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In this paper we give a suitable notion of entropy solution of parabolic$p-$laplacian type equations with $1\leq p<2$ which blows up at the boundary ofthe domain. We prove existence and uniqueness of this type of solutions whenthe initial data is locally integrable (for $1<p<2$) or integrable (for $p=1$;i.e the Total Variation Flow case).

Large solutions for nonlinear parabolic equations without absorption terms