@article{59258,
author = {{Winkler, Michael}},
issn = {{0095-4616}},
journal = {{Applied Mathematics & Optimization}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters}}},
doi = {{10.1007/s00245-025-10243-9}},
volume = {{91}},
year = {{2025}},
}
@article{63344,
abstract = {{Abstract
A Neumann-type initial-boundary value problem for
$$\begin{aligned} \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma (\Theta ) \nabla u_t) + a \nabla \cdot (\gamma (\Theta ) \nabla u) + \nabla \cdot f(\Theta ), \\ \Theta _t = D\Delta \Theta + \Gamma (\Theta ) |\nabla u_t|^2 + F(\Theta )\cdot \nabla u_t, \end{array} \right. \end{aligned}$$
u
tt
=
∇
·
(
γ
(
Θ
)
∇
u
t
)
+
a
∇
·
(
γ
(
Θ
)
∇
u
)
+
∇
·
f
(
Θ
)
,
Θ
t
=
D
Δ
Θ
+
Γ
(
Θ
)
|
∇
u
t
|
2
+
F
(
Θ
)
·
∇
u
t
,
is considered in a smoothly bounded domain
$$\Omega \subset \mathbb {R}^n$$
Ω
⊂
R
n
,
$$n\ge 1$$
n
≥
1
. In the case when
$$n=1$$
n
=
1
,
$$\gamma \equiv \Gamma $$
γ
≡
Γ
and
$$f\equiv F$$
f
≡
F
, this system coincides with the standard model for heat generation in a viscoelastic material of Kelvin-Voigt type, well-understood in situations in which
$$\gamma =const$$
γ
=
c
o
n
s
t
. Covering scenarios in which all key ingredients
$$\gamma ,\Gamma ,f$$
γ
,
Γ
,
f
and F may depend on the temperature
$$\Theta $$
Θ
here, for initial data which merely satisfy
$$u_0\in W^{1,p+2}(\Omega )$$
u
0
∈
W
1
,
p
+
2
(
Ω
)
,
$$u_{0t}\in W^{1,p}(\Omega )$$
u
0
t
∈
W
1
,
p
(
Ω
)
and
$$\Theta _0\in W^{1,p}(\Omega )$$
Θ
0
∈
W
1
,
p
(
Ω
)
with some
$$p\ge 2$$
p
≥
2
such that
$$p>n$$
p
>
n
, a result on local-in-time existence and uniqueness is derived in a natural framework of weak solvability.}},
author = {{Winkler, Michael}},
issn = {{0095-4616}},
journal = {{Applied Mathematics & Optimization}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters}}},
doi = {{10.1007/s00245-025-10243-9}},
volume = {{91}},
year = {{2025}},
}