Heat equation in a multidimensional domain with a general stochastic measure

2017 ◽  
Vol 93 ◽  
pp. 1-17 ◽  
Author(s):  
I. M. Bodnarchuk ◽  
G. M. Shevchenko
Keyword(s):  
Heat Equation ◽  
Stochastic Measure ◽  
Multidimensional Domain ◽  
General Stochastic

Existence and uniqueness of solutions to a concentrated capacity problem with change of phase

1990 ◽  
Vol 1 (4) ◽  
pp. 339-351 ◽  
Author(s):  
Daniele Andreucci
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Weak Sense ◽  
Uniqueness Of Solutions ◽  
Limiting Case ◽  
Multidimensional Domain ◽  
Diffusion Problems

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

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