Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev-Petviashvili Equation

1999 ◽  
Vol 201 (3) ◽  
pp. 577-590 ◽  
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin ◽  
Jean-Claude Saut
Keyword(s):  
Large Time ◽  
Global Solutions ◽  
Petviashvili Equation ◽  
Asymptotics For Large Time

Very slow grow-up of solutions of a semi-linear parabolic equation

2011 ◽  
Vol 54 (2) ◽  
pp. 381-400 ◽  
Author(s):  
Marek Fila ◽  
John R. King ◽  
Michael Winkler ◽  
Eiji Yanagida
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Large Time ◽  
Global Solutions ◽  
Linear Parabolic Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Supercritical Nonlinearity ◽  
Singular Steady State

AbstractWe consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

Nonlinear diffraction and caustic formation

1990 ◽  
Vol 430 (1878) ◽  
pp. 69-88 ◽  
Keyword(s):  
Exact Solutions ◽  
Burgers Equation ◽  
Global Solutions ◽  
Similarity Solutions ◽  
New Family ◽  
Nonlinear Diffraction ◽  
Generalized Burgers Equation ◽  
Petviashvili Equation

This paper reports a new family of symmetries to the Zabolotskaya-Khokhlov, dissipative Zabolotskaya-Khokhlov, and Kadomtsev-Petviashvili equations. It also reports the details of the corresponding set of exact similarity solutions to the Zabolotskaya-Khokhlov equation, and the corresponding reduction of the dissipative Zabolotskaya-Khokhlov equation onto the generalized Burgers’ equation, and implies that of the Kadomtsev-Petviashvili equation onto a simpler equation. The bearing that the symmetries and exact solutions have on other work is discussed. The first non-trivial smooth global solutions to the Zabolotskaya-Khokhlov equation are presented, answering a conjecture as to the existence of such. The formation of line caustics is examined, using the exact solutions quasi-statically, giving new results.

Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

2018 ◽  
Vol 15 (02) ◽  
pp. 259-290 ◽  
Author(s):  
Weixuan Shi ◽  
Jiang Xu
Keyword(s):  
Large Time ◽  
Characteristic Root ◽  
Global Solutions ◽  
Strong Solutions ◽  
Parabolic Systems ◽  
Large Time Behavior ◽  
Optimal Time ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Mhd System

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.

Sign in / Sign up

Export Citation Format

Share Document