Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

Carmen Cortázar; Fernando Quirós; Noemí Wolanski; Carmen Cortázar; Fernando Quirós; Noemí Wolanski

doi:10.3934/mine.2022022

Mathematics in Engineering

2022, Volume 4, Issue 3: 1-17. doi: 10.3934/mine.2022022

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Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

1.
Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain
3.
Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UCM-UC3M), 28049-Madrid, Spain
4.
IMAS-UBA-CONICET, Ciudad Universitaria, Pab. I, (1428) Buenos Aires, Argentina

Received: 17 April 2021 Accepted: 07 July 2021 Published: 09 August 2021

We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N > 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.
- heat equation with nonlocal time derivative,
- Caputo derivative,
- fully nonlocal heat equations,
- fractional Laplacian,
- large-time behavior
Citation: Carmen Cortázar, Fernando Quirós, Noemí Wolanski. Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions[J]. Mathematics in Engineering, 2022, 4(3): 1-17. doi: 10.3934/mine.2022022

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Abstract

We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N > 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.

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