---
_id: '63297'
abstract:
- lang: eng
text: "We consider the no-flux initial-boundary value problem for the cross-diffusive
evolution system:\r\n\\begin{eqnarray*}
\ \\left\\{ \\begin{array}{ll} u_t = u_{xx} - \\chi \\big(\\frac{u}{v}
\\partial_x v \\big)_x - uv +B_1(x,t), \\qquad & x\\in \\Omega, \\
t>0, \\\\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \\qquad &
x\\in \\Omega, \\ t>0, \\end{array} \\right. \\end{eqnarray*}\r\nwhich
was introduced by Short et al. in [40] with \r\n$\\chi=2$\r\n
to describe the dynamics of urban crime.In bounded intervals
\r\n$\\Omega\\subset\\mathbb{R}$\r\n
and with prescribed suitably regular non-negative functions \r\n$B_1$\r\n
and \r\n$B_2$\r\n,
we first prove the existence of global classical solutions for any choice of \r\n$\\chi>0$\r\n
and all reasonably regular non-negative initial data.We next
address the issue of determining the qualitative behaviour of solutions under
appropriate assumptions on the asymptotic properties of \r\n$B_1$\r\n
and \r\n$B_2$\r\n.
Indeed, for arbitrary \r\n$\\chi>0$\r\n,
we obtain boundedness of the solutions given strict positivity of the average
of \r\n$B_2$\r\n
over the domain; moreover, it is seen that imposing a mild decay assumption on
\r\n$B_1$\r\n
implies that u must decay to zero in the long-term
limit. Our final result, valid for all \r\n$\\chi\\in\\left(0,\\frac{\\sqrt{6\\sqrt{3}+9}}{2}\\right),$\r\n
which contains the relevant value \r\n$\\chi=2$\r\n,
states that under the above decay assumption on \r\n$B_1$\r\n,
if furthermore \r\n$B_2$\r\n
appropriately stabilises to a non-trivial function \r\n$B_{2,\\infty}$\r\n,
then (u,v) approaches the
limit \r\n$(0,v_\\infty)$\r\n,
where \r\n$v_\\infty$\r\n
denotes the solution of \r\n\\begin{eqnarray*}
\ \\left\\{ \\begin{array}{l} -\\partial_{xx}v_\\infty + v_\\infty
= B_{2,\\infty}, \\qquad x\\in \\Omega, \\\\[1mm] \\partial_x v_{\\infty}=0,
\ \\qquad x\\in\\partial\\Omega. \\end{array} \\right. \\end{eqnarray*}\r\nWe
conclude with some numerical simulations exploring possible effects that may arise
when considering large values of \r\n$\\chi$\r\n
not covered by our qualitative analysis. We observe that when \r\n$\\chi$\r\n
increases, solutions may grow substantially on short time intervals, whereas only
on large timescales diffusion will dominate and enforce equilibration."
author:
- first_name: NANCY
full_name: RODRIGUEZ, NANCY
last_name: RODRIGUEZ
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: RODRIGUEZ N, Winkler M. On the global existence and qualitative behaviour of
one-dimensional solutions to a model for urban crime. European Journal of Applied
Mathematics. 2021;33(5):919-959. doi:10.1017/s0956792521000279
apa: RODRIGUEZ, N., & Winkler, M. (2021). On the global existence and qualitative
behaviour of one-dimensional solutions to a model for urban crime. European
Journal of Applied Mathematics, 33(5), 919–959. https://doi.org/10.1017/s0956792521000279
bibtex: '@article{RODRIGUEZ_Winkler_2021, title={On the global existence and qualitative
behaviour of one-dimensional solutions to a model for urban crime}, volume={33},
DOI={10.1017/s0956792521000279},
number={5}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
University Press (CUP)}, author={RODRIGUEZ, NANCY and Winkler, Michael}, year={2021},
pages={919–959} }'
chicago: 'RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative
Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European
Journal of Applied Mathematics 33, no. 5 (2021): 919–59. https://doi.org/10.1017/s0956792521000279.'
ieee: 'N. RODRIGUEZ and M. Winkler, “On the global existence and qualitative behaviour
of one-dimensional solutions to a model for urban crime,” European Journal
of Applied Mathematics, vol. 33, no. 5, pp. 919–959, 2021, doi: 10.1017/s0956792521000279.'
mla: RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative
Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European
Journal of Applied Mathematics, vol. 33, no. 5, Cambridge University Press
(CUP), 2021, pp. 919–59, doi:10.1017/s0956792521000279.
short: N. RODRIGUEZ, M. Winkler, European Journal of Applied Mathematics 33 (2021)
919–959.
date_created: 2025-12-18T19:23:28Z
date_updated: 2025-12-18T20:08:49Z
doi: 10.1017/s0956792521000279
intvolume: ' 33'
issue: '5'
language:
- iso: eng
page: 919-959
publication: European Journal of Applied Mathematics
publication_identifier:
issn:
- 0956-7925
- 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: On the global existence and qualitative behaviour of one-dimensional solutions
to a model for urban crime
type: journal_article
user_id: '31496'
volume: 33
year: '2021'
...
