scholarly journals Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Yanchao Gao ◽  
Wenjie Gao
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Variable Exponent ◽  
Nonlinear Parabolic Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Parabolic

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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Quincy Stévène Nkombo ◽  
Fengquan Li ◽  
Christian Tathy
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Parabolic Equations ◽  
Radon Measure ◽  
Decay Estimate ◽  
Young Measure ◽  
Nonlinear Parabolic Equations ◽  
Bounded Interval ◽  
Nonlinear Parabolic ◽  
Alpha Beta

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

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