Blow-up Solutions Research Articles

Global vs Blow-Up Solutions and Optimal Threshold for Hyperbolic ODEs with Possibly Singular Nonlinearities

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbations or large initial values, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold.

On Blow-up and Global Existence of Weak Solutions to Cauchy Problem for Some Nonlinear Equation of the Pseudoparabolic Type

On Blow-up and Global Existence of Weak Solutions to Cauchy Problem for Some Nonlinear Equation of the Pseudoparabolic Type

Erratum by editorial office: Minimal mass blow-up solutions for nonlinear Schrödinger equations with a potential (Tohoku Math.J. 75 (2023), 215--232)

Erratum by editorial office: Minimal mass blow-up solutions for nonlinear Schrödinger equations with a potential (Tohoku Math.J. 75 (2023), 215--232)

Delayed blow-up and enhanced diffusion by transport noise for systems of reaction–diffusion equations

This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the Lp(Lq)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(L^q)$$\\end{document}-approach to stochastic PDEs, highlighting new connections between the two areas.

Well-Posedness and Blow-Up of Solutions for a Variable Exponent Nonlinear Petrovsky Equation

Well-Posedness and Blow-Up of Solutions for a Variable Exponent Nonlinear Petrovsky Equation

Life span of blowing-up solutions to the Cauchy problem for a time-fractional Schrödinger equation

Life span of blowing-up solutions to the Cauchy problem for a time-fractional Schrödinger equation

Blow-up solution to an abstract non-local stochastic heat equation with Lévy noise

This paper investigates explosive solutions to a class of non-local stochastic heat equations driven by Lévy noise. We establish sufficient conditions to show that the positive solution will blow-up in finite time.

Global existence of solutions in two-species chemotaxis system with two chemicals with sub-logistic sources in 2d

This paper investigates the global existence and boundedness of solutions in a two-species chemotaxis system with two chemicals and sub-logistic sources. The presence of a sub-logistic source in only one cell density equation effectively prevents the occurrence of blow-up solutions, even in fully parabolic chemotaxis systems.

Blow-Up of Solution of Lamé Wave Equation with Fractional Damping and Logarithmic Nonlinearity Source Terms

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Beam vibrations with allowance for deformation effects in the transverse direction are modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem is investigated in the space of continuous functions. Conditions for solution blow-up on a finite time interval are considered.

Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

This article is concerned with the initial-value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass in the L 2 -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criterion in the L 2 -critical and L 2 -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions in the L 2 -critical case. The main ingredients of the proofs are the variational characterisation of the ground state, a suitably refined compactness lemma, and scaling techniques. Our conclusions extend and compensate for some previous results.

B-methods for the numerical solution of evolution problems with blow-up solutions part II: Splitting methods

B-methods are numerical methods which are especially tailored to solve non-linear partial differential equation that have blow up solutions. We have presented in Part I a systematic construction of B-methods based on the variation of constants formula. Here, we use splitting methods as a second way to construct B-methods, and we prove several special properties of such methods. We illustrate our analysis with numerical experiments.

Infinite-time blowup and global solutions for a semilinear Klein–Gordan equation with logarithmic nonlinearity

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein–Gordon equation characterized by logarithmic nonlinearity, specifically in the form . Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions and in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein–Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations.

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations

Abstract The Keller–Segel–Navier–Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller–Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.

A Priori Estimates for Finite-Energy Sign-Changing Blowing-Up Solutions of Critical Elliptic Equations

Abstract We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger–Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^{0}$-theory for positive solutions of Schrödinger–Yamabe-type equations. To deal with the sign-changing case, we develop a method of proof that combines an a priori bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp a priori blow-up estimates for a linear problem.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

Finite-time blow-up of weak solutions to a chemotaxis system with gradient dependent chemotactic sensitivity

This paper deals with the parabolic–elliptic chemotaxis system with gradient dependent chemotactic sensitivity,{ut=Δu−χ∇⋅(u|∇v|p−2∇v),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0, where Ω:=BR(0)⊂Rn(n≥2) is a ball with some R>0, χ>0, p∈(nn−1,∞) and μ:=1|Ω|∫Ωu0, where u0 is an initial datum of arbitrary size. In the case that p∈(1,nn−1), Negreanu and Tello (2018) [23] established global existence and uniform boundedness of solutions, whereas when p∈(nn−1,2), Tello (2022) [31] showed that solutions blow up in finite time under the condition that μ>6 and χ is large enough. These works imply that the number p=nn−1 certainly plays the role of a critical blow-up exponent, and it is expected that when p>nn−1, for arbitrary μ>0 the system admits at least one solution which blows up in finite time. The purpose of this paper is to prove that this conjecture is true within a framework of weak solutions with a moment inequality.

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.

Simple blow-up solutions of singular Liouville equations

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.