The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When $ m\in(1,2] $, it is shown that there exists a smooth solution for $ t>0 $ when the initial data belongs to $ L_{\text{loc}}^p, p>1 $. When $ m>2 $, the same results holds when the initial data belongs to $ W_{\text{loc}}^{1,p}, p\geq m-1 $. An example is displayed to show that a smooth solution may not exist when the initial data is merely in $ L^p_{\text{loc}}, p>1 $. Solvability of the weak solution is also studied.
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