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<p/> <p>We study the nonlinear parabolic problem with <inline-formula> <graphic file="1029-242X-2010-163269-i2.gif"/></inline-formula>-growth conditions in the space <inline-formula> <graphic file="1029-242X-2010-163269-i3.gif"/></inline-formula> and give a local boundedness theorem of weak solutions for the following equation <inline-formula> <graphic file="1029-242X-2010-163269-i4.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2010-163269-i5.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-163269-i6.gif"/></inline-formula> satisfy <inline-formula> <graphic file="1029-242X-2010-163269-i7.gif"/></inline-formula>-growth conditions with respect to <inline-formula> <graphic file="1029-242X-2010-163269-i8.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-163269-i9.gif"/></inline-formula>.</p>