Sufficient Conditions For Blow-up Research Articles

Sufficient conditions of blowup to a shallow water wave equation

The blow-up features of a shallow water wave equation on the line R are investigated. The L2 conservation law is utilized to derive several estimates of solutions for the equation. Sufficient conditions for wave breaking and lifespan of the solutions are established. Our main results contain parts of the wave breaking conditions for the Fornberg–Whitham and Degasperis–Procesi equations in the previous literatures.

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.

Enhancement of the Zakharov–Glassey’s method for Blow-up in nonlinear Schrödinger equations

In this paper we give a sharper sufficient condition for blow-up of the solution to a nonlinear Schrödinger equation with free/Stark/quadratic potential by improving the well known Zakharov–Glassey’s method.

Wave-breaking phenomena and global existence for the generalized Fornberg-Whitham equation

We consider the initial value problem for the generalized Fornberg-Whitham equation, which consists of p-order nonlinear advection term and nonlocal dispersion term. In the case of p=2, we study sufficient conditions for blow-up of solutions and some relations to the Burgers equation. Also, for p≥5, we show the global existence of the solutions with small initial data.

An inverse problem for the pseudo-parabolic equation with p-Laplacian

<p style='text-indent:20px;'>In this article, we study the inverse problem of determining the right side of the pseudo-parabolic equation with a p-Laplacian and nonlocal integral overdetermination condition. The existence of solutions in a local and global time to the inverse problem is proved by using the Galerkin method. Sufficient conditions for blow-up (explosion) of the local solutions in a finite time are derived. The asymptotic behavior of solutions to the inverse problem is studied for large values of time. Sufficient conditions are obtained for the solution to disappear (vanish to identical zero) in a finite time. The limits conditions that which ensure the appropriate behavior of solutions are considered.</p>

Blow-up instability in non-linear wave models with distributed parameters

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space . Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine’s method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time.

Inverse Problem for an Equation with A Nonstandard Growth Condition

This paper describes an inverse problem for determining the right side of a parabolic equation with a nonstandard growth condition and integral overdetermination condition. The Galerkin method is used to prove the existence of two solutions of the inverse problem and their uniqueness, one of them being local and the other one being global in time. Sufficient blow-up conditions for the local solution for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary are obtained. The blow-up of the solution is proven using the Kaplan method. The asymptotic behavior of the inverse problem solutions for large time values is investigated. Sufficient conditions for vanishing of the solution for a finite time are obtained. Boundary conditions ensuring the corresponding behavior of the solutions are considered.

Blowup of solutions for nonlinear nonlocal heat equations

Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. Sufficient conditions for blowup are expressed in terms of some Morrey space norms. A comparison of these with recent results on global-in-time solutions is discussed.

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732‐2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd.

Numerical detection of blow-up: a new sufficient condition for blow-up

In many evolution equations, solutions may become unbounded in finite time. This phenomenon is often called blow-up. There are many mathematical tools used to establish blow-up. In this paper, we derive a new sufficient condition for blow-up from a numerical viewpoint, which provides us a new approach for the computation of blow-up solutions and the detection of blow-up.

A new sufficient condition for blow-up of solutions to a class of parabolic equations

In this paper, a new sufficient condition under which the solutions to a classical parabolic equation blow up is established. During this process, through constructing a new functional which falls between the energy and Nehari functionals, we also get a larger blow-up set.

Blow-Up of Solution to Cauchy Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.

Global solvability for the porous medium equation with boundary flux governed by nonlinear memory

We introduce the study of global existence and blow-up in finite time for nonlinear diffusion equations with flux at the boundary governed by memory. Via a simple transformation, the memory term arises out of a corresponding model introduced in previous studies of tumor-induced angiogenesis. The present study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Specifically, we establish an identical set of necessary and sufficient conditions for blow-up in finite time as previously established in the case of local flux conditions at the boundary.

Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations

We unify a few of the best known results on wave breaking for the Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.

Local solvability and solution blowup for the Benjamin-Bona-Mahony-Burgers equation with a nonlocal boundary condition

We consider a model initial-boundary value problem for the Benjamin-Bona-Mahony-Burgers equation with initial conditions having a physical meaning. We prove the unique local solvability in the classical sense and obtain sufficient conditions for blowup and an estimate of the blowup time. To prove the blowup, we use the known test function method developed in papers by V. A. Galaktionov, E. L. Mitidieri, and S. I. Pohozaev. We note that this is one of the first results toward the blowup for the considered equation.

Blow-up of solutions of strongly dissipative generalized Klein–Gordon equations

We study the question of sufficient conditions for blow-up of solutions of a model initial-boundary value problem. We use a modified form of Levine's method to obtain sufficient conditions for blow-up in finite time.

Blow-up of solutions of an inhomogeneous system of Sobolev-type equations

We consider a system of two inhomogeneous non-linear Sobolev-type equations of order 6 with second derivative with respect to the time variable. We prove that the problem is soluble locally in time, state sufficient conditions for blow-up of a solution at finite time, and give an upper bound for the blow-up time.

Existence and blow-up of solutions of a pseudoparabolic equation

We obtain sufficient blow-up conditions for the solution of a nonlinear differential problem with given initial and boundary conditions. We prove the solvability of this problem in any finite cylinder under some restrictions on the nonlinear operators.

Blow-up criteria for the 3D cubic nonlinear Schrödinger equation

We consider solutions u to the 3D nonlinear Schrödinger equation i∂tu + Δu + |u|2u = 0. In particular, we are interested in finding criteria on the initial data u0 that predict the asymptotic behaviour of u(t), e.g., whether u(t) blows up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H1 data) (Duyckaerts–Holmer–Roudenko) if M[u]E[u] ⩽ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u and Q denotes the ground state solution to −Q + ΔQ + |Q|2Q = 0. Here we consider the complementary case M[u]E[u] > M[Q]E[Q]. In the first (analytical) part of the paper, we present a result due to Lushnikov, based on the virial identity and the generalized uncertainty principle, giving a sufficient condition for blow-up. By replacing the uncertainty principle in his argument with an interpolation-type inequality, we obtain a new blow-up condition that in some cases improves upon Lushnikov's condition. Our approach also allows for an adaptation to radial infinite-variance initial data that has a conceptual interpretation: for real-valued initial data, if a certain fraction of the mass is contained within the ball of radius M[u], then blow up occurs. We also show analytically (if one takes the numerically computed value of ) that there exist Gaussian initial data u0 with negative quadratic phase such that but the solution u(t) blows up. In the second (numerical) part of the paper, we examine several different classes of initial data—Gaussian, super Gaussian, off-centred Gaussian, and oscillatory Gaussian—and for each class give the theoretical predictions for scattering or blow-up provided by the above theorems as well as the results of numerical simulation. We find that depending upon the form of the initial conditions, any of the three analytical criteria for blow-up can be optimal. We formulate several conjectures, among them that for real initial data, the quantity provides the threshold for scattering.