We study the existence of the global solutions in a model describing the evolution of density of bacteria and oxygen dissolved in water filling a capillary. In the proof of local existence of classical solutions we use Amann theory. The Moser-Alikakos technique is the main tool for the proof of L ∞ boundedness of local solutions.1. E is a nonnegative C 3 function on R with bounded first derivative, 2. E(p) ≡ 0 for p ≤ 0 and E(p) > 0 for p > 0, 3. E has only one local maximum at a point p * , 4. lim p→∞ E(p) = 0.2010 Mathematics Subject Classification: Primary 35B45; Secondary 92C45.

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