On a problem with free boundary for parabolic equations

This paper considers the problem of determining a solution of the parabolic equation
$$ L\theta\equiv D_t\theta-\sum^2_{i,j=1}D_i(a_{ij}(x,t,\theta)\cdot D_j\theta)+a(x,t,\theta,D\theta)=0 $$
and the boundary of the two-dimensional region in which a solution of the equation is sought in the case where on the free boundary the value of the desired function and the additional condition
$$ \sum^2_{i,j=1}a_{ij}D_i\theta\cdot D_j\theta=g(x,t) $$
are satisfied.
For this problem a theorem asserting the existence of a smooth solution on a small time interval is proved. If $L\theta=0$ is the heat equation, then the solution exists on any time interval, and it is unique.
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