---
_id: '63318'
abstract:
- lang: eng
text: In a planar smoothly bounded domain$\Omega$,
we consider the model for oncolytic virotherapy given by$$\left\{ \begin{array}{l}
u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm]
w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array}
\right.$$with positive
parameters$
D_w $,$ D_z $and$\beta$.
It is firstly shown that whenever$\beta \lt 1$,
for any choice of$M \gt 0$,
one can find initial data such that the solution of an associated no-flux initial-boundary
value problem, well known to exist globally actually for any choice of$\beta \gt 0$,
satisfies$$u\ge M \qquad \mbox{in
} \Omega\times (0,\infty).$$If$\beta \gt 1$,
however, then for arbitrary initial data the corresponding is seen to have the
property that$$\liminf_{t\to\infty}
\inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$This
may be interpreted as indicating that$\beta$plays
the role of a critical virus replication rate with regard to efficiency of the
considered virotherapy, with corresponding threshold value given by$\beta = 1$.
author:
- first_name: YOUSHAN
full_name: TAO, YOUSHAN
last_name: TAO
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: TAO Y, Winkler M. A critical virus production rate for efficiency of oncolytic
virotherapy. European Journal of Applied Mathematics. 2020;32(2):301-316.
doi:10.1017/s0956792520000133
apa: TAO, Y., & Winkler, M. (2020). A critical virus production rate for efficiency
of oncolytic virotherapy. European Journal of Applied Mathematics, 32(2),
301–316. https://doi.org/10.1017/s0956792520000133
bibtex: '@article{TAO_Winkler_2020, title={A critical virus production rate for
efficiency of oncolytic virotherapy}, volume={32}, DOI={10.1017/s0956792520000133},
number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
University Press (CUP)}, author={TAO, YOUSHAN and Winkler, Michael}, year={2020},
pages={301–316} }'
chicago: 'TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for
Efficiency of Oncolytic Virotherapy.” European Journal of Applied Mathematics
32, no. 2 (2020): 301–16. https://doi.org/10.1017/s0956792520000133.'
ieee: 'Y. TAO and M. Winkler, “A critical virus production rate for efficiency of
oncolytic virotherapy,” European Journal of Applied Mathematics, vol. 32,
no. 2, pp. 301–316, 2020, doi: 10.1017/s0956792520000133.'
mla: TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency
of Oncolytic Virotherapy.” European Journal of Applied Mathematics, vol.
32, no. 2, Cambridge University Press (CUP), 2020, pp. 301–16, doi:10.1017/s0956792520000133.
short: Y. TAO, M. Winkler, European Journal of Applied Mathematics 32 (2020) 301–316.
date_created: 2025-12-18T19:33:01Z
date_updated: 2025-12-18T20:06:35Z
doi: 10.1017/s0956792520000133
intvolume: ' 32'
issue: '2'
language:
- iso: eng
page: 301-316
publication: European Journal of Applied Mathematics
publication_identifier:
issn:
- 0956-7925
- 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: A critical virus production rate for efficiency of oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 32
year: '2020'
...
---
_id: '63314'
abstract:
- lang: eng
text: We propose and study a class of parabolic-ordinary differential equation
models involving chemotaxis and haptotaxis of a species following signals indirectly
produced by another, non-motile one. The setting is motivated by cancer invasion
mediated by interactions with the tumour microenvironment, but has much wider
applicability, being able to comprise descriptions of biologically quite different
problems. As a main mathematical feature constituting a core difference to both
classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis
systems, the considered model accounts for certain types of indirect signal production
mechanisms. The main results assert unique global classical solvability under
suitably mild assumptions on the system parameter functions in associated spatially
two-dimensional initial-boundary value problems. In particular, this rigorously
confirms that at least in two-dimensional settings, the considered indirectness
in signal production induces a significant blow-up suppressing tendency also in
taxis systems substantially more general than some particular examples for which
corresponding effects have recently been observed.
author:
- first_name: CHRISTINA
full_name: SURULESCU, CHRISTINA
last_name: SURULESCU
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: SURULESCU C, Winkler M. Does indirectness of signal production reduce the explosion-supporting
potential in chemotaxis–haptotaxis systems? Global classical solvability in a
class of models for cancer invasion (and more). European Journal of Applied
Mathematics. 2020;32(4):618-651. doi:10.1017/s0956792520000236
apa: SURULESCU, C., & Winkler, M. (2020). Does indirectness of signal production
reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global
classical solvability in a class of models for cancer invasion (and more). European
Journal of Applied Mathematics, 32(4), 618–651. https://doi.org/10.1017/s0956792520000236
bibtex: '@article{SURULESCU_Winkler_2020, title={Does indirectness of signal production
reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global
classical solvability in a class of models for cancer invasion (and more)}, volume={32},
DOI={10.1017/s0956792520000236},
number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
University Press (CUP)}, author={SURULESCU, CHRISTINA and Winkler, Michael}, year={2020},
pages={618–651} }'
chicago: 'SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal
Production Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis
Systems? Global Classical Solvability in a Class of Models for Cancer Invasion
(and More).” European Journal of Applied Mathematics 32, no. 4 (2020):
618–51. https://doi.org/10.1017/s0956792520000236.'
