Geodesic
On~global solvability of nonlinear parabolic boundary-value problems
Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 89-104

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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. 
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     author = {A. V. Babin},
     title = {On~global solvability of nonlinear parabolic boundary-value problems},
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A. V. Babin. On~global solvability of nonlinear parabolic boundary-value problems. Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 89-104. http://geodesic.mathdoc.fr/item/SM_1975_26_1_a4/