Finite Time Blow-up Research Articles

On the blow-up for a Kuramoto–Velarde type equation

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters γ1 and γ2 involved in the non-linear terms verify γ2=γ12 or γ2=0. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework γ2≠γ12 and γ2≠0, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm. As a complement to it, we address an alternative result on the finite-time blow-up of smooth solutions by considering a virial-type estimate.

Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,p}$$\\end{document}-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations

On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations

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.

Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

Finite time blowup of solutions of the hydrostatic Euler equations

Finite time blowup of solutions of the hydrostatic Euler equations

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

Boundedness and finite-time blow-up in a Keller–Segel chemotaxis-growth system with flux limitation

Boundedness and finite-time blow-up in a Keller–Segel chemotaxis-growth system with flux limitation

Different aspects of blow-up property for a nonlinear wave equation

In this work, we are concerned with a nonlinear wave equation with variable exponents. In the presence of the logarithmic nonlinear source, we established a global nonexistence result with negative initial data and without imposing the Sobolev Logarithmic Inequality. The blow-up time is established with upper bound and lower bound. In addition, under some conditions on the initial data and for a specific class of relaxation functions, we established an infinite time blow-up result.

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.

Existence of global and explosive mild solutions of fractional reaction–diffusion system of semilinear SPDEs with fractional noise

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction–diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by [Formula: see text] for [Formula: see text], along with [Formula: see text] where [Formula: see text] is the fractional power [Formula: see text] of the Laplacian, [Formula: see text] and [Formula: see text] and [Formula: see text] are real constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that [Formula: see text] with Hurst index [Formula: see text] we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solution to our considered system.

Prevention of infinite-time blowup by slightly super-linear degradation in a Keller–Segel system with density-suppressed motility

An initial-Neumann boundary value problem for a Keller–Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension N⩾2 . In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of slog⁡s at most, the classical solution is uniformly-in-time bounded when N⩽3 , thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equationi∂tu−(−i∇+α|x|2(−x2,x1))2u=±|x|−ϱ|u|p−1u,(t,x)∈R×R2, where α∈R∖Z,ϱ>0,p>1. We establish a dichotomy between global existence and scattering versus finite-time blow-up for energy solutions below the ground state threshold in the inter-critical regime. The scattering result is obtained by employing the novel approach developed by Dodson and Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. (2017)). We employ Tao's scattering criteria and Morawetz estimates as the foundation of this method. The novelty of our approach lies in two aspects: firstly, we explore the case of nonzero ϱα, and secondly, we consider general energy initial data that need not be radially symmetric. The specific instance of α=0, known as INLS, has garnered significant attention in recent years. Additionally, X. Gao and C. Xu recently investigated the case of ϱ=0 in the homogeneous regime, establishing scattering theory for spherically symmetric data in their work (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. (2020)). In the radial framework, the aforementioned problem can be translated to the INLS with an inverse square potential, which has been extensively studied in higher-dimensional spaces. The Hardy inequality ‖|x|−1f‖L2(RN)≤2N−2‖∇f‖L2(RN), which establishes the norm equivalence ‖f‖H1≃‖f‖H1+‖|x|−1f‖L2, does not hold in two space dimensions. Consequently, it remains unclear how to handle the NLS with an inverse square potential in the H1 framework for two-dimensional space. This article stands out as the first one to address the NLS with Aharonov-Bohm magnetic potential within the inhomogeneous regime, specifically when ϱ≠0.

Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations

Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations

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.

Blow-up of solutions for a system of nonlocal singular viscoelastic equations with sources and distributed delay terms

In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution.

Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system We consider the critical mass case ∫R2u0(x)dx=8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{{\\mathbb {R}}^2} u_0(x)\\, \ extrm{d}x = 8\\pi $$\\end{document}, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} with mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document} such that for any initial condition u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document} sufficiently close to u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} and mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document}, the solution u(x, t) of (∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) is globally defined and blows-up in infinite time. As t→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow +\\infty $$\\end{document} it has the approximate profile u(x,t)≈1λ2(t)Ux-ξ(t)λ(t),U(y)=8(1+|y|2)2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u(x,t) \\approx \\frac{1}{\\lambda ^2(t)} U\\left( \\frac{x-\\xi (t)}{\\lambda (t)} \\right) , \\quad U(y)= \\frac{8}{(1+|y|^2)^2}, \\end{aligned}$$\\end{document}where λ(t)≈clogt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (t) \\approx \\frac{c}{\\sqrt{\\log t}}$$\\end{document}, ξ(t)→q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi (t)\\rightarrow q$$\\end{document} for some c>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c>0$$\\end{document} and q∈R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in {\\mathbb {R}}^2$$\\end{document}. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Abstract In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.