ieee: 'C. SURULESCU and M. Winkler, “Does indirectness of signal production reduce
the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical
solvability in a class of models for cancer invasion (and more),” European
Journal of Applied Mathematics, vol. 32, no. 4, pp. 618–651, 2020, doi: 10.1017/s0956792520000236.'
mla: SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal Production
Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis Systems? Global
Classical Solvability in a Class of Models for Cancer Invasion (and More).” European
Journal of Applied Mathematics, vol. 32, no. 4, Cambridge University Press
(CUP), 2020, pp. 618–51, doi:10.1017/s0956792520000236.
short: C. SURULESCU, M. Winkler, European Journal of Applied Mathematics 32 (2020)
618–651.
date_created: 2025-12-18T19:31:21Z
date_updated: 2025-12-18T20:06:05Z
doi: 10.1017/s0956792520000236
intvolume: ' 32'
issue: '4'
language:
- iso: eng
page: 618-651
publication: European Journal of Applied Mathematics
publication_identifier:
issn:
- 0956-7925
- 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Does indirectness of signal production reduce the explosion-supporting potential
in chemotaxis–haptotaxis systems? Global classical solvability in a class of models
for cancer invasion (and more)
type: journal_article
user_id: '31496'
volume: 32
year: '2020'
...
---
_id: '63371'
abstract:
- lang: eng
text: Adhesion between cells and other cells (cell–cell adhesion) or other
tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon.
Consequently, a number of recently developed mathematical models for cell adhesion
have taken the form of non-local partial differential equations, where the non-local
term arises inside a spatial derivative. The mathematical properties of such a
non-local gradient term are not yet well understood. Here we use sophisticated
estimation techniques to show local and global existence of classical solutions
for such examples of adhesion-type models, and we provide a uniform upper bound
for the solutions. Further, we discuss the significance of these results to applications
in cell sorting and in cancer invasion and support the theoretical results through
numerical simulations.
author:
- first_name: T.
full_name: HILLEN, T.
last_name: HILLEN
- first_name: K. J.
full_name: PAINTER, K. J.
last_name: PAINTER
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: HILLEN T, PAINTER KJ, Winkler M. Global solvability and explicit bounds for
non-local adhesion models. European Journal of Applied Mathematics. 2017;29(4):645-684.
doi:10.1017/s0956792517000328
apa: HILLEN, T., PAINTER, K. J., & Winkler, M. (2017). Global solvability and
explicit bounds for non-local adhesion models. European Journal of Applied
Mathematics, 29(4), 645–684. https://doi.org/10.1017/s0956792517000328
bibtex: '@article{HILLEN_PAINTER_Winkler_2017, title={Global solvability and explicit
bounds for non-local adhesion models}, volume={29}, DOI={10.1017/s0956792517000328},
number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
University Press (CUP)}, author={HILLEN, T. and PAINTER, K. J. and Winkler, Michael},
year={2017}, pages={645–684} }'
chicago: 'HILLEN, T., K. J. PAINTER, and Michael Winkler. “Global Solvability and
Explicit Bounds for Non-Local Adhesion Models.” European Journal of Applied
Mathematics 29, no. 4 (2017): 645–84. https://doi.org/10.1017/s0956792517000328.'
ieee: 'T. HILLEN, K. J. PAINTER, and M. Winkler, “Global solvability and explicit
bounds for non-local adhesion models,” European Journal of Applied Mathematics,
vol. 29, no. 4, pp. 645–684, 2017, doi: 10.1017/s0956792517000328.'
mla: HILLEN, T., et al. “Global Solvability and Explicit Bounds for Non-Local Adhesion
Models.” European Journal of Applied Mathematics, vol. 29, no. 4, Cambridge
University Press (CUP), 2017, pp. 645–84, doi:10.1017/s0956792517000328.
short: T. HILLEN, K.J. PAINTER, M. Winkler, European Journal of Applied Mathematics
29 (2017) 645–684.
date_created: 2025-12-19T11:03:50Z
date_updated: 2025-12-19T11:03:57Z
doi: 10.1017/s0956792517000328
intvolume: ' 29'
issue: '4'
language:
- iso: eng
page: 645-684
publication: European Journal of Applied Mathematics
publication_identifier:
issn:
- 0956-7925
- 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Global solvability and explicit bounds for non-local adhesion models
type: journal_article
user_id: '31496'
volume: 29
year: '2017'
...