Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u(x, t) ≥ 0, defined in a domain D ⊂ RN × (0, T ) and such that ∆u+ ∑ ai uxi − ut = 0 in D ∩ {u > 0}. We also assume that the interior boundary of the positivity set, D∩∂{u > 0}, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

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