Blow-up Result Research Articles

Curvature blow-up and peakons for the Camassa–Holm type equation

In this paper we are concerned with a periodic Camassa–Holm type equation, which includes several integrable systems as particular cases such as the Camassa–Holm equation, the Novikov equation and the modified Camassa–Holm equation. By applying the method of characteristics, the blow-up criterion and the precise blow-up scenario are established. Then the curvature blow-up result is presented. Finally, periodic peakons are expressed in an explicit function formula.

Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.

Blow-up in a $ p $-Laplacian mutualistic model based on graphs

<abstract><p>In this paper, we study a $ p\, $-Laplacian ($ p > 2 $) reaction-diffusion system based on weighted graphs that is used to describe a network mutualistic model of population ecology. After overcoming difficulties caused by the nonlinear $ p\, $-Laplacian, we develop a new strong mutualistic condition, and the blow-up properties of the solution for any nontrivial initial data are proved under this condition. In this sense, we extend the blow-up results of models with a graph Laplacian ($ p = 2 $) to a general graph $ p\, $-Laplacian.</p></abstract>

On the global existence and blow-up for the double dispersion equation with exponential term

<abstract><p>This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively.</p></abstract>

Existence of global solutions and blow-up results for a class of p(x)−Laplacian heat equations with logarithmic nonlinearity

This paper?s main objective is to examine an initial boundary value problem of a quasilinear parabolic equation of non-standard growth and logarithmic nonlinearity by utilizing the logarithmic Sobolev inequality and potential well method. Results of global existence, estimates of polynomial decay, and blowing up of weak solutions have been obtained under certain conditions that will be stated later. Our results extend those of a recent paper that appeared in the literature.

Global existence and blow-up of solution to a class of fourth-order equation with singular potential and logarithmic nonlinearity

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo–Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time.

Improvement on the blow-up for a weakly coupled wave equations with scale-invariant damping and mass and time derivative nonlinearity

An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of the blow-up region. Furthermore, the techniques used in this article may be extended to other systems and interestingly they simplify the proof of the blow-up result in [3] which is concerned with the single wave equation in the same context as in the present work.

Blow-up time analysis of parabolic equations with variable nonlinearities

The blow-up of a class of nondegenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient allowed us to discuss our blow-up results differently. We prove blow-up results and give a new lower and upper bounds of the blow-up time to a class of semilinear reaction-diffusion equations and a nonlinear equation governed by the -Laplacian, for .

On a strongly damped semilinear wave equation with time-varying source and singular dissipation

Abstract This paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.

Higher order evolution inequalities with convection terms in an exterior domain of [formula omitted

We establish new blow-up results for a higher order (in time) evolution inequality involving a convection term in an exterior domain of RN. We study two types of inhomogeneous boundary conditions: Dirichlet and Neumann. Using a unified approach, we obtain optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary problems. We also investigate the effect of a nonlinear memory term on the critical behaviors of the considered problems. Notice that no restriction on the sign of solutions is imposed.

A blow-up result for attraction-repulsion system with nonlinear signal production and generalized logistic source

This paper is devoted to studying the following chemotaxis system{ut=∇⋅(φ(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇ϖ)+λu(1−uα),x∈Ω,t>0,0=Δv−m1(t)+f1(u),m1(t)=1|Ω|∫Ωf1(u),x∈Ω,t>0,0=Δϖ−m2(t)+f2(u),m2(t)=1|Ω|∫Ωf2(u),x∈Ω,t>0,∂u∂ν=∂v∂ν=∂ϖ∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω, where Ω=BR(0)⊂Rn(n≥2) with R>0, ν denotes the outward unit normal vector on ∂Ω, χ,ξ,λ,α are positive constants, and φ(u),f1(u) and f2(u) are suitably regular functions satisfying φ(u)≤C0(1+u)−m, f1(u)≥k1(u+1)γ1 and f2(u)≤k2(u+1)γ2 for all u≥0 with C0,k1,k2,γ1,γ2>0 and m∈R. It is proved that if γ1>max⁡{α,γ2} and{γ1>2n(α+1),ifm≥0,γ1+m>2n(α+1),ifm<0, then there exists a suitable initial data u0 such that the corresponding solution (u,v,ϖ) of the system blows up in finite time. The results of this paper extend the blow-up criteria established in Liu et al. (2021) [28].

On the Cauchy problem for a weakly dissipative Camassa-Holm equation in critical Besov spaces

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces with and Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.

A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type

In this paper, we consider a weakly coupled system of a wave and damped Klein–Gordon equation with nonlinearities of derivative type. We prove a blow-up result for the Cauchy problem associated with this system for nonnegative and compactly supported data by means of an iteration argument.

