Finite Time Blow-up Of Solutions Research Articles

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.

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the solution to a linear Schr\"odinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solution $e^{it}Q(x)$, where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} < \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{it}Q(x)$, for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} > \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS.

The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System

We study the gravitational Vlasov Poisson system \({f_t + v \cdot \nabla_{x} f - E \cdot \nabla_{v} f = 0}\) = 0 where \({E(x) = \nabla_{x}\phi(x)}\), \({\Delta_{x}\phi = \rho(x)}\), \({\rho(x) = \int_{\mathbb{R}^{N}} f(x, v)\,{\rm d}x\,{\rm d}v}\), in dimension N = 3, 4. In dimension N = 3 where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies, in particular, the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in [11, 16, 19]. In dimension N = 4 where the problem is L1 critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy. We then derive using an explicit pseudo-conformal symmetry the existence of critical mass finite time blow-up solutions, and prove more generally a mass concentration phenomenon for finite time blow up solutions. This is the first result of description of a singularity formation in a Vlasov setting. The global structure of the problem is reminiscent of the one for the focusing nonlinear Schrödinger equation iu t = −Δu−|u|p−1u in the energy space \({H^1(\mathbb{R}^N)}\) .

Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every $$u_0 \in L^1 (\mathbb {R}^2)$$ . The local existence time is characterized for $$u_0 \in L^1 \cup L^{q*}(\mathbb {R}^2)$$ with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption $$||u_0||_{L^{1}} > 8 \pi$$ and $$||x|^2u_0||{L^1} < \frac {1}{\gamma}.g (||u_0||{L^1}/8\pi)$$ , where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)$ where u is the polar angle on the sphere, $Q(r)=2\arctan r$ is the ground state harmonic map, λ(t)=t -1-ν, and $\mathcal{R}(t,r)$ is a radiative error with local energy going to zero as t→0. The number $\nu>\frac{1}{2}$ can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.

Global nonexistence of solutions to a semilinear wave equation in the Minkowski space

This work presents the finite-time blow-up of solutions to the equation u t t − Δ u = a − k | u | p , in the Minkowski space. We extend the previous result of Belchev, Kepka and Zhou [E. Belchev, M. Kepka, Z. Zhou, Finite-time blow-up of solutions to semilinear wave equations, J. Funct. Anal. 190 (1) (2002) 233–254] comprehensively. Due to a modification of the so-called method of conformal compactification used by Belchev, Kepka and Zhou, we show that the solutions blow up in finite time with more relaxed initial data and extended index p .

Existence and Stability of the log–log Blow-up Dynamics for the L 2-Critical Nonlinear Schrödinger Equation in a Domain

Let\(iu_{t} = -\Delta u-|u|^{\frac{4}{N}}u\) be the L2-critical nonlinear Schrodinger equation, in a domain \(\Omega \subset \mathbb{R}^{N}\) with initial data in \(H^{1}_{0}(\Omega)\) (Dirichlet boundary condition) and \(N \leq 4\). We prove existence and stability of finite time blow-up dynamics with the log-log blow-up speed \({|\Delta u(t)|_{{L}^{2}}}\sim\sqrt{\frac{\rm{log|log}({\it T-t})|}{T-t}}.\) Moreover, for a suitable class of finite time blow-up solutions, we derive global rigidity properties which turn out to be modeled after the \(\mathbb{R}^{N}\) ones.