In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: {ut=αuxx+β[φ(u)]xx+f(u)inQ:=Ω×(0,T),u=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}=\\alpha u_{xx}+\\beta [\\varphi (u) ]_{xx}+f(u) &\\text{in} \\ Q:=\\Omega \\times (0,T), \\\\ u=0 &\\text{on} \\ \\partial \\Omega \\times (0,T), \\\\ u(x,0)=u_{0}(x) &\\text{in} \\ \\Omega , \\end{cases} $$\\end{document} where T>0, Omega subset mathbb{R} is a bounded interval, u_{0} is nonnegative bounded Radon measure on Ω, and alpha , beta geq 0, under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.