Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

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Differentiability of the blow-up curve for one dimensional nonlinear wave equations

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00280224 ◽

1985 ◽

Vol 91 (1) ◽

pp. 83-98 ◽

Cited By ~ 32

Author(s):

Luis A. Caffarelli ◽

Avner Friedman

Keyword(s):

Nonlinear Wave ◽

Wave Equations ◽

Blow Up ◽

Nonlinear Wave Equations ◽

One Dimensional

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Existence and nonexistence of entire solutions to the logistic differential equation

Abstract and Applied Analysis ◽

10.1155/s1085337503305020 ◽

2003 ◽

Vol 2003 (17) ◽

pp. 995-1003 ◽

Cited By ~ 16

Author(s):

Marius Ghergu ◽

Vicentiu Radulescu

Keyword(s):

Blow Up ◽

Entire Solutions ◽

One Dimensional ◽

Nonexistence Result ◽

Nonnegative Solutions ◽

Existence And Nonexistence ◽

Logistic Problem ◽

Whole Space ◽

The Mean ◽

The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

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FREE-BOUNDARY PROBLEM OF THE ONE-DIMENSIONAL EQUATIONS FOR A VISCOUS AND HEAT-CONDUCTIVE GASEOUS FLOW UNDER THE SELF-GRAVITATION

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202513500127 ◽

2013 ◽

Vol 23 (08) ◽

pp. 1377-1419 ◽

Cited By ~ 3

Author(s):

MORIMICHI UMEHARA ◽

ATUSI TANI

Keyword(s):

Blow Up ◽

A Priori Estimates ◽

A Priori ◽

Global Solvability ◽

The Self ◽

Fundamental Result ◽

System Of Equations ◽

One Dimensional ◽

Unique Existence ◽

The One

In this paper we consider a system of equations describing the one-dimensional motion of a viscous and heat-conductive gas bounded by the free-surface. The motion is driven by the self-gravitation of the gas. This system of equations, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian-mass transformation. For smooth initial data we first establish the temporally global solvability of the problem based on both the fundamental result for local in time and unique existence of the classical solution and a priori estimates of its solution. Second it is proved that some estimates of the global solution are independent of time under a certain restricted, but physically plausible situation. This gives the fact that the solution does not blow up even if time goes to infinity under such a situation. Simultaneously, a temporally asymptotic behavior of the solution is established.

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