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.
A High-order Exponential Integrator for Nonlinear Parabolic Equations with Nonsmooth Initial Data
