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We prove an approximation result for a class of functionals G% %EndExpansion (u)=_Ω φ (x,Du) defined on BV( Ω ) where φ (· ,Du)L^1( Ω ) , Ω R ^N bounded, φ (x,p) convex, radially symmetric and of the form equation* φ (x,p)={ arrayll g(x,p) & if |p|≤ β ψ (x)|p|+k(x) & if |p|>β .% array% . equation*% We show for each uBV( Ω ) L^p( Ω ) , 1≤ p<∞ , there exist u_kW^1,1( Ω ) C^∞ ( Ω ) L^p( Ω ) so that G% %EndExpansion (u_k)→ %TCIMACROmathcal GG % %BeginExpansion G% %EndExpansion (u). Approximation theorems in BV are used to prove existence results for the strong solution to the time flow u_t=div( abla _pφ (x,Du) ) in L^1((0,∞ );BV( Ω ) L^p( Ω ) ), typically with additional boundary condition or penalty term in u to ensure uniqueness. The functions in this work are not covered by previous approximation theorems since for fixed p we have φ (x,p)L^1( Ω ) which do not in general hold for assumptions on φ in earlier work. , number=4, journal=European Journal of Pure and Applied Mathematics, year=2023, month=Oct., pages=2025–2034