On~a~problem with free boundary for parabolic equations

A. M. Meirmanov

Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 473-484

Résumé

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. Bibliography: 7 titles.