Finite Time Blow-up Of Solutions Research Articles

Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone

In this paper, we establish the existence of a family of surfaces (Γ(t))0<t≤T that evolve by the vanishing mean curvature flow in Minkowski space and, as t tends to 0, blow up towards a surface that behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of a finite-time blow-up solution to this hyperbolic equation under the form u(t,x)∼tν+1Q(xtν+1), where Q is a stationary solution and ν is an irrational number strictly larger than 1/2. Our strategy roughly follows that of Krieger, Schlag and Tataru in [7–9]. However, contrary to these articles, the equation to be handled in this work is quasilinear. This induces a number of difficulties to face.

Instability of standing waves for nonlinear Schrödinger equation with delta potential

We consider nonlinear Schrodinger equation with an attractive delta potential in one space dimension. We review some recent results on relations between orbitally unstable standing wave solutions and finite time blowup solutions.

Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation∂tu=(um)xx+a(x)(um)x+b(x)um, posed for x∈R, t≥0 and m>1, where a, b are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium typef(y)∂τθ=(θm)yy, posed in the half-line y∈[0,∞) with τ≥0, m>1 and suitable density functions f(y). We apply this correspondence to the case of constant coefficients a(x)=1 and b(x)=K>0. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as x→−∞. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when f(y)=y−γ with γ>2.

Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction

We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions d≤3. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension d=4. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.

Global phase diagrams of run-and-tumble dynamics: equidistribution, waves, and blowup

For spatially one-dimensional run-and-tumble dynamics with mass conservation we develop a coarse phase diagram, that discriminates between global decay to equidistributed constant states, existence of spatially non-trivial waves, and finite time blowup of solutions. Motivated by counter-migrating ripples of high and low population density and fruiting body formation in myxobacteria, we identify phase boundaries as particular critical tumbling dynamics that allow for switching between these spatio-temporal phases upon slight changes in mass densities or parameter values.

Prescribing the center of mass of a multi-soliton solution for a perturbed semilinear wave equation

We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer k≥2 and ζ0∈R, we construct a blow-up solution with a characteristic point a, such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs, whose centers (in the hyperbolic geometry) have ζ0 as a center of mass, for all times. Although the result is similar to the unperturbed case in its statement, our method is new. Indeed, our perturbed equation is not invariant under the Lorentz transform, and this requires new ideas. In fact, the main difficulty in this paper is to prescribe the center of mass ζ0∈R. We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by Côte and Zaag [9], as far as the center of mass prescription is concerned.

Existence and uniqueness of solutions to initial boundary value problem for the higher order Boussinesq equation

This paper studies the initial and boundary value problem for the higher order Boussinesq equation. The existence of a local and global solution are proved. The sufficient conditions for finite time blow-up solutions are provided by using the concavity method.

Asymptotic behavior of solutions for a free boundary problem with a nonlinear gradient absorption

This paper deals with the free boundary problem for a parabolic equation, $$u_t-u_{xx}=u^{p}-\lambda |u_x|^{q}$$ , $$t>0$$ , $$0<x<s(t)$$ , with $$p, q>1$$ . It is well known that global existence or blowup of solutions of nonlinear parabolic equations depends on which one dominating the model, the source or absorption, and on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of p, q, initial data and free boundary on the asymptotic behavior of solutions. At first, the ecological meaning of this model is explained by deriving the equation and the free boundary condition. Then, local existence and uniqueness are discussed, and the continuous dependence on initial data and comparison principle are proved. Furthermore, the finite time blowup and global solution are given by constructing sub- and super-solutions. In the case of $$ p>q>1 $$ , our results show that the solution blows up in finite time for the initial value sufficiently large; but if the initial data are small, there exist global fast solutions which decay exponentially. Also, with a suitable initial value, the existence of global slow solution with at most polynomial decaying is established. While, in the case of $$q\ge p>1$$ , all solutions with nonnegative initial data of exponential decay exist globally. Finally, the problem with double free boundaries, $$u_t-u_{xx}=u^p-\lambda |u_x|^q, t>0, g(t)<x<h(t)$$ , is also discussed and the similar conclusions are obtained.

Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $ \mathbb{R}^d $, $ d = 5,6 $, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $ \mathbb{R}^d $, $ d = 5, 6 $, in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

Lower bounds for the finite-time blow-up of solutions of a cancer invasion model

Lower bounds for the finite-time blow-up of solutions of a cancer invasion model

Blow-up solutions to nonlinear Schrödinger system at multiple points

In this paper, we consider finite-time blow-up solutions to the following N coupled nonlinear Schrodinger system in \(\mathbb {R}^2\): $$\begin{aligned}i\partial _t\phi _j+\Delta \phi _j+\sum _{k=1}^{N}a_{jk}|\phi _k|^2\phi _j=0, \quad j=1,2,\ldots ,N,\quad (0.1) \end{aligned}$$ where \(N\ge 2\) and the coefficient matrix A satisfies \(a_{jk}=a_{kj}\) and \(a_{jj}>0\) for \(1\le j,k\le N\). We construct finite-time blow-up solutions concentrating at arbitrary K different points with \(K\ge N\).

A p $p$ -Laplace Equation with Logarithmic Nonlinearity at High Initial Energy Level

In this paper the authors investigate a class of $p$ -Laplace equations with logarithmic nonlinearity, which were considered in Le and Le (Acta Appl. Math. 151:149–169, 2017), where, among other things, global existence and finite time blow-up of solutions were proved when the initial energy is subcritical and critical, that is, initial energy smaller than or equal to the depth of the potential well. Their results are complemented in this paper in the sense that an abstract criterion is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time, when the initial energy is supercritical. As a byproduct it is shown that the problem admits a finite time blow-up solution for arbitrarily high initial energy.

EXISTENCE AND BLOW-UP OF SOLUTIONS TO A PARABOLIC EQUATION WITH NONSTANDARD GROWTH CONDITIONS

We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.

A new blow-up criterion for non-Newton filtration equations with special medium void

This paper deals with the finite time blow-up of solutions to the initial boundary value problem of a non-Newton filtration equation with special medium void. A new criterion for the solutions to blow up in finite time is established by using the Hardy inequality. Moreover, the upper and lower bounds for the blow-up time are also estimated. The results solve an open problem proposed by Liu in 2016.

A Class of Fourth Order Damped Wave Equations with Arbitrary Positive Initial Energy

AbstractIn this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy

In this paper the authors deal with a class of parabolic type Kirchhoff equations, which were considered in Han and Li (2018), where global existence and finite time blow-up of solutions were studied when the initial energy was subcritical, critical and supercritical. Their results are complemented in this paper in the sense that a new blow-up criterion will be given for nonnegative initial energy and upper and lower bounds for blow-up time will be derived.

Wave breaking and global existence for a family of peakon equations with high order nonlinearity

This paper is concerned with the Cauchy problem for a family of peakon equations with high order nonlinearity. To begin with, the local well-posedness results in Besov and Sobolev spaces are discussed. Then we obtain the precise blow-up scenario for strong solutions to the equation, which is conjectured by Anco et al., (2015). Moreover, we establish some global existence results for the strong solutions by deriving two useful conservation laws. Furthermore, wave breaking phenomenon is studied and we construct a new finite time blow-up solution to the equation with respect to the initial data.

Properties of blow-up solutions and their initial data for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type{ut=∇⋅(∇um−uq−1∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂RN (N≥3), m≥1, q≥2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q>m+2N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case.

Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [ 18 ]. At the end of the paper, a numerical example satisfying the theory is provided.