New probabilistic approach to the classical heat equation

1988 ◽  
pp. 205-220 ◽  
Author(s):  
J. C. Zambrini
Keyword(s):  
Heat Equation ◽  
Probabilistic Approach

A probabilistic approach to blow-up of a semilinear heat equation

1996 ◽  
Vol 126 (6) ◽  
pp. 1235-1245 ◽  
Author(s):  
Alison M. Etheridge
Keyword(s):  
Markov Process ◽  
Power Series ◽  
Heat Equation ◽  
Series Expansion ◽  
Blow Up ◽  
Probabilistic Approach ◽  
Power Series Expansion ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Special Case

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.

scholarly journals A probabilistic approach to the Yang–Mills heat equation

2002 ◽  
Vol 81 (2) ◽  
pp. 143-166 ◽  
Author(s):  
Marc Arnaudon ◽  
Robert O. Bauer ◽  
Anton Thalmaier
Keyword(s):  
Heat Equation ◽  
Probabilistic Approach ◽  
Yang Mills

SPECTRAL PROBLEMS ARISING IN THE STABILIZATION PROBLEM FOR THE LOADED HEAT EQUATION: A TWO-DIMENSIONAL AND MULTI-POINT CASES

2019 ◽  
Vol 7 (1) ◽  
pp. 23-37
Author(s):  
Jenaliyev M.T. ◽  
◽  
Imanberdiyev K.B. ◽  
Kassymbekova A.S. ◽  
Sharipov K.S. ◽  
...  
Keyword(s):  
Heat Equation ◽  
Two Dimensional ◽  
Stabilization Problem ◽  
Spectral Problems

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

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