Blow-up Phenomena Research Articles

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

Some special self-similar solutions for a model of inviscid liquid-gas two-phase flow

In this article, we are concerned with analytical solutions for a model of inviscid liquid-gas two-phase flow. On the basis of Yuen’s works [25, 27–29] on self-similar solutions for compressible Euler equations, we present some special self-similar solutions for a model of inviscid liquid-gas two-phase flow in radial symmetry with and without rotation, and in elliptic symmetry without rotation. Some blowup phenomena and the global existence of the solutions obtained are classified.

Blow-up phenomena for a Kirchhoff-type parabolic equation with logarithmic nonlinearity

A Kirchhoff-type parabolic equation with logarithmic nonlinearity is studied in this paper. By employing the potential well theory and some differential inequality techniques, a new blow-up condition, the upper bound of the blow-up time, and the lower bound of the growth rate of blow-up solutions are obtained. The effect of the parameters a,b,θ in the assumption of Kirchhoff function M is simultaneously considered, while Ding and Zhou only considered the effect of the parameters b,θ. Thus, the main result in the present paper improves a recent blow-up result.

Global existence and blow-up phenomena for a periodic modified Camassa–Holm equation (MOCH)

ABSTRACT In this paper, we study global existence and blow-up for a periodic modified Camassa–Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm inflation and ill-posedness problem for the equation in the critical Sobolev space.

“Local-in-space” Blowup Criterion for a Weakly Dissipative Dullin–Gottwald–Holm Equation

We are concerned with the blowup phenomena of a dissipative Dullin–Gottwald–Holm equation which can describe unidirectional propagation of surface waves in a shallow water regime. A “local-in-space” blowup criterion is obtained by delicate analysis on the evolution of some linear combinations of the solution u and its derivative $$u_x$$ . The result is the improvement of E. Novruzov’s theorem on the dissipative Dullin–Gottwald–Holm equation with some special coefficients.

Blow-up phenomena in nonlocal eigenvalue problems: When theories of L1 and L2 meet

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu=λu+f involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators.Without singular boundary datum and when λ is not an eigenvalue of the operator, we construct an L2-projected theory of solutions, which we extend to the optimal space of data for the operator L. We present a Fredholm alternative as λ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function.We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value λ, there exist “large eigenfunctions” that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as λ approaches the values of the spectrum.We also obtain a maximum principle for weighted L1 solutions when the operator is L2-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below.Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green functions. Thus “large eigenfunctions” represent a purely nonlocal phenomenon.

Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production

The purpose of this paper is to study radially symmetric solutions of a parabolic–elliptic chemotaxis system (0.1)ut=Δu−∇⋅(f(u)∇v)inBR×(0,+∞),0=Δv−μ(t)+g(u)inBR×(0,+∞)subject to homogeneous Neumann boundary conditions, where BR=BR(0)⊂Rn with n⩾1, R>0 and μ(t)=1|BR|∫BRg(u(x,t))dx denotes the time-dependent spatial mean of g(u). The chemosensitivity f and nonlinear signal production g are suitably regular functions. We show that the blow-up phenomenon of radially symmetric solution (u,v) of (0.1) in finite time for some initial datum u0, when f(ξ)⩾Kξk and g(ξ)⩾Lξl for all ξ⩾1 with K,k,L,l>0 and k+l−1>2n. However, system (0.1) admits a global bounded classical solution for arbitrary initial datum when f(ξ)⩽Kξk and g(ξ)⩽Lξl for all ξ⩾1 with k+l−1<2n.

Multi-bump analysis for Trudinger–Moser nonlinearities. I. Quantification and location of concentration points

In this paper, we carefully investigate the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension 2 containing a nonlinearity with Trudinger–Moser growth. A quantification result was obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers of Del Pino–Musso–Ruf [12],Malchiodi–Martinazzi [30] and Martinazzi [34]. We answer all of them, proving in particular that the blow-up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving results on existence of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what was done for the Liouville equation in the beautiful work of Chen–Lin [8].

A hydrodynamic model for synchronization phenomena

We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena.

