Geodesic
On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity
Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 399-410

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For linear parabolic equations of second order it is proved that the solution of the first boundary value problem may remain bounded at interior points in spite of the fact that the boundary function tends to infinity together with the time variable if there are lower order terms having definite signs and increasing sufficiently fast in absolute value. For quasilinear parabolic (possibly degenerate) equations of second order it is established that decay of the lower order coefficients as the spatial coordinates tend to infinity may entail the disappearance of effects of total stabilization in finite time and of instantaneous compactification of the support of the solution. Bibliography: 11 titles. 
@article{SM_1986_53_2_a6,
     author = {A. S. Kalashnikov},
     title = {On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity},
     journal = {Sbornik. Mathematics},
     pages = {399--410},
     publisher = {mathdoc},
     volume = {53},
     number = {2},
     year = {1986},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/}
}
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A. S. Kalashnikov. On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity. Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 399-410. http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/