@article{63672,
author = {{Black, Tobias and Kohatsu, Shohei and Wu, Duan}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption}}},
doi = {{10.1007/s00028-025-01163-w}},
volume = {{26}},
year = {{2026}},
}
@article{63249,
abstract = {{Abstract
The model
$$\begin{aligned} \left\{ \begin{array}{l}u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + au_{xx} - \big (f(\Theta )\big )_x, \\[1mm] \Theta _t = \Theta _{xx} + \gamma (\Theta ) u_{xt}^2 - f(\Theta ) u_{xt}, \end{array} \right. \end{aligned}$$
u
tt
=
(
γ
(
Θ
)
u
xt
)
x
+
a
u
xx
-
(
f
(
Θ
)
)
x
,
[
1
m
m
]
Θ
t
=
Θ
xx
+
γ
(
Θ
)
u
xt
2
-
f
(
Θ
)
u
xt
,
for thermoviscoelastic evolution in one-dimensional Kelvin–Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if
$$\gamma _0>0$$
γ
0
>
0
is fixed, then there exists
$$\delta =\delta (\gamma _0)>0$$
δ
=
δ
(
γ
0
)
>
0
with the property that for suitably regular initial data of arbitrary size an associated initial boundary value problem posed in an open bounded interval admits a global classical solution whenever
$$\gamma \in C^2([0,\infty ))$$
γ
∈
C
2
(
[
0
,
∞
)
)
and
$$f\in C^2([0,\infty ))$$
f
∈
C
2
(
[
0
,
∞
)
)
are such that
$$f(0)=0$$
f
(
0
)
=
0
and
$$|f(\xi )| \le K_f \cdot (\xi +1)^\alpha $$
|
f
(
ξ
)
|
≤
K
f
·
(
ξ
+
1
)
α
for all
$$\xi \ge 0$$
ξ
≥
0
and some
$$K_f>0$$
K
f
>
0
and
$$\alpha <\frac{3}{2}$$
α
<
3
2
, and that
$$\begin{aligned} \gamma _0 \le \gamma (\xi ) \le \gamma _0 + \delta \qquad \hbox {for all } \xi \ge 0. \end{aligned}$$
γ
0
≤
γ
(
ξ
)
≤
γ
0
+
δ
for all
ξ
≥
0
.
This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.
}},
author = {{Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{4}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities}}},
doi = {{10.1007/s00028-025-01144-z}},
volume = {{25}},
year = {{2025}},
}
@article{53316,
abstract = {{AbstractThe quasilinear Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$
u
t
=
∇
·
(
D
(
u
)
∇
u
)
-
∇
·
(
S
(
u
)
∇
v
)
,
v
t
=
Δ
v
-
v
+
u
,
endowed with homogeneous Neumann boundary conditions is considered in a bounded domain $$\Omega \subset {\mathbb {R}}^n$$
Ω
⊂
R
n
, $$n \ge 3$$
n
≥
3
, with smooth boundary for sufficiently regular functions D and S satisfying $$D>0$$
D
>
0
on $$[0,\infty )$$
[
0
,
∞
)
, $$S>0$$
S
>
0
on $$(0,\infty )$$
(
0
,
∞
)
and $$S(0)=0$$
S
(
0
)
=
0
. On the one hand, it is shown that if $$\frac{S}{D}$$
S
D
satisfies the subcritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$
S
(
s
)
D
(
s
)
≤
C
s
α
for
all
s
≥
1
with
some
α
<
2
n
and $$C>0$$
C
>
0
, then for any sufficiently regular initial data there exists a global weak energy solution such that $${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$
ess
sup
t
>
0
‖
u
(
t
)
‖
L
p
(
Ω
)
<
∞
for some $$p > \frac{2n}{n+2}$$
p
>
2
n
n
+
2
. On the other hand, if $$\frac{S}{D}$$
S
D
satisfies the supercritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$
S
(
s
)
D
(
s
)
≥
c
s
α
for
all
s
≥
1
with
some
α
>
2
n
and $$c>0$$
c
>
0
, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value $$\alpha = \frac{2}{n}$$
α
=
2
n
for $$n \ge 3$$
n
≥
3
, without any additional assumption on the behavior of D(s) as $$s \rightarrow \infty $$
s
→
∞
, in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type $$Q(s) = \exp (-s^\beta )$$
Q
(
s
)
=
exp
(
-
s
β
)
, $$s \ge 0$$
s
≥
0
, for global solvability the exponent $$\beta = \frac{n-2}{n}$$
β
=
n
-
2
n
is seen to be critical.
}},
author = {{Stinner, Christian and Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
keywords = {{Mathematics (miscellaneous)}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects}}},
doi = {{10.1007/s00028-024-00954-x}},
volume = {{24}},
year = {{2024}},
}
@article{53542,
abstract = {{AbstractThis work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$
L
1
initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$
L
1
asymptotic convergence without the assumption of bi-K-invariance.}},
author = {{Papageorgiou, Efthymia}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
keywords = {{Mathematics (miscellaneous)}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces}}},
doi = {{10.1007/s00028-024-00959-6}},
volume = {{24}},
year = {{2024}},
}
@article{63257,
abstract = {{AbstractThe quasilinear Keller–Segel system$$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u,endowed with homogeneous Neumann boundary conditions is considered in a bounded domain$$\Omega \subset {\mathbb {R}}^n$$Ω⊂Rn,$$n \ge 3$$n≥3, with smooth boundary for sufficiently regular functionsDandSsatisfying$$D>0$$D>0on$$[0,\infty )$$[0,∞),$$S>0$$S>0on$$(0,\infty )$$(0,∞)and$$S(0)=0$$S(0)=0. On the one hand, it is shown that if$$\frac{S}{D}$$SDsatisfies the subcritical growth condition$$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$S(s)D(s)≤Csαforalls≥1withsomeα<2nand$$C>0$$C>0, then for any sufficiently regular initial data there exists a global weak energy solution such that$${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$esssupt>0‖u(t)‖Lp(Ω)<∞for some$$p > \frac{2n}{n+2}$$p>2nn+2. On the other hand, if$$\frac{S}{D}$$SDsatisfies the supercritical growth condition$$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$S(s)D(s)≥csαforalls≥1withsomeα>2nand$$c>0$$c>0, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value$$\alpha = \frac{2}{n}$$α=2nfor$$n \ge 3$$n≥3, without any additional assumption on the behavior ofD(s) as$$s \rightarrow \infty $$s→∞, in particular without requiring any algebraic lower bound forD. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type$$Q(s) = \exp (-s^\beta )$$Q(s)=exp(-sβ),$$s \ge 0$$s≥0, for global solvability the exponent$$\beta = \frac{n-2}{n}$$β=n-2nis seen to be critical.}},
author = {{Stinner, Christian and Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{2}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects}}},
doi = {{10.1007/s00028-024-00954-x}},
volume = {{24}},
year = {{2024}},
}
@article{63295,
abstract = {{AbstractWe introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.}},
author = {{Lankeit, Johannes and Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects}}},
doi = {{10.1007/s00028-022-00768-9}},
volume = {{22}},
year = {{2022}},
}
@article{63381,
author = {{Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{3}},
pages = {{1267--1289}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components}}},
doi = {{10.1007/s00028-018-0440-8}},
volume = {{18}},
year = {{2018}},
}
@article{34665,
author = {{Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
keywords = {{Mathematics (miscellaneous)}},
number = {{2}},
pages = {{561--581}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Singular sensitivity in a Keller–Segel-fluid system}}},
doi = {{10.1007/s00028-017-0411-5}},
volume = {{18}},
year = {{2017}},
}