Diffeomorphisms of function spaces corresponding to quasilinear parabolic equations

This paper considers a boundary value problem for a quasilinear parabolic equation. In terms of Sobolev and Besov spaces the author determines a solution space $A$ and a space $B$ of initial conditions and right hand members such that the operator corresponding to the boundary value problem is a diffeomorphism, analytic in the Frechet sense, of the whole space $A$ and a domain $\mathscr O$ in the space $B$. The behavior of the inverse operator of the problem around the boundary of $\mathscr O$ is studied, and it is shown that for different problems the domain $\mathscr O$ can coincide with the whole function space or be a strict subset of it.
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