Self Similar Solutions for a Degenerate Cauchy Problem.

1984 ◽  
Author(s):  
K. Hoellig ◽  
J. A. Nohel
Keyword(s):  
Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions ◽  
Degenerate Cauchy Problem

ON THE CAUCHY PROBLEM IN BESOV SPACES FOR A NON-LINEAR SCHRÖDINGER EQUATION

2000 ◽  
Vol 02 (02) ◽  
pp. 243-254 ◽  
Author(s):  
FABRICE PLANCHON
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Initial Value ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions ◽  
Well Posed

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.

On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

2011 ◽  
Vol 22 (3) ◽  
pp. 245-265 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Free Boundary ◽  
Boundary Problem ◽  
Similarity Solutions ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

On self-similar solutions of Barenblatt's nonlinear filtration equation

1996 ◽  
Vol 7 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Julian D. Cole ◽  
Barbara A. Wagner
Keyword(s):  
Cauchy Problem ◽  
Linear Problem ◽  
Asymptotic Form ◽  
Similarity Solutions ◽  
The Self ◽  
Nonlinear Filtration ◽  
Filtration Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions

We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

A NOTE ON THE NONLINEAR SCHRÖDINGER EQUATION IN WEAK LpSPACES

2001 ◽  
Vol 03 (01) ◽  
pp. 153-162 ◽  
Author(s):  
THIERRY CAZENAVE ◽  
LUIS VEGA ◽  
MARI CRUZ VILELA
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
The Self ◽  
Initial Value ◽  
Common Framework ◽  
Self Similar ◽  
Similar Solutions ◽  
Global Cauchy Problem

We study the global Cauchy problem for the equation iut+Δu+λ|u|αu=0 in ℝN. Using generalized Strichartz' inequalities we show that, under some restrictions on α, if the initial value is sufficiently small in some weak Lpspace then there exists a global solution. This result provides a common framework to the "classical" Hssolutions and to the self-similar solutions, thereby extending previous results by Planchon.

scholarly journals SPECTRAL PROPERTIES AND LINEAR STABILITY OF SELF-SIMILAR WAVE MAPS

2009 ◽  
Vol 06 (02) ◽  
pp. 359-370 ◽  
Author(s):  
ROLAND DONNINGER ◽  
PETER C. AICHELBURG
Keyword(s):  
Cauchy Problem ◽  
Spectral Properties ◽  
Well Posedness ◽  
Wave Maps ◽  
Growth Estimate ◽  
Three Sphere ◽  
Rotational Wave ◽  
Linear Perturbations ◽  
Self Similar ◽  
Similar Solutions

We study co-rotational wave maps from (3 + 1)-Minkowski space to the three-sphere S3. It is known that there exists a countable family {fn} of self-similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well-posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from fn by letting n → ∞.

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