. First, we establish a stochastic Strichartz estimate related to the fractional Schrödinger propagator. Local well-posedness for the Cauchy problem is proved by using stochastic and radial deterministic Strichartz estimates. Then, based on Itô’s formula and stopping time arguments, the existence of a global solution is studied. Finally, we investigate the blow-up phenomenon and give a criterion via localized virial estimates.

Formation of Singularity for Isentropic Irrotational Compressible Euler Equations

The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper is concerned with the Cauchy problem of isentropic compressible Euler equations away from the vacuum. We use the integration method with the general test function f=f(r), proving that there exist the corresponding blowup results of C1 irrotational solutions for Euler equations and Euler equations with time-dependent damping in Rn (n≥2), provided the density-independent initial functional is sufficiently large. We also provide two simple and explicit test functions f(r)=r and f(r)=1+r, to demonstrate the blowup phenomenon in the one-dimensional case. In particular, our results are applicable to the non-radial system.

Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D

Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D

Blow-up for a Porous Medium Equation with Local Linear Boundary Dissipation

This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation. Making use of the concavity method, we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon. Meanwhile, we show the existence of the finite time blow-up solutions for arbitrarily high initial energy. Finally, we derive the life span bounds (i.e., the lower and upper bounds of the blow-up time).

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.

Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup

In this work, we explore the use of several discontinuous Galerkin (DG) methods for simulating soliton dynamics in the generalized Korteweg-de Vries (gKdV) equation. The presence of high-order nonlinearity in this model can lead to the finite-time blowup phenomenon, which challenges every aspect of a numerical scheme. The considered DG schemes encompass popular methods derived from the energy and/or Hamiltonian conservations of the gKdV equation. To evaluate the performance of these DG schemes from various angles, we present a series of numerical experiments. Through comprehensive comparisons, we aim to identify the scheme that exhibits the best performance. To further enhance the accuracy and efficiency of DG methods, particularly in the context of blowup simulations, we incorporate the arbitrary Lagrangian-Eulerian method for adaptive mesh movement.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point

Abstract A singular nonlinear differential equation z σ d w d z = a w + z w f ( z , w ) , {z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ > 1 \sigma \gt 1 , is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C {\mathbb{C}} if σ \sigma is a natural number, in a Riemann surface of a rational function if σ \sigma is a rational number, or in the Riemann surface of logarithmic function if σ \sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a ∈ C ⧹ { 0 } a\in {\mathbb{C}}\setminus \left\{0\right\} , and that the function f f is analytic in a neighbourhood of the origin in C × C {\mathbb{C}}\times {\mathbb{C}} . Considering σ \sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w ( z ) w=w\left(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C {\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z → 0 w ( z ) = 0 {\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w ( z ) w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

Nonlinear logarithmic wave equations: Blow-up phenomena and the influence of fractional damping, infinite memory, and strong dissipation

Nonlinear logarithmic wave equations: Blow-up phenomena and the influence of fractional damping, infinite memory, and strong dissipation

On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>

Blow-up phenomena for pseudo-parabolic Kirchhoff equations with logarithmic nonlinearity

Blow-up phenomena for pseudo-parabolic Kirchhoff equations with logarithmic nonlinearity

Blow-up phenomena for a singular nonlocal viscoelastic problem with logarithmic nonlinearity

This article deals with the blow-up phenomena for a singular nonlocal viscoelastic equation with logarithmic nonlinearity subject to nonlocal boundary conditions u(α,t)=0 and ∫0αxu(x,t)dx=0. Under the three different initial energy levels, we established the finite time blow-up criterion, and then give the upper bounds for blow-up time T∗ depending on the sign and range of initial energy E(0).

Singularities in finite time of the full compressible Euler equations in [formula omitted

The present article is concerned with the study of blow-up phenomena of the smoooth solutions for the full compressible Euler equations in Rd,d≥1, which have always been a great concern to physicists and mathematicians throughout history. The approach is to construct exact explicit function to study singularities in finite time of the spherically symmetric solutions, provided the initial data satisfy some conditions. Compared with the results obtained by Sideris in 1984 (CMP), in this article, the initial velocity field is not required to have a compact support and the initial density and entropy is not equal to a constant outside the support of the initial velocity field.