Fourth Order Schrödinger Equation with Mixed Dispersion on Certain Cartan-Hadamard Manifolds

This paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on MN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^N$$\\end{document}, an N-dimensional Cartan-Hadamard manifold. Namely we consider 4NLSi∂tψ=-ΔM2ψ+βΔMψ+λ|ψ|2σψinR×M,ψ(0,·)=ψ0∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} i\\partial _t\\psi = -\\Delta _M^2\\psi +\\beta \\Delta _M \\psi +\\lambda |\\psi |^{2\\sigma }\\psi \\quad \ ext {in }\\mathbb {R}\ imes M,\\\\ \\psi (0,\\cdot )=\\psi _0\\in X, \\end{array}\\right. } \\end{aligned}$$\\end{document}where β≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\ge 0$$\\end{document}, λ={-1,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =\\{-1,1\\}$$\\end{document}, 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document}, ΔM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _M$$\\end{document} is the Laplace-Beltrami operator on M and X=L2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (M)$$\\end{document} or X=H2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (M)$$\\end{document}. At first, we focus on the case where M is the hyperbolic space HN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^N$$\\end{document}. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that N≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 4$$\\end{document} and 0<σ<4/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/N$$\\end{document} if X=L2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (\\mathbb {H}^N)$$\\end{document} or 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document} if X=H2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (\\mathbb {H}^N)$$\\end{document}. Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.

Nonexistence of global solutions for an inhomogeneous pseudo-parabolic equation

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity ut−kΔut−Δu=I0+γ(|u|p)+ω(x),(t,x)∈(0,∞)×RN,where p>1,k≥0, ω(x)≠0 and I0+γ is the left Riemann–Liouville fractional integral of order γ∈(0,1). Based on the test function method, we have proved the blow-up result for the critical case γ=0,p=pc for N≥3, which answers an open question posed by Zhou (2020), and in particular when k=0 it improves the result obtained by Bandle et al. (2000). An interesting fact is that in the case γ>0, the problem does not admit global solutions for any p>1 and ∫RNω(x)dx>0.

Sharp remainder of the Poincaré inequality for Baouendi–Grushin vector fields

In this note, we establish a sharp remainder formula for the Poincaré inequality for the Baouendi–Grushin vector fields. We give a simple proof for it without using the variational principle. As an application, we obtain a blow-up result for solutions to the Dirichlet initial-boundary value problem for the Baouendi–Grushin heat operator.

General decay and blow-up results for a class of nonlinear pseudo-parabolic equations with viscoelastic term

In this paper, we consider initial boundary value problem of the generalized pseudo-parabolic equation contain viscoelastic term. We establish firstly the local existence of solutions by Banach fixed point theorem. Then we prove blow-up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow-up rate for the weak solution via finding the upper bound and the lower bound for the blow-up times and the upper bound and the lower bound for the blow-up rate. For negative energy, we introduce a new method to prove blow-up results with sharper estimate for upper bound for the blow-up times. Finally, we prove the global existence of the solution and a general decay estimate under some suitable assumptions.

Finite time blow-up in the higher dimensional parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system

In this work, we mainly study nonnegative classical solutions to a Neumann initial-boundary value problem for the following parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded and smooth domain Ω⊂Rn(n≥1) with χ,ξ≥0, nonnegative initial data (u0,w0) and homogeneous Neumann boundary conditions.In this setup, we first show pure haptotaxis (χ=0) cannot induce any blow-up and pattern for any n≥1, showing haptotaxis is overbalanced by diffusion on boundedness and longtime behavior. Then, in the radial setting with Ω=BR⊂Rn(n≥3), it is well-known that small mass of ‖u0‖L1(Ω) can lead to blow-up in the corresponding Keller-Segel chemotaxis-only model obtained by setting w≡0, known as generic mass blow-up phenomenon [23]. Herein, in the presence of the temporal nonlocality brought by haptotaxis, we show, in an explicit realm of the form χ‖u0‖L1(Ω)>A(n,R)ξ‖w0‖L∞(Ω), that the aforementioned generic mass finite time blow-up phenomenon preserves. This seems to be the first rigorous blow-up result in relevant chemotaxis-haptotaxis models. The blow-up result also suggests that haptotatic cross-diffusion may not be negligible compared to chemotactic aggregation in ≥3D, in contrast to the recently detected 2D negligibility of haptotaxis in a minimal chemotaxis-haptotaxis model [16].

Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities

This paper is concerned with the Cauchy problem for semilinear wave equation with space-dependent scattering damping and combined nonlinearities. The blowup results of solution are established by introducing proper test functions. Moreover, upper bound lifespan estimates of a solution to the Cauchy problem with small initial values are derived. To the best of our knowledge, the results in Theorems 1.1–1.2 are new.

Lifespan estimates for local solutions to the semilinear wave equation in Einstein–de Sitter spacetime

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument, and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions.