On global solvability of nonlinear parabolic boundary-value problems

In this paper one considers nonlinear parabolic boundary-value problems of a general form. It is known that the solution of such problems can go to infinity in a finite interval of time. One shows that this effect is in a certain sense of a finite-dimensional character. Namely, one shows that if the solution is considered on the segment $[0,T]$, while the right-hand sides are bounded in the norm by a constant $R$ and satisfy a finite number of conditions, then the problem admits a solution which is smooth for $0\leqslant t\leqslant T$ (the number of conditions depends on $R$ and $T$).
Bibliography: 11 titles.