Conditions For Blow-up Research Articles

Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system

This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to −∞ as time goes to +∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution (u,v), u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ>0 there exists M(τ)>0 with M(τ)→∞ as τ→∞ such that all radial solutions (u,v) with initial mass of u larger than M(τ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ=1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ>0.

Finite-time singularity formation for C 1 solutions to the compressible Euler equations with time-dependent damping

In this paper, the initial-boundary value problem of the multidimensional compressible Euler equations with time-dependent damping in radial symmetry is considered. It is shown that finite-time singularity will be developed for the solutions of the compressible Euler equations with time-dependent damping coefficients if the initial value of a newly introduced functional, with a time-dependent parameter is sufficiently large. The blowup conditions imply that the initial kinetic energy of the fluid must not be less than a given constant.

Global Existence and Blow-Up for a Parabolic Problem of Kirchhoff Type with Logarithmic Nonlinearity

In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u_t} +{M([u]^2_s){\mathcal {L}}_Ku}={|u|^{p-2}u\log |u|},\ \ \ &{}\hbox { in } \Omega \times (0,+\infty ),\\ \displaystyle u(x,t)=0,&{}\hbox { in }({\mathbb {R}}^N\setminus \Omega )\times (0,+\infty ),\\ \displaystyle u(x,0)=u_0(x),&{}\hbox { in }\Omega , \end{array}\right. \end{aligned}$$ where $$[u]_s$$ is the Gagliardo seminorm of u, $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary, $$0<s<1$$ , $${\mathcal {L}}_K$$ is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator $$(-\Delta )^s$$ , $$u_0$$ is the initial function, and $$M:[0,+\infty )\rightarrow [0,+\infty )$$ is continuous. Let $$J(u_0)$$ be the initial energy (see (2.1) for the definition of J), $$d>0$$ be the mountain-pass level given in (2.4), and $${\widetilde{M}}\in (0,d]$$ be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for $$J(u_0)\le d$$ . Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for $$J(u_0)\le {\widetilde{M}}$$ , the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. In addition, for $$J(u_0)\le d$$ , we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.

Quasilinear parabolic equations with a degenerate absorption potential

Quasilinear parabolic equations with a degenerate absorption potential are considered. The estimates of all weak solutions of such equations, including large solutions satisfying the blow-up conditions on the parabolic boundary of a domain, are obtained.

Optimal condition for blow-up of the critical Lq norm for the semilinear heat equation

We shed light on a long-standing open question for the semilinear heat equation ut=Δu+|u|p−1u. Namely, without any restriction on the exponent p>1 nor on the smooth domain Ω, we prove that the critical Lq norm blows up whenever the solution undergoes type I blow-up. A similar property is also obtained for the local critical Lq norm near any blow-up point.In view of recent results of existence of type II blow-up solutions with bounded critical Lq norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical Lq norm blow-up appears to be a completely new observation.Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with backward uniqueness and unique continuation properties for parabolic equations.As a by-product, we obtain the nonexistence of self-similar profiles in the critical Lq space. Such properties were up to now only known for p≤pS and in radially symmetric case for p>pS, where pS is the Sobolev exponent.

Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.

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.

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some sufficient initial conditions of finite time blowup and global existence are obtained, respectively.

Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

We study the dynamics for the focusing nonlinear Klein–Gordon equation, [Formula: see text] with positive radial potential [Formula: see text] and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on [Formula: see text]. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.

Short note on conditional collapse of self-gravitating system with positive total energy

In this study, we first show a finite time collapse of a three-dimensional Smoluchowski–Poisson system in the micro-canonical ensemble (i.e., μSP system) with positive total energy. Using a precise estimation of the bound of the gravitational contribution to total energy, we provide blow-up conditions on the initial state of the μSP system. The conditions indicate that highly clustered self-gravitating particles in the μSP system induce a finite time collapse, even for positive total energy.

Global asymptotical behavior and some new blow-up conditions of solutions to a thin-film equation

We consider a thin-film equation. By exploiting the boundary condition and the variational structure of the equation, we consider the global existence and blow-up of the solutions. For the global solutions, we study the asymptotic behavior, and for the blow-up solutions, we study the upper and lower bounds of the blow-up time. Especially, we have derived the necessary and sufficient conditions for the solution blowing up in finite time when the initial energy J(u0)≤d, where d is the mountain-pass level. For J(u0)>d, we also study the conditions for the solution exists globally and blows up in finite time, and we also obtain some necessary and sufficient conditions for the solution blowing up in finite time when the initial energy at arbitrary level. Furthermore, we study the decay behavior of both the global solutions and its corresponding energy functional, and the decay ratios are given specially. The results generalize the former studies on this equation, such as the papers: Li et al. (2016) [16], Dong and J. Zhou (2017) [4] and Sun et al. (2018) [23].

Extremely low order time-fractional differential equation and application in combustion process

Fractional blow-up model, especially which is of very low order of fractional derivative, plays a significant role in combustion process. The order of time-fractional derivative in diffusion model essentially distinguishes the super-diffusion and sub-diffusion processes when it is relatively high or low accordingly. In this paper, the blow-up phenomenon and condition of its appearance are theoretically proved. The blow-up moment is estimated by using differential inequalities. To numerically study the behavior around blow-up point, a mixed numerical method based on adaptive finite difference on temporal direction and highly effective discontinuous Galerkin method on spatial direction is proposed. The time of blow-up is calculated accurately. In simulation, we analyze the dynamics of fractional blow-up model under different orders of fractional derivative. It is found that the lower the order, the earlier the blow-up comes, by fixing the other parameters in the model. Our results confirm the physical truth that a combustor for explosion cannot be too small.

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.

Blow-up of solutions of a full non-linear equation of ion-sound waves in a plasma with non-coercive non-linearities

We consider a series of initial- boundary value problems for the equation of ion- sound waves in a plasma. For each of them we prove the local (in time) solubility and perform an analytical- numerical study of the blow-up of solutions. We use the method of test functions to obtain sufficient conditions for finite-time blow-up and an upper bound for the blow-up time. In concrete numerical examples we improve these bounds numerically using the mesh refinement method. Thus the analytical and numerical parts of the investigation complement each other. The time interval for the numerical modelling is chosen in accordance with the analytically obtained upper bound for the blow-up time. In return, numerical calculations specify the moment and pattern of this blow-up.

Blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations

The blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations is investigated. More precisely, we consider a functional F(t) associated with the momentum and weighted by a general test function f and show that if F(0) is sufficiently large, then the finite time blowup of the solutions of the non-isentropic compressible Euler equations occurs. As the test function f is a general function with only mild conditions imposed, a class of blowup conditions is established.

On blow-up conditions for solutions of higher order differential inequalities

ABSTRACTFor differential inequalities of the form where and b are some functions, we obtain conditions guaranteeing that any solution is identically equal to zero. We construct examples which show that the obtained conditions are sharp.

Blowup of a free boundary problem with a nonlocal reaction term

We consider a blowup problem of a reaction–diffusion equation with a nonlocal reaction term. Such a problem arises in the description of the species inhabiting in a region surrounded by an inhospitable area with the free boundary representing the spreading front of the species. Firstly, we give some sufficient conditions for finite time blowup. Then we show that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial data is small and the result is different for the positive and non-positive growth rate.

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.