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In this paper we consider the $\bar\partial$-problem in fiber bundles (fibersbiholomorphic to $\mathbb C^k$, $k\geq 1$), namely the equation$\bar\partial\sigma =\omega$ for $(0,1)$-forms $\omega$ which decrease alongthe fibers. The order of decrease is slightly more than one. The important factis that we do not assume that $\omega$ has compact support. The main theoremsays that the equation has a solution which also decreases along fibers,however, not necessarily with the order as the original form. Existence ofsolution of the above mentioned $\bar\partial$-problem can be applied invarious situations in Complex Analysis, in particular, to the Hartogs extensionphenomenon.

$\bar\partial$-problem in fiber bundles for decreasing $(0,1)$-forms