Conditions for positivity of solutions of a class of dissipative partial differential equations

1998 ◽  
Vol 11 (6) ◽  
pp. 71-74 ◽  
Author(s):  
M.V. Bartuccelli ◽  
S.A. Gourley ◽  
C.J. Woolcock
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Positivity Of Solutions ◽  
Dissipative Partial Differential Equations

On the crest factor for dissipative partial differential equations

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190322
Author(s):  
Michele V. Bartuccelli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Positive Parameter ◽  
Crest Factor ◽  
Partial Differential ◽  
Large Potential ◽  
Dissipative Partial Differential Equations

In this work, we have introduced and then computed the so-called crest factor associated with solutions of dissipative partial differential equations (PDEs). By taking two paradigmatic dissipative PDEs, we estimated in an explicit and accurate manner the values of the crest factor of their solutions. We then analysed and compared the estimates as a function of the positive parameter which appears in the PDEs in space dimensions one and two. These estimates shed some light on the dynamics of the fluctuations of the solutions of the two model PDEs, and therefore provide a criterion for discerning between small and large potential excursions in space for the solution of any dissipative PDE. Being able to detect between small and large intermittent fluctuations is one of the hallmarks of turbulence. We believe that the crest factor is an appropriate tool for extracting space fluctuation features in solutions of dissipative PDEs.

The implicit function theorem and multi-bump solutions of periodic partial differential equations

2006 ◽  
Vol 136 (3) ◽  
pp. 559-583 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Implicit Function Theorem ◽  
Superposition Principle ◽  
Implicit Function ◽  
Partial Differential ◽  
Degenerate Solution ◽  
Positivity Of Solutions ◽  
Perturbation Arguments

A modification of the implicit function theorem is advanced for cases where the continuity of the derivative fails. It is applied to a superposition principle for periodicpartial differential equations. The assumption of the principle, that there should exist a non-degenerate solution, is studied and instances of it realized using perturbation arguments and scaling. The positivity of solutions is considered.

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