Wave Breaking, Global Existence and Persistent Decay for the Gurevich–Zybin System

In this paper, the blow-up phenomenon, global existence and persistent decay of the solutions to the Gurevich–Zybin system are studied. We show that the system possesses a so called critical threshold phenomena, that is, global smoothness versus finite time breakdown depends on whether the initial configuration crosses an intrinsic critical threshold. We prove that a finite maximal life span for a solution necessarily implies wave breaking for this solution, and show some conditions which are local-in-space on the initial data to ensure wave-breaking for this system by making use of the characteristics method, otherwise, the system has a global smooth solution. Furthermore, we establish the persistence properties for the system in weighted $$L^p$$ spaces.

Some Results on Blow-up Phenomenon for Nonlinear Porous Medium Equations with Weighted Source

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

New blow-up phenomena for hyperbolic inequalities with combined nonlinearities

We study for the first time the large-time behavior of solutions to certain hyperbolic inequalities with nonlinearities involving the function and its gradient. Namely, we consider the problem utt−Δu≥|u|p+|∇u|q+f(t,x) in (0,∞)×RN, where p,q>1 and f≥0, f≢0. We obtain general criteria for the nonexistence of global solutions to this problem. Next, we discuss some special cases of the inhomogeneous term f. In particular, when N≥3 and f depends only on the variable space, we obtain a discontinuous Fujita critical exponent p⁎(N,q). Namely, p⁎(N,q)=1+2N−2 if q>1+1N−1, and p⁎(N,q)=∞ if q<1+1N−1. Next, we extend our study to the exterior problem utt−Δu≥|u|p+|∇u|q in (0,∞)×Dc, under the boundary condition ∂u∂n+≥f(t,x) in (0,∞)×∂D, where D is the unit open ball in RN, Dc=RN\\D and n+ is the outward (relative to Dc) unit normal of ∂D.

Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

Abstract In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

Blow-up phenomena for a reaction diffusion equation with special diffusion process

ABSTRACT This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, by combining Hardy inequality, ‘moving’ potential well methods with some differential inequalities. Moreover, the upper and lower bounds for the blow-up time are also derived when blow-up occurs.

Bounds for below-up time in a nonlinear generalized heat equation

ABSTRACT This paper deals with the blow-up phenomena of the solutions to an evolution heat equation with nonlinearities of variable exponent type. For a certain solution with positive initial energy, an upper bound for blow-up time is determined if the solutions blow-up.

Blowup solutions for stochastic parabolic equations

In this short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using the comparison principle and the results for deterministic parabolic equations, we obtain blowup results of solutions for stochastic parabolic equations.

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of “lower energy” solutions to the above equation fails when the dimension of the manifold is not less than 62.

Revisit to wave breaking phenomena for the periodic Dullin–Gottwald–Holm equation

In this paper, we mainly devote to investigate the periodic Dullin–Gottwald–Holm equation. By overcoming the difficulties caused by the complicated mixed nonlinear structure, a very useful priori estimate is derived in Lemma 2.7. Based on Hα1-conservation and L∞-estimate of solution, some new blow-up phenomena are derived for the periodic Dullin–Gottwald–Holm equation under different initial conditions.

Blow-Up Solution of a Porous Medium Equation with Nonlocal Boundary Conditions

In this paper, we devote to studying the blow-up phenomena for a porous medium equation under nonlocal boundary conditions. Based on auxiliary functions and differential inequality technique, we derive the sufficient conditions to guarantee the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R3 and in the higher-dimensional space Rnn≥3, respectively. At last, two examples are given to illustrate the applications of main results.

Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form utt(x,t)−Δu(x,t)+∫0tg(t−s)Δu(x,s)ds−Δut(x,t)=|u(x,t)|p−2u(x,t)ln|u(x,t)|\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}(x,t) - \\Delta u (x,t) + \\int ^{t}_{0} g(t-s) \\Delta u(x,s)\\,ds - \\Delta u_{t} (x,t) = \\bigl\\vert u(x,t) \\bigr\\vert ^{p-2} u(x,t) \\ln \\bigl\\vert u(x,t) \\bigr\\vert $$\\end{document} in a bounded domain varOmega subset {mathbb{R}}^{n} , where g is a nonincreasing positive function. Firstly, we prove the existence and uniqueness of local weak solutions by using Faedo–Galerkin’s method and contraction mapping principle. Then we establish a finite time blow-up result for the solution with positive initial energy as well as nonpositive initial energy.