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Difference between revisions of "Parabolic partial differential equation"

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An equation (cf. [[Differential equation, partial|Differential equation, partial]]) of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712101.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712102.png" /></td> </tr></table>
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An equation (cf. [[Differential equation, partial|Differential equation, partial]]) of the form
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712103.png" /> is a positive-definite quadratic form. The variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712104.png" /> is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the [[Heat equation|heat equation]]
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712105.png" /></td> </tr></table>
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$$
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u _ {t} - \sum _ { i,j=1}^n  a _ {ij} ( x, t) u _ {x _ {i}  x _ {j} } - \sum
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_ { i=1}^ { n }  a _ {i} ( x, t) u _ { x _ i } - a( x, t) u =
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f( x, t),
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$$
  
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where  $  \sum a _ {ij} \xi _ {i} \xi _ {j} $
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is a positive-definite quadratic form. The variable  $  t $
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is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the [[Heat equation|heat equation]]
  
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$$
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u _ {t} - \sum _ { i=1}^{ n }  u _ {x _ {i}  x _ {i} }  =  0.
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$$
  
 
====Comments====
 
====Comments====
The above defines second-order linear parabolic differential equations. There also exist notions of non-linear parabolic equations. For instance, in [[#References|[a2]]] equations are studied of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712107.png" /> is a function of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712108.png" /> such that for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p0712109.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p07121010.png" /> on the domain under consideration.
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The above defines second-order linear parabolic differential equations. There also exist notions of non-linear parabolic equations. For instance, in [[#References|[a2]]] equations are studied of the form $  \phi _ {t} = F( \phi _ {x _ {i}  x _ {j} } , \phi _ {x _ {i}  } , \phi , x _ {i} , t ) $,  
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where $  F $
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is a function of variables $  ( u _ {ij} , u _ {i} , u , x _ {i} , t ) $
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such that for a certain $  \epsilon > 0 $
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one has $  \epsilon  \| \lambda \|  ^ {2} \leq  \sum _ {i,j} F _ {u _ {ij}  } \lambda _ {i} \lambda _ {j} $
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on the domain under consideration.
  
A semi-linear partial differential equation of the second order, i.e. one of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p07121011.png" />, is said to be of parabolic type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071210/p07121012.png" /> at each point of the domain under consideration.
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A semi-linear partial differential equation of the second order, i.e. one of the form $  \sum _ {i,j = 1 }  ^ {n} a _ {ij} ( \partial  ^ {2} \phi ) / ( \partial  x _ {i} \partial  x _ {j} )= f ( x , \phi , \phi _ {x _ {i}  } ) $,  
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is said to be of parabolic type if $  \mathop{\rm det}  ( a _ {ij} ) = 0 $
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at each point of the domain under consideration.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)  {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.V. Krylov,  "Nonlinear elliptic and parabolic equations of the second order" , Reidel  (1987)  (Translated from Russian)  {{MR|0901759}} {{ZBL|0619.35004}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)  {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.V. Krylov,  "Nonlinear elliptic and parabolic equations of the second order" , Reidel  (1987)  (Translated from Russian)  {{MR|0901759}} {{ZBL|0619.35004}} </TD></TR></table>

Latest revision as of 01:00, 24 December 2020


An equation (cf. Differential equation, partial) of the form

$$ u _ {t} - \sum _ { i,j=1}^n a _ {ij} ( x, t) u _ {x _ {i} x _ {j} } - \sum _ { i=1}^ { n } a _ {i} ( x, t) u _ { x _ i } - a( x, t) u = f( x, t), $$

where $ \sum a _ {ij} \xi _ {i} \xi _ {j} $ is a positive-definite quadratic form. The variable $ t $ is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the heat equation

$$ u _ {t} - \sum _ { i=1}^{ n } u _ {x _ {i} x _ {i} } = 0. $$

Comments

The above defines second-order linear parabolic differential equations. There also exist notions of non-linear parabolic equations. For instance, in [a2] equations are studied of the form $ \phi _ {t} = F( \phi _ {x _ {i} x _ {j} } , \phi _ {x _ {i} } , \phi , x _ {i} , t ) $, where $ F $ is a function of variables $ ( u _ {ij} , u _ {i} , u , x _ {i} , t ) $ such that for a certain $ \epsilon > 0 $ one has $ \epsilon \| \lambda \| ^ {2} \leq \sum _ {i,j} F _ {u _ {ij} } \lambda _ {i} \lambda _ {j} $ on the domain under consideration.

A semi-linear partial differential equation of the second order, i.e. one of the form $ \sum _ {i,j = 1 } ^ {n} a _ {ij} ( \partial ^ {2} \phi ) / ( \partial x _ {i} \partial x _ {j} )= f ( x , \phi , \phi _ {x _ {i} } ) $, is said to be of parabolic type if $ \mathop{\rm det} ( a _ {ij} ) = 0 $ at each point of the domain under consideration.

References

[a1] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) MR0181836 Zbl 0144.34903
[a2] N.V. Krylov, "Nonlinear elliptic and parabolic equations of the second order" , Reidel (1987) (Translated from Russian) MR0901759 Zbl 0619.35004
How to Cite This Entry:
Parabolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_partial_differential_equation&oldid=28257
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article