Finite Time Blow-up Of Solutions Research Articles

Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem

In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann–Robin boundary conditions, for p -Laplacian, has finite time blow-up solutions.

Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level

In this paper we study the initial boundary value problem of a class of semilinear parabolic equation. Our main tools are the comparison principle and variational methods. In this paper, we will find both finite time blow-up and global solutions at high energy level.

Nonlinear Schrödinger equations with coupled Hartree-type terms and rotation

We consider the Cauchy problem for a class of magnetic Schrödinger system with local and nonlocal nonlinearities. The problem stems from a typical model describing the mean-field dynamics of rotating many-body bosons in a confining trap. We present sufficient conditions which yield global well-posedness or finite time blowup solutions to the system.

Optimal lower bound estimates for the blow-up rate for the Zakharov system in a nonhomogeneous medium

In this paper we obtain lower bound estimates for the blow-up rate of finite time blow-up solutions to the Cauchy problem for the Zakharov system in a nonhomogeneous medium in two space dimensions. By introducing suitable scale transformations of space and time, and the use of compactness arguments, we derive an optimal lower bound estimate in the energy space H 2 ( R 2 ) × L 2 ( R 2 ) × H 1 ( R 2 ) for the blow-up rate for t near the finite blow-up time T. Also we give an application to the virial identity for the Zakharov system under study.

Precise bounds for finite time blowup of solutions to very general one-space-dimensional nonlinear Neumann problems

In this paper we analyze the asymptotic finite time blowup of solutions to the heat equation with a nonlinear Neumann boundary flux in one space dimension. We perform a detailed examination of the nature of the blowup, which can occur only at the boundary, and we provide tight upper and lower bounds for the blowup rate for “arbitrary” nonlinear functions F F , subject to very mild restrictions.

Shock formation in a traffic flow model with Arrhenius look-ahead dynamics

We consider a nonlocal traffic flow model with Arrhenius look-ahead dynamics. We provide a complete local theory and give the blowup alternative of solutions to the conservation law with a nonlocal flux. We show that the finite time blowup of solutions must occur at the level of the first order derivative of the solution. Furthermore, we prove that finite time singularities do occur for several types of physical initial data by analyzing the solutions on different characteristic lines. These results are new and are consistent with the blowups observed in previous numerical simulations on the nonlocal traffic flow model [6].

Global existence and blow-up of solutions for generalized Pochhammer-Chree equations

In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation u t t - u x x - u x x t - u x x t t = f ( u ) x x , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( 0 , t ) = u ( 1 , t ) = 0 , t ≥ 0 , where Ω =(0,1). First, we obtain the existence of local W k,p solutions. Then, we prove that, if f ( s ) ∈ C k + 1 ( R ) is nondecreasing, f(0) = 0 and | f ( u ) | ≤ C 1 | u | ∫ 0 u f ( s ) d s + C 2 , u 0 ( x ) , u 1 ( x ) ∈ W k , p ( Ω ) ∩ W 0 1 , p ( Ω ) , k ≥ 1 , 1 < p ≤ ∞ , then for any T > 0 the problem admits a unique solution u ( x , t ) ∈ W 2 , ∞ ( 0 , T ; W k , p ( Ω ) ∩ W 0 1 , p ( Ω ) ) . Finally, the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.

Solutions of the Generalized Rotation-Modified Kadomtsev-Petviashvili Equation

Abstract The rotation-modified Kadomtsev-Petviashvili equation is a modification of the Kadomtsev-Petviashvili (KP) equation widely used to describe the effects of rotation on small-amplitude, long waves propagating in one primary direction. In this paper we investigate conditions for the finite-time blow-up solutions and the uniform bounds of the solution of the generalized rotation-modified Kadomtsev-Petviashvili equation. The conditions are expressed in terms of the energy and the best constant of the anisotropic Sobolev inequality.

Large Solutions for Semilinear Parabolic Equations Involving Some Special Classes of Nonlinearities

We consider a new class of nonlinearities for which a nonlocal parabolic equation with Neumann boundary conditions has finite time blow‐up solutions. Our approach is inspired by previous work done by Jazar and Kiwan (2008) and El Soufi et al. (2007).

