@article{63250,
abstract = {{Abstract
An initial-boundary value problem for
$$\begin{aligned} \left\{ \begin{array}{ll}u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + au_{xx} - \big (f(\Theta )\big )_x, \qquad & x\in \Omega , \ t>0, \\[1mm] \Theta _t = \Theta _{xx} + \gamma (\Theta ) u_{xt}^2 - f(\Theta ) u_{xt}, \qquad & x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$
u
tt
=
(
γ
(
Θ
)
u
xt
)
x
+
a
u
xx
-
(
f
(
Θ
)
)
x
,
x
∈
Ω
,
t
>
0
,
[
1
m
m
]
Θ
t
=
Θ
xx
+
γ
(
Θ
)
u
xt
2
-
f
(
Θ
)
u
xt
,
x
∈
Ω
,
t
>
0
,
is considered in an open bounded real interval
$$\Omega $$
Ω
. Under the assumption that
$$\gamma \in C^0([0,\infty ))$$
γ
∈
C
0
(
[
0
,
∞
)
)
and
$$f\in C^0([0,\infty ))$$
f
∈
C
0
(
[
0
,
∞
)
)
are such that
$$f(0)=0$$
f
(
0
)
=
0
, and
$$k_\gamma \le \gamma \le K_\gamma $$
k
γ
≤
γ
≤
K
γ
as well as
$$\begin{aligned} |f(\xi )| \le K_f \cdot (\xi +1)^\alpha \qquad \hbox {for all } \xi \ge 0 \end{aligned}$$
|
f
(
ξ
)
|
≤
K
f
·
(
ξ
+
1
)
α
for all
ξ
≥
0
with some
$$k_\gamma>0, K_\gamma>0, K_f>0$$
k
γ
>
0
,
K
γ
>
0
,
K
f
>
0
and
$$\alpha <\frac{3}{2}$$
α
<
3
2
, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. By particularly covering the thermodynamically consistent choice
$$f\equiv id$$
f
≡
i
d
of predominant physical relevance, this appears to go beyond previous related literature which seems to either rely on independence of
$$\gamma $$
γ
on
$$\Theta $$
Θ
, or to operate on finite time intervals.
}},
author = {{Winkler, Michael}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
number = {{5}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities}}},
doi = {{10.1007/s00033-025-02582-y}},
volume = {{76}},
year = {{2025}},
}
@article{63283,
abstract = {{AbstractThe parabolic problem $$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta \big (u\phi (v)\big ), \\ v_t=\Delta v-uv, \end{array} \right. \end{aligned}$$
u
t
=
Δ
(
u
ϕ
(
v
)
)
,
v
t
=
Δ
v
-
u
v
,
is considered in smoothly bounded subdomains of $${\mathbb {R}}^n$$
R
n
with arbitrary $$n\ge 1$$
n
≥
1
. Under the assumptions that $$\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))$$
ϕ
∈
C
0
(
[
0
,
∞
)
)
∩
C
3
(
(
0
,
∞
)
)
is positive on $$(0,\infty )$$
(
0
,
∞
)
and satisfies $$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^\alpha }>0 \quad {\text{ and }} \quad \limsup _{\xi \searrow 0} \big \{ \xi ^\beta |\phi '(\xi )| \big \}<\infty \end{aligned}$$
lim inf
ξ
↘
0
ϕ
(
ξ
)
ξ
α
>
0
and
lim sup
ξ
↘
0
{
ξ
β
|
ϕ
′
(
ξ
)
|
}
<
∞
with some $$\alpha >0$$
α
>
0
and $$\beta >0$$
β
>
0
, for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices $$\alpha =0$$
α
=
0
and $$\alpha \in (0,2)$$
α
∈
(
0
,
2
)
, or were restricted to low-dimensional settings by requiring that $$n\le 2$$
n
≤
2
.}},
author = {{Winkler, Michael}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration–consumption interaction}}},
doi = {{10.1007/s00033-022-01925-3}},
volume = {{74}},
year = {{2023}},
}
@article{34673,
author = {{Black, Tobias and Fuest, Mario and Lankeit, Johannes}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
keywords = {{Applied Mathematics, General Physics and Astronomy, General Mathematics}},
number = {{3}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems}}},
doi = {{10.1007/s00033-021-01524-8}},
volume = {{72}},
year = {{2021}},
}
@article{34675,
author = {{Black, Tobias and Wu, Chunyan}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
keywords = {{Applied Mathematics, General Physics and Astronomy, General Mathematics}},
number = {{4}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation}}},
doi = {{10.1007/s00033-021-01565-z}},
volume = {{72}},
year = {{2021}},
}
@article{63377,
author = {{Winkler, Michael}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation}}},
doi = {{10.1007/s00033-018-0935-8}},
volume = {{69}},
year = {{2018}},
}