Finite-time Blow-up Result Research Articles

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.

Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

We consider the nonlinear pseudoparabolic equation with a memory termut-Δu-Δut+∫0tλt-τΔuτdτ=div∇up-2u+u1+α,x∈Ω,t>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions onp,α, and the relaxation functionλ(t), we prove a finite-time blow-up result by using the concavity method.

A blow-up result in a system of nonlinear viscoelastic wave equations with arbitrary positive initial energy

In this paper we consider a system of nonlinear viscoelastic wave equations. Under arbitrary positive initial energy and standard conditions on the relaxation functions, we prove a finite-time blow-up result.

Fifth-order complex Korteweg–de Vries-type equations

This paper studies spatially periodic complex-valued solutions of the fifth-order Korteweg–de Vries (KdV)-type equations. The aim is at several fundamental issues including the existence, uniqueness and finite-time blowup problems. Special attention is paid to the Kawahara equation, a fifth-order KdV-type equation. When a Burgers dissipation is attached to the Kawahara equation, we establish the existence and uniqueness of the Fourier series solution with the Fourier modes decaying algebraically in terms of the wave numbers. We also examine a special series solution to the Kawahara equation and prove the convergence and global regularity of such solutions associated with a single mode initial data. In addition, finite-time blowup results are discussed for the special series solution of the Kawahara equation.

A blowup result for the periodic NLS without gauge invariance

In this Note, we prove a finite-time blowup result for the periodic nonlinear Schrödinger equation on Td with nonlinearity |u|p for p>1. In particular, our blowup result holds above the Strauss exponent. This is in contrast with the non-periodic setting, where global existence for small data is known above the Strauss exponent.

Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

This paper deals with blowing up of solutions to the Cauchy problem for a class of generalized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α ≥ 0 and \(1 + \tfrac{4} {N} \leqslant p < \tfrac{{N + 2}} {{N - 2}}\) or α < 0 and \(1 < p < 1 + \tfrac{4} {N}\) (N = 2, 3); the other is established under the condition N = 3, \(1 < p < \tfrac{{N + 2}} {{N - 2}}\) and α(p − 3) ≥ 0. On the other hand, for c0 < +∞ and α(p − 3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.

Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms

In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms. Under suitable assumptions on the initial data and the relaxation function, we establish a finite-time blow-up result and a global existence result.

Finite-time blowup for Zakharov system with combined power-type nonlinearities

This article is concerned with finite-time blowup results for solutions of a class of generalized Zakharov system with combined power-type nonlinearities in three space dimensions: On the one hand, for c 0 = +∞ (in the subsonic limit), if α ≥ 0 and or α < 0 and , we obtain that the solution of (GZS) blows up in finite time in the energy space H 1(R 3). On the other hand, for 0 < c 0 < +∞, if α ≥ 0, and the solution (ϵ, n, v)(t) of (GZS) are radially symmetric for all t, we prove a new finite-time blowup result of solutions to (GZS) in the energy space H 1(R 3) × L 2(R 3) × L 2(R 3).

A blow-up result in a Cauchy viscoelastic problem

In this work we consider a Cauchy problem for a nonlinear viscoelastic equation. Under suitable conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.

Uniform blow-up profile for a degenerate parabolic system with nonlocal source

In this paper, we establish the local existence of the solution and the finite-time blow-up result for the following system: u t=u r1 δu+a∫ ωu pdx , forxϵohm;, t>0, u t=u r2 δu+b∫ ωu qdx , forxϵohm;, t>0, where p, q ≥ 1 and 0 < r 1, r 2 < 1. Moreover, it is proved that the solution has global blow-up and lim t←T ∗ (T ∗:−t) ( p+1−r 2) y u(x,t)=(1−r1) −1 1−r1) (a∥aω∥y) −(p+1−r2) y (q+1−−r1) p y x a(p+1−r2) b 1−r2) y lim t←T ∗ (T ∗:−t) ( q+1−r 1) y u(x,t)=(1−r2) −1 1−r2) (b∥aω∥y) −(q+1−r1) y (p+1−−r2) q y x b(q+1−r1) a 1−r1) y uniformly on compact subsets of Ω, where γ = pq - (1 − r 1)(1 − r 2) and T∗ is the blow-up time.

Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

We study the Cauchy problem tor the nonlinear wave equation with nonlinear damping au t | u t | m−1 and a nonlinear source term of type ba| a| p−1 . When existence result is established for any initial data. When 1 < m < p. a finite time blow-up result is proved for compactly supported data with negative initial energy.

Analysis of Blowup for a Nonlinear Degenerate Parabolic Equation

In this paper the nonlinear degenerate parabolic equation u t = u α (Δ u + u) subject to Dirichlet boundary conditions is studied. The finite time blowup result, together with an estimate of the blowup time, is found. The blowup set with positive measure and the asymptotic behavior of solution near the blowup time are analyzed in some detail via the similarity solutions.