Blow-up Criterion Research Articles

Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy

In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, a linear weak damping and superlinear source. Finite time blow-up results have been obtained for the case in which the initial energy $$E(0)\le M$$ , where M is a positive constant. By utilizing Levine’s classical concavity method, we give a new blow-up criterion which includes the case of $$E(0)>M$$ and derive an explicit upper bound for the blow-up time. By using the Fountain Theorem, we show that the problem with arbitrary positive initial energy always admits weak solutions blowing up in finite time.

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity

<p style='text-indent:20px;'>In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].</p>

A note on blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

In this note, we study the Cauchy problem for a class of nonlinear dispersive wave equation with dissipative term on the real line. We establish a new local-in-space blow-up criterion. Our results improve the corresponding ones in the previous paper.

A class of nonlinear parabolic equations with anisotropic nonstandard growth conditions

In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with the solution. Second, we give two new blow-up criteria that one of them is obtained to remove the exponent restriction by using an extended form of the concavity method and the other one is a blow-up result of weak solutions with arbitrarily high initial energy. Moreover, an extinction result is also showed by establishing a new auxiliary lemma. The present results of this paper complement and improve a recent paper investigated by Liu et al. [J. Math. Phys. 59, 121504 (2018)].

Blow-up criterion for the chemotaxis-fluid equations in a 3D unbounded domain with mixed boundary conditions

In this paper, we consider a coupled chemotaxis-fluid system in a 3D unbounded domain with mixed boundary conditions. A blow-up criterion for such a system is established by using the proper elliptic estimates and Stokes estimates under some assumptions on the chemotactic sensitivity function.

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.

Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

Abstract In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i ∂ t ψ − ( − Δ ) s ψ + ( I α ∗ | ψ | p ) | ψ | p − 2 ψ = 0. $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L 2-critical and L 2-supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in Hs . Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.

On a Stochastic Camassa\u2013Holm Type Equation with Higher Order Nonlinearities

The subject of this paper is a generalized Camassa–Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces H^s with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.

Blow-up criteria for linearly damped nonlinear Schrödinger equations

We consider the Cauchy problem for linearly damped nonlinear Schrodinger equations \begin{document}$ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $\end{document} where \begin{document}$ a>0 $\end{document} and \begin{document}$ \alpha>0 $\end{document} . We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up \begin{document}$ H^1 $\end{document} solutions to the focusing problem in the mass-critical and mass-supercritical cases.

Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions

<p style='text-indent:20px;'>In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{H}}^m $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ m \geq 3 $\end{document}</tex-math></inline-formula>. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumption on the tangential trace of the solution. The proof is based on detailed apriori estimates and the regularity theory for the corresponding linear problem which we also develop here.

Normalized ground states for the NLS equation with combined nonlinearities

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities−Δu=λu+μ|u|q−2u+|u|p−2uin RN, N≥1, having prescribed mass∫RN|u|2=a2. Under different assumptions on q<p, a>0 and μ∈R we prove several existence and stability/instability results. In particular, we consider cases when2<q≤2+4N≤p<2⁎,q≠p, i.e. the two nonlinearities have different character with respect to the L2-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature.We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.

On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.

On the Cauchy problem for a modified Camassa–Holm equation

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

A Blow-Up Criterion of Strong Solutions to the Quantum Hydrodynamic Model

In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density.

On the continuation principle of local smooth solution for the Hall-MHD equations

ABSTRACT This work deals with the blow-up criterion of local smooth solution to the 3D Hall-magnetohydrodynamics equations in Besov spaces .

Blowup Criterion via Only the Middle Eigenvalue of the Strain Tensor in Anisotropic Lebesgue Spaces to the 3D Double-Diffusive Convection Equations

In this paper, we consider the three-dimensional (3D) incompressible double-diffusive convection equations. By making use of a anisotropic Sobolev’s inequality, we obtain a blowup criterion for the strong solution to the 3D double-diffusive convection equations involving only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces. This result improves a result established in a recent work by Miller (Arch Ration Mech Anal 235:99–139, 2019).

Singularity formation and global Well-posedness for the generalized Constantin–Lax–Majda equation with dissipation

We study a generalization due to De Gregorio and Wunsch et al of the Constantin–Lax–Majda equation (gCLM) on the real linewhere H is the Hilbert transform and . We use the method in Chen J et al (2019 (arXiv:1905.06387)) to prove finite time self-similar blowup for a close to and by establishing nonlinear stability of an approximate self-similar profile. For a > −1, we discuss several classes of initial data and establish global well-posedness and an one-point blowup criterion for different initial data. For , we prove global well-posedness for gCLM with critical and supercritical dissipation.

Biharmonic wave maps: local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

Abstract In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].