Finite Time Blow-up Of Solutions Research Articles

Concentration for blow-up solutions of semi-relativistic Hartree equations of critical type

This paper concerns the semi-relativistic Hartree equation i ∂ t u = − Δ + m 2 u − ( | ⋅ | − 1 ∗ | u | 2 ) u in R 3 . We prove the concentration results for finite time blow-up solutions with general H x 1 ∕ 2 ( R 3 ) data, and show the relation between the concentration rate and the blow-up order.

A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions

This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: $$u_{t}=-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u(1-u^k)$$ , $$0=\Delta v+\alpha u^q-\beta v$$ , $$ 0=\Delta w+\gamma u^r-\delta w$$ , in a bounded domain $$\Omega \subset \mathbb {R}^n$$ , $$n\ge 1$$ , subject to the non-flux boundary condition. We at first establish the local existence of solutions (the so-called strong $$W^{1,p}$$ -solutions, satisfying the hyperbolic equation weakly and solving the elliptic ones classically) to the model via applying the viscosity vanishing method and then give criteria on global boundedness versus finite- time blowup for them. It is proved that if the attraction is dominated by the logistic source or the repulsion with $$\max \{r,k\}>q$$ , the solutions would be globally bounded; otherwise, the finite-time blowup of solutions may occur whenever $$\max \{r,k\}<q$$ . Under the balance situations with $$q=r=k$$ , $$q=r>k$$ or $$q=k>r$$ , the boundedness or possible finite-time blowup would depend on the sizes of the coefficients involved.

Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals**Supported by the National Natural Science Foundation of China (11171048), the Science Foundation of Liaoning Education Department grant (LYB201601) and the Fundamental Research Funds for the Central Universities (DUT16LK24).

This paper considers the two-species chemotaxis system with two chemicalsin a smooth bounded domain , subject to the non-flux boundary condition, and . We obtain a blow-up criterion that if , then there exist finite time blow-up solutions to the system with and . When , the blow-up criterion becomes , and the global boundedness of solutions is furthermore established with under the condition that . This improves the current results for finite time blow-up with and global boundedness with respectively in Tao and Winkler (2015 Discrete Contin. Dyn. Syst. Ser. B 20 3165–83)

Nonlinear profile decomposition and the concentration phenomenon for supercritical generalized KDV equations

A nonlinear profile decomposition is established for solutions of supercritical generalized Korteweg-de Vries equations. As a consequence, we obtain a concentration result for finite time blow-up solutions that are of Type II.

Minimal mass blow-up solutions for the [formula omitted] critical NLS with inverse-square potential

We study minimal mass blow-up solutions of the focusing L2 critical nonlinear Schrödinger equation with inverse-square potential, i∂tu+Δu+c|x|2u+|u|4Nu=0,with N≥3 and 0<c<(N−2)24. We first prove a sharp global well-posedness result: all H1 solutions with a mass (i.e. L2 norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.

Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms

In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term \(|u_t|^{m-2}u_t\) and source term \(|u|^{p-2}u\). We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when \(p>m\), we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time \(T_{max}\) is also established. It worth to mention that our obtained results extend the recent results of Wang (J Math Anal Appl 418(2):713–733, 2014) to the nonlinear damping case.

Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term

ABSTRACTIn this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term where is a bounded domain in () with a Dirichlet boundary condition. Under suitable assumptions on the initial data and the relaxation function g, we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. ), by using the Galerkin method, the concavity method and an improved potential well method involving time t. We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.

Global existence and finite time blow-up of the solution for a thin-film equation with high initial energy

This paper is devoted to the study of a thin-film equation, which was considered extensively in recent years. We obtain the conditions for global existence and finite time blow-up of solution for the high initial energy case, i.e., J(u0)>d, while the previous works only considered the case J(u0)≤d, here d is the mountain pass level. Moreover, for the blow-up solution with J(u0)<d, we obtain the lower bound estimate of the blow-up time and blow-up rate.

Nonlinear damped wave equation: Existence and blow-up

In this paper, we consider the following nonlinear wave equation with variable exponents: utt−Δu+aut|ut|m(⋅)−2=bu|u|p(⋅)−2,where a,b are positive constants. By using the Faedo–Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p. We also prove the finite time blow-up of solutions and give a two-dimension numerical example to illustrate the blow up result.

Stabilization by noise of a ℂ2-valued coupled system

D. Herzog and J. Mattingly have shown that a [Formula: see text]-valued polynomial ODE with finite-time blow-up solutions may be stabilized by the addition of [Formula: see text]-valued Brownian noise. In this paper, we extend their results to a [Formula: see text]-valued system of coupled ODEs with finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved in our setting by adding a suitable Brownian noise, and that the resulting system of SDEs is ergodic. The proof uses the Girsanov theorem to induce a time change from our [Formula: see text]-system to a quasi-[Formula: see text]-system similar to the one studied by Herzog and Mattingly.

Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration

In this paper, we study the semilinear pseudo-parabolic equations $$\displaystyle u_{t} - \triangle _{{\mathbb {B}}}u - \triangle _{{\mathbb {B}}}u_{t} = \left| u\right| ^{p-1}u$$ on a manifold with conical singularity, where $$\triangle _{{\mathbb {B}}}$$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $$x_{1} = 0$$ . Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy $$J(u_{0})<d$$ , the solution is global in time with $$I(u_{0}) >0$$ or $$\displaystyle \Vert \nabla _{{\mathbb {B}}}u_{0}\Vert _{L_{2}^{\frac{n}{2}}({\mathbb {B}})} = 0$$ and blows up in finite time with $$I(u_{0}) < 0$$ ; for the critical initial energy $$J(u_{0}) = d$$ , the solution is global in time with $$I(u_{0}) \ge 0$$ and blows up in finite time with $$I(u_{0}) < 0$$ . The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

Blow-up phenomena for p-adic semilinear heat equations

The problem of existence of solutions to p-adic semilinear heat equations with particular nonlinear terms has already been studied in the literature but the occurrence of blow-up phenomena has not been considered yet. We initiate the study of finite time blow-up for solutions of this kind of p-adic semilinear equations, proving that this phenomenon always arises under appropriate assumptions in the case when the exponent of nonlinearity times the dimension is strictly less than the order of the operator.

On the blow-up criterion of magnetohydrodynamics equations in homogeneous Sobolev spaces

In this paper, we obtain new lower bounds of the maximal solution of the magnetohydrodynamics equations in the homogeneous Sobolev space , for , when solutions blow-up in finite time. This gives us some new insight on the blow-up behavior of the solution.

The initial-boundary value problem for the generalized double dispersion equation

We consider the initial-boundary value problem for the generalized double dispersion equation in all space dimension. Under the suitable assumptions on the initial data and the parameters in the equation, we establish several results concerning local existence, global existence, uniqueness, and finite time blowup property. The exponential decay rate of the energy is proved for global solutions. The sufficient and necessary conditions of global solutions and finite time blowup of solutions are given, respectively.

Uniqueness of minimal blow-up solutions to nonlinear Schrödinger system

In this paper, we are concerned with the finite time blow-up solutions to the following N coupled nonlinear Schrödinger system in R2: (0.1)i∂tϕj+Δϕj+∑k=1Najk|ϕk|2ϕj=0, where N≥2 and the coefficient matrix A satisfies ajk=akj>0. We study the properties of the blow-up solutions which obtain the minimal L2-norm, and prove that such solution is unique, up to the symmetries of the system.

Instability of an integrable nonlocal NLS

In this note, we discuss the global dynamics of an integrable nonlocal NLS on R, which has been the object of a recent investigation by integrable systems methods. We prove two results that are in striking contrast with the case of the local cubic focusing NLS. First, finite-time blow-up solutions exist with arbitrarily small initial data in Hs(R), for any s⩾0. On the other hand, the solitons of the local NLS, which are also solutions to the nonlocal equation, are unstable by blow-up for the latter.

Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution

In this paper, we study the global well-posedness and scattering problem for the nonlinear wave equation with a convolution utt−Δu+(|x|−γ∗|u|2)u=0 in dimensions d≥6. We show that if the solution u is apriorily bounded in the critical homogeneous Sobolev space, that is, u∈Lt∞(Ḣxsc×Ḣxsc−1), with sc≔γ−22>1, then u is global and it scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. Our analysis derived from the concentration compactness method to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. We note that, authors preclude the finite time blow-up solution to wave equation usually by the property of the finite speed of propagation in the previous literature, however, the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity. For this, we will initially establish the low regularity results for almost periodic solutions including the finite time blow-up solutions and initially preclude the finite time blow-up solutions by lowering the regularity.

Asymptotic behavior and blowup for two generalized Ginzburg–Landau type equations with several nonlinear source terms

In this paper, the long-time behavior and blowup of solutions for two generalized Ginzburg–Landau type equations with several nonlinear source terms are investigated. First, the conditions in which the solutions of the two equations are positive are analyzed by using Green’s function. According to the characteristics of nonlinear terms, four different functionals are constructed for solving the problems. Finally, we obtain the long-time behavior and finite-time blowup of solutions for the initial and boundary value problem by using these functionals.

Local existence and singularities formation for isothermal relativistic radiation hydrodynamics equations

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

Critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. Our aim is to obtain the threshold, to classify the global existence of solutions for small initial data or the finite time blow-up of solutions, under the growth condition of the nonlinearity.