A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We prove that the initial value problem for a non-linear Schrödinger equation is well-posed in the Besov space [Formula: see text], where the nonlinearity is of type |u|αu. This allows to obtain self-similar solutions, and to recover previous results under weaker smallness assumptions on the data.

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ON SELF-SIMILAR SOLUTIONS, WELL-POSEDNESS AND THE CONFORMAL WAVE EQUATION

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000658 ◽

2002 ◽

Vol 04 (02) ◽

pp. 211-222 ◽

Cited By ~ 4

Author(s):

FABRICE PLANCHON

Keyword(s):

Wave Equation ◽

Linear Wave ◽

Well Posedness ◽

Initial Value ◽

Linear Wave Equation ◽

Conformally Invariant ◽

Self Similar ◽

Similar Solutions ◽

Well Posed ◽

Homogeneous Data

We prove that the initial value problem for the conformally invariant semi-linear wave equation is well-posed in the Besov space [Formula: see text]. This induces the existence of (non-radially symmetric) self-similar solutions for homogeneous data in such Besov spaces.

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Annular self-similar solutions in ideal magnetogasdynamics

Journal of Plasma Physics ◽

10.1017/s0022377808007101 ◽

2008 ◽

Vol 74 (4) ◽

pp. 531-554 ◽

Cited By ~ 14

Author(s):

R. M. LOCK ◽

A. J. MESTEL

Keyword(s):

Magnetic Field ◽

Initial Value Problem ◽

Numerical Solutions ◽

Initial Conditions ◽

The Self ◽

Initial Value ◽

Azimuthal Magnetic Field ◽

Self Similar ◽

Similar Solutions ◽

Z Pinch

AbstractWe consider the possibility of self-similar solutions describing the implosion of hollow cylindrical annuli driven by an azimuthal magnetic field, in essence a self-similar imploding liner z-pinch. We construct such solutions for gasdynamics, for ideal ‘β=0’ plasma and for ideal magnetogasdynamics (MGD). In the latter two cases some quantities are singular at the annular boundaries. Numerical solutions of the full ideal MGD initial value problem indicate that the self-similar solutions are not attractive for arbitrary initial conditions, possibly as a result of flux-freezing.

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Crystalline flow starting from a general polygon

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021182 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Mi-Ho Giga ◽

Yoshikazu Giga ◽

Ryo Kuroda ◽

Yusuke Ochiai

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Value Problem ◽

Initial Value ◽

Singular Initial Value Problem ◽

Self Similar

<p style='text-indent:20px;'>This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.</p>

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Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

Nonautonomous Dynamical Systems ◽

10.1515/msds-2018-0008 ◽

2018 ◽

Vol 5 (1) ◽

pp. 102-112 ◽

Cited By ~ 5

Author(s):

Shekhar Singh Negi ◽

Syed Abbas ◽

Muslim Malik

Keyword(s):

Time Scales ◽

Initial Value Problem ◽

Dynamic Equation ◽

Nonlinear Dynamic ◽

Type Inequality ◽

Second Order ◽

Oscillation Criterion ◽

Initial Value ◽

Nonlinear Dynamic Equation ◽

Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

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