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FastML Science Benchmarks:
Accelerating Real-Time Scientific Edge Machine Learning

###### Abstract

Applications of machine learning (ML) are growing by the day for many unique
and challenging scientific applications. However, a crucial challenge facing
these applications is their need for ultra low-latency and on-detector ML
capabilities. Given the slowdown in Moore’s law and Dennard scaling, coupled
with the rapid advances in scientific instrumentation that is resulting in
growing data rates, there is a need for ultra-fast ML at the extreme edge.
Fast ML at the edge is essential for reducing and filtering scientific data in
real-time to accelerate science experimentation and enable more profound
insights. To accelerate real-time scientific edge ML hardware and software
solutions, we need well-constrained benchmark tasks with enough specifications
to be generically applicable and accessible. These benchmarks can guide the
design of future edge ML hardware for scientific applications capable of
meeting the nanosecond and microsecond level latency requirements. To this
end, we present an initial set of scientific ML benchmarks, covering a variety
of ML and embedded system techniques.

††footnotetext: . Correspondence to: Anonymous Author<EMAIL_ADDRESS>
Preliminary work. Under review by the Machine Learning and Systems (MLSys)
Conference. Do not distribute.

##  1 Introduction

In pursuit of scientific discovery across many domains, experiments are
becoming exceedingly sophisticated to probe physical systems at increasingly
smaller spatial resolutions and shorter timescales. These order of magnitude
advancements have led to explosions in both data volumes and richness, leaving
domain scientists to develop novel methods to handle growing data processing
needs. Figure 1 shows the volume of data ($y$-axis) that is generated in
scientific applications such as those at the CERN Large Hadron Collider (LHC)
and in particle accelerator controls. They produce tens of terabytes of data
every second, as discussed below.

Figure 1: Reference latencies and streaming input data rates for common
benchmarks and those proposed in this paper. The horizontal error bar
represents the range of acceptable latencies for the various domains, while
the vertical error bar denotes the range of streaming data rates typical for
those domains. The real-time scientific application domain, or the FastML
Science domain, produces a staggering volume of data and the inference latency
requirements are orders of magnitude far more stringent than they are for more
traditional consumer-facing applications and their benchmarks.

As scientific ecosystems snowball in their speed and scale, new data
processing and reduction paradigms need to be integrated into the system-level
design. The large volume of data needs to be rapidly reduced to a sustainable
level by a real-time event filter system on whether the data should be kept
for further analysis or discarded. Fortunately, this coincides with the rise
of machine learning (ML), or the use of algorithms that can learn directly
from data. Recent advancements demonstrate that ML architectures based on
structured deep neural networks are versatile and capable of solving a broad
range of complex scientific problems. Fast ML is leading to rapid advancements
across many different scientific domains Carleo et al. (2019); Deiana et al.
(2022).

However, a crucial challenge in applying ML to scientific domains is the need
for ultra-fast inference speeds on the order of microseconds. Compared to
traditional ML inference latencies, the scientific application domain requires
nearly 1000$\times$ faster inference performance, as shown in Figure 1. Many
of the consumer facing edge ML deployments like MLPerf Mobile and Tiny are at
least an order of magnitude off from the scientific application latency
requirements.

Moreover, while ML can power scientific advances that can lead to future
paradigm shifts in a broad range of scientific domains, including particle
physics, accelerator physics, plasma physics, astronomy, neuroscience,
chemistry, material science, and biomedical engineering, each application
brings its own domain-specific knowledge and application-specific constraints.
Furthermore, there can be a broad range of different challenges even within a
given domain. This leaves us with two primary types of challenges. The first
is how can we define generically applicable ML benchmarks tasks for bespoke
domain problems which can attract interest from a broad community of system
and ML experts? The second is how can we design benchmark tasks to satisfy
challenging system-level scientific requirements that can, at its core, have
common elements which overlap with a number of system architectures? To
address these issues, we introduce a domain-specific benchmark for fast ML
science.

In this paper, we address the aforementioned challenges by defining an initial
set of scientific “edge” ML benchmarks. These benchmarks strike a balance
between a universality common to many scientific ML applications while also
being uniquely challenging to push designers to further the state-of-the-art
ML and system architecture beyond traditional industrial applications. We
choose our benchmark tasks to cover the three different areas of ML, also we
select domain tasks that, while seeming quite specific, can be generically
mapped onto other scientific applications. In generic terms, the initial
benchmark tasks are the following:

  * •

Supervised learning for physics event triggering: feature extraction of point
cloud inputs for rare data classification and filtering.

  * •

Unsupervised learning for lossy compression of particle detector data: near-
sensor image data compression for custom sensor geometries using image
similarity metrics based on energy distribution profiles.