On the blow-up phenomenon for the mass-critical focusing Hartree equation in R4

We characterize the dynamics of the finite time blow-up solutions with minimal mass for the focusing mass-critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo&#8211;Nirenberg in

On Stability of Pseudo-Conformal Blowup for L 2-critical Hartree NLS

We consider L 2-critical focusing nonlinear Schrödinger equations with Hartree type nonlinearity $$i \partial_{t} u = - \Delta u - \left(\Phi \ast |u|^2 \right) u \quad {\rm in}\, \mathbb {R}^4,$$where Φ(x) is a perturbation of the convolution kernel |x|−2. Despite the lack of pseudo-conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions u(t, x) that exhibit the pseudo-conformal blowup rate $$\| \nabla u(t) \|_{L^2_x}\sim \frac{1}{|t|} \quad {\rm as}\, t \nearrow 0.$$Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see Bourgain and Wang in Ann. Scuola Norm Sup Pisa Cl Sci (4) 25(1–2), 197–215, 1997/1998) to L 2-critical Hartree NLS.

ON GLOBAL SOLUTIONS AND BLOW-UP OF SOLUTIONS FOR A NONLINEARLY DAMPED PETROVSKY SYSTEM

We consider the initial boundary value problem for a Petrovsky system with nonlinear damping \begin{equation*} u_{tt}+\Delta ^{2}u+a\left| u_{t}\right| ^{m-2}u_{t}=b\left| u\right| ^{p-2}u, \end{equation*} in a bounded domain. We showed that the solution is global in time under some conditions without the relation between $m$ and $p$. We also prove that the local solution blows-up in finite time if $p\gt m$ and the initial energy is nonngeative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of ([6]).

Blow-up solutions of the Chern–Simons–Schrödinger equations

We study finite time blow-up solutions of the Chern–Simons–Schrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.

GROUND STATE MASS CONCENTRATION IN THE L2-CRITICAL NONLINEAR HARTREE EQUATION BELOW H1

We consider finite time blowup solutions of the -critical focusing Hartree equation on , below .

Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources

This paper investigates the global existence of the nonnegative solution and the finite time blow-up of solutions of nonlinear parabolic equation with a more complicated source term, which is a product of localized source, local source, and weight function; we also study the blow-up rate of solution to this problem.

Well-posedness of modified Camassa–Holm equations

The Camassa–Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H 1 metric. We derive the geodesic equations on that group with respect to the invariant H k metric, which we call the modified Camassa–Holm equation, and then study the well-posedness and dynamics of a modified Camassa–Holm equation on the unit circle S, which has some significant difference from that of Camassa–Holm equation, e.g., it does not admit finite time blowup solutions.

ON GLOBAL SOLUTIONS AND BLOW-UP OF SOLUTIONS FOR A NONLINEARLY DAMPED PETROVSKY SYSTEM

We consider the initial boundary value problem for a Petrovsky system with nonlinear damping \begin{equation*} u_{tt}+\Delta ^{2}u+a\left| u_{t}\right| ^{m-2}u_{t}=b\left| u\right| ^{p-2}u, \end{equation*} in a bounded domain. We showed that the solution is global in time under some conditions without the relation between $m$ and $p$. We also prove that the local solution blows-up in finite time if $p>m$ and the initial energy is nonngeative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of ([6]).

Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering

We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type.We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.

Refined blowup criteria and nonsymmetric blowup of an aggregation equation

We consider an aggregation equation in R d , d ⩾ 2 , with fractional dissipation: u t + ∇ ⋅ ( u ∇ K ∗ u ) = − ν Λ γ u , where ν ⩾ 0 , 0 < γ < 1 , and K ( x ) = e − | x | . We prove a refined blowup criteria by which the global existence of solutions is controlled by its L x q norm, for any d d − 1 ⩽ q ⩽ ∞ . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν = 0 and the supercritical case ν > 0 and 0 < γ < 1 . Additionally, we present new proofs of blowup which does not use free energy arguments.

S-I-R Model with Directed Spatial Diffusion

A S-I-R epidemic model is described in which susceptible individuals move away from foci of infection, and all individuals move away from overcrowded regions. It consists of hyperbolic partial differential equations, the sum of these equations being parabolic. Positivity and regularity of solutions are discussed and finite time blow-up of some solutions is illustrated through numerical simulations. A numerical test of the finite time blow-up of solutions is proposed.