Finite Time Blow-up Of Solutions Research Articles

Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping R2 to S2 with arbitrary given coefficients ρ1∈R,ρ2>0. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.

Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

Finite time blowup of solutions of the hydrostatic Euler equations

Finite time blowup of solutions of the hydrostatic Euler equations

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations

Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

A special form of solution to complex valued Benjamin–Ono equations

We investigate a special form of solution to the complex valued Benjamin–Ono (cBO) equation. Under the special form of solution involving Hilbert transform, the complex valued (modified) Benjamin–Ono equations are equivalent to the quadratic (cubic) derivative Schrödinger equations. The soliton interaction dynamics is used to construct a finite time blowup solution of the cBO equation on R. We also find static solutions. Applying Cole–Hopf transformation to quadratic derivative Schrödinger equation, we derive a linear Schrödinger equation with a special form of initial data from which some solutions to cBO equation can be obtained by an inverse transform.

Blow-up of solutions to the Keller–Segel model with tensorial flux in high dimensions

Over the course of the last decade, there has been a significant level of interest in the analysis of Keller–Segel models incorporating tensorial flux. Despite this interest, the question of whether finite-time blowup solutions exist remains a topic of ongoing research. Our study provides evidence that solutions of this nature are indeed possible in dimensions n≥3, when utilizing a tensorial flux expressed in the form of A∇v, where A denotes a matrix with constant components.

L2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

This article studies the Schrödinger equation with an inhomogeneous combined term i∂tu−(−Δ)su+λ1|x|−b|u|pu+λ2|u|qu=0, where s∈(12,1),λ1,λ2=±1,0<b<{2s,N} and p,q>0. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p,q,λ1 and λ2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

Blow-up for a Porous Medium Equation with Local Linear Boundary Dissipation

This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation. Making use of the concavity method, we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon. Meanwhile, we show the existence of the finite time blow-up solutions for arbitrarily high initial energy. Finally, we derive the life span bounds (i.e., the lower and upper bounds of the blow-up time).

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Global well-posedness of solutions for 2-D Klein–Gordon equations with exponential nonlinearity

This paper considers the global well-posedness of two-dimensional Klein–Gordon equations with exponential nonlinearity. By employing the potential well method, we conduct a comprehensive study on the global existence and finite time blowup of solutions by the requirement of the initial energy at three different initial energy levels.

On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

Abstract In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given.

On the regularity of the De Gregorio model for the 3D Euler equations

We study the regularity of the De Gregorio (DG) model \omega_{t} + u\omega_{x} = u_{x} \omega on S^{1} for initial data \omega_{0} with period \pi and in class X : \omega_{0} is odd and \omega_{0} \leq 0 (or \omega_{0} \geq 0 ) on [0,\pi/2] . These sign and symmetry properties are the same as those of the smooth initial data that lead to singularity formation of the De Gregorio model on \mathbb{R} or the generalized Constantin–Lax–Majda (gCLM) model on \mathbb{R} or S^{1} with a positive parameter. Thus, to establish global regularity of the DG model for general smooth initial data, which is a conjecture on the DG model, an important step is to rule out potential finite time blowup from smooth initial data in X . We accomplish this by establishing a one-point blowup criterion and proving global well-posedness for initial data \omega_{0} \in H^{1} \cap X with \omega_{0}(x) x^{-1}\in L^{\inf} . On the other hand, for any \alpha \in (0,1) , we construct a finite time blowup solution from a class of initial data with \omega_{0} \in C^{\alpha}\cap C^{\infty}(S^{1} \setminus \{0\}) \cap X . Our results imply that singularities developed in the DG model and the gCLM model on S^{1} can be prevented by stronger advection.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \left\{\begin{array}{ll} \partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) & \hbox{ in }~ \mathbb{H}^{n}\times (0, T), \\ \quad u =u_{0} & \hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \tag{0.1} \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, (0.1) exhibits Fujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e., there exists a critical exponent $ \mu^*$ such that if $\mu > \mu^*$, then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$, there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that (0.1) with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system ut=Δu−χ∇⋅(uf(|∇v|)∇v),0=Δv−M+u,with homogeneous Neumann boundary condition is considered in a radially symmetric domain Ω=BR(0)⊂RN(N≥3), where f(ξ)=(ξp−2(1+ξp)q−pp),ξ≥0,p≥2,1<q≤p<∞,and BR(0) is a N-dimensional ball of radius R>0. We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when NN−1<q<2.

Optimal mass on the parabolic-elliptic-ODE minimal chemotaxis-haptotaxis in

In this paper, we study the Cauchy problem to a chemotaxis-haptotaxis model describing cancer invasion in . The main feature is to prove that 8π is the critical mass on initial data for distinguishing existence and blow-up of solutions to the model. Namely, when the initial mass is less than 8π, we prove global existence of solutions by constructing a proper free energy and using the Brezis-Merle type inequality. On the contrary, the finite time blow-up of solutions may occur if the initial mass is greater than 8π and the initial second moment is small enough. Comparing to the classical Keller-Segel model, we find that the haptotaxis has no effect on the critical mass of the model.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.