Strong solutions to a nonlocal-in-time semilinear heat equation
Author:
Christoph Walker
Journal:
Quart. Appl. Math. 79 (2021), 265-272
MSC (2010):
Primary 35K91
DOI:
https://doi.org/10.1090/qam/1579
Published electronically:
September 14, 2020
MathSciNet review:
4246493
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Abstract: The existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder’s fixed point theorem and semigroup theory.
References
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385
- Anna Sh. Lyubanova, On nonlocal problems for systems of parabolic equations, J. Math. Anal. Appl. 421 (2015), no. 2, 1767–1778. MR 3258348, DOI https://doi.org/10.1016/j.jmaa.2014.08.027
- C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl. 195 (1995), no. 3, 702–718. MR 1356638, DOI https://doi.org/10.1006/jmaa.1995.1384
- V. V. Shelukhin, A variational principle in problems that are nonlocal with respect to time for linear evolution equations, Sibirsk. Mat. Zh. 34 (1993), no. 2, 191–207, 230, 236 (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 2, 369–384 (1994). MR 1223770, DOI https://doi.org/10.1007/BF00970965
- V. N. Starovoitov and B. N. Starovoitova, Modeling the dynamics of polymer chains in water solution. Application to sensor design, IOP Conf. Series: Journal of Physics: Conf. Series 894 (2017), 012088.
- V. N. Starovoitov, Initial boundary value problem for a nonlocal in time parabolic equation, Sib. Èlektron. Mat. Izv. 15 (2018), 1311–1319. MR 3873768, DOI https://doi.org/10.30757/alea.v15-49
- V. N. Starovoitov. Boundary value problem for a global-in-time parabolic equation. Preprint (2020), https://arxiv.org/abs/2001.04058.
- Christoph Walker, On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions, J. Reine Angew. Math. 660 (2011), 149–179. MR 2855823, DOI https://doi.org/10.1515/crelle.2011.074
- Christoph Walker, Some results based on maximal regularity regarding population models with age and spatial structure, J. Elliptic Parabol. Equ. 4 (2018), no. 1, 69–105. MR 3797184, DOI https://doi.org/10.1007/s41808-018-0010-9
References
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I: Abstract linear theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 1345385
- Anna Sh. Lyubanova, On nonlocal problems for systems of parabolic equations, J. Math. Anal. Appl. 421 (2015), no. 2, 1767–1778. MR 3258348, DOI https://doi.org/10.1016/j.jmaa.2014.08.027
- C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl. 195 (1995), no. 3, 702–718. MR 1356638, DOI https://doi.org/10.1006/jmaa.1995.1384
- V. V. Shelukhin, A variational principle in problems that are nonlocal with respect to time for linear evolution equations, Sibirsk. Mat. Zh. 34 (1993), no. 2, 191–207, 230, 236 (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 2, 369–384 (1994). MR 1223770, DOI https://doi.org/10.1007/BF00970965
- V. N. Starovoitov and B. N. Starovoitova, Modeling the dynamics of polymer chains in water solution. Application to sensor design, IOP Conf. Series: Journal of Physics: Conf. Series 894 (2017), 012088.
- V. N. Starovoitov, Initial boundary value problem for a nonlocal in time parabolic equation, Sib. Èlektron. Mat. Izv. 15 (2018), 1311–1319. MR 3873768, DOI https://doi.org/10.30757/alea.v15-49
- V. N. Starovoitov. Boundary value problem for a global-in-time parabolic equation. Preprint (2020), https://arxiv.org/abs/2001.04058.
- Christoph Walker, On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions, J. Reine Angew. Math. 660 (2011), 149–179. MR 2855823, DOI https://doi.org/10.1515/crelle.2011.074
- Christoph Walker, Some results based on maximal regularity regarding population models with age and spatial structure, J. Elliptic Parabol. Equ. 4 (2018), no. 1, 69–105. MR 3797184, DOI https://doi.org/10.1007/s41808-018-0010-9
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Additional Information
Christoph Walker
Affiliation:
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, D–30167 Hannover, Germany
MR Author ID:
695761
Email:
walker@ifam.uni-hannover.de
Keywords:
Semilinear heat equation,
nonlocal in time,
existence of global solutions
Received by editor(s):
June 18, 2020
Received by editor(s) in revised form:
July 13, 2020
Published electronically:
September 14, 2020
Article copyright:
© Copyright 2020
Brown University
