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

Abstract

Abstract 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: { 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 , $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ where T > 0 $T>0$ , Ω ⊂ R $\Omega \subset \mathbb{R}$ is a bounded interval, u 0 $u_{0}$ is nonnegative bounded Radon measure on Ω, and α , β ≥ 0 $\alpha , \beta \geq 0$ , 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.

Keywords

• Radon measure-valued solution
• Nonlinear parabolic equations
• Young measure
• Asymptotic behavior