Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

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: $$ {cases} uₜ=α uₓₓ+β [φ (u) ]ₓₓ+f(u) &in \ Q:=Ω × (0,T), \\ u=0 &on \ ∂ Ω × (0,T), \\ u(x,0)=u₀(x) &in \ Ω , {cases} u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , whereT>0$ T > 0 , $Ω ⊂ R$ Ω ⊂ R is a bounded interval, $u₀$ u 0 is nonnegative bounded Radon measure on Ω, and $α , β ≥ 0$ α , β ≥ 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.