  * •

Reinforcement learning for accelerator beam control: optimal real-time system
controls based on time-series data from multiple sensors.

With this first suite of benchmarks, we provide suitable coverage for
scientific tasks that span a range of different ML areas and include unique,
meaningful, and challenging system-level requirements that go beyond those
from industry. Future work would include finding areas not covered by these
tasks and broadening the benchmarks.

##  2 Related Work

Table 1 compares characteristics of some of the existing science-related
benchmarks and initiatives to the real-time scientific edge ML benchmarks we
present in this work. Most of the prior work included in the table has been
formalized as an official benchmark, but a couple of related initiatives
remain as standalone datasets or competitions.

| Formalized | Scientific | Edge | Real-Time
---|---|---|---|---
| Benchmark | Workload(s) | Computing | Constraints
FastML Science Benchmarks (this work) | ✓ | ✓ | ✓ | ✓
SciMLBench Thiyagalingam et al. (2021) | ✓ | ✓ | ✓ | $\times$
LHC New Physics Dataset Govorkova et al. (2021) | $\times$ | ✓ | ✓ | ✓
MLPerf HPC Farrell et al. (2021) | ✓ | ✓ | $\times$ | $\times$
BenchCounil AIBench HPC BenchCouncil (2018) | ✓ | ✓ | $\times$ | $\times$
MLCommons Science MLCommons (2020) | ✓ | ✓ | $\times$ | $\times$
ITU Modulation Classification ITU (2021) | $\times$ | $\times$ | ✓ | ✓
Table 1: Comparison of existing machine learning benchmarks and related
initiatives.

Among existing AI benchmarks, the community-driven MLPerf benchmarks from
MLCommons Mattson et al. (2020) are well-established. The benchmarks are run
under predefined conditions and evaluate the performance of training and
inference for hardware, software, and services. MLPerf regularly conducts new
tests and adds new workloads to adapt to the latest industry trends and state
of the art in AI across various domains including high performance computing
(HPC) Farrell et al. (2021), datacenter MLCommons (2021a), edge MLCommons
(2021b), mobile Reddi et al. (2020), and tiny Banbury et al. (2021).
Additionally, BenchCouncil AIBench is a comprehensive AI benchmark suite
including AI Scenario, Training, Inference, Micro, and Synthetic Benchmarks
across datacenter, HPC, IoT and edge BenchCouncil (2018). Other benchmarks
have also been developed by academia and industry. Additional examples of
prior art and initiatives include AI Benchmark Ignatov et al. (2019), EEMBC
MLMark Torelli & Bangale (2019), AIMatrix Alibaba (2018), AIXPRT Principled
Technologies (2019), DeepBench Baidu (2017), TBD Zhu et al. (2018), Fathom
Adolf et al. (2016), RLBench James et al. (2020), and DAWNBench Coleman et al.
(2017).

However, scientific applications (i.e. cosmology, particle physics, biology,
clean energy, etc.) are innately distinct from traditional industrial
applications with respect to the type and volume of data and the resulting
model complexity Farrell et al. (2021). The MLCommons Science Working Group
MLCommons (2020) has a suite of benchmarks that focus on such scientific
workloads including application examples across several domains such as
climate, materials, medicine, and earthquakes. SciMLBench Thiyagalingam et al.
(2021) from the Rutherford Appleton Laboratory is another benchmark suite
specifically focused on scientific machine learning and aimed towards the “AI
for Science” domain. The suite currently contains three benchmarks that
represent problems taken from the material and environmental sciences. MLPerf
HPC and AIBench HPCAI500 are two more benchmarks that include scientific
workloads. In general, HPC is being leveraged by the scientific community for
accelerating scientific insights and discovery. MLPerf HPC aims to
systematically understand how scientific applications perform on diverse
supercomputers, focusing on the time to train for three representative
scientific machine learning applications with massive datasets (i.e.
cosmology, extreme weather analytics, and molecular dynamics). Similarly,
AIBench HPCAI500 also includes a benchmark on extreme weather analytics.
However, all of these existing benchmarks fail to capture the unique
constraints and demands required by scientific “edge” computing in which the
processing must occur near the data source in real-time.

The LHC physics dataset for new physics detection Govorkova et al. (2021) is
the only dataset to our knowledge that retains such characteristics, requiring
unsupervised detection at 40 MHz. However, a related dataset for charged
particle tracking at the LHC, the throughput phase of the TrackML Challenge
Amrouche et al. (2021) emphasized balancing the accuracy of the solution with
the speed of inference. Similarly, one of the tasks in the International
Telecommunication Union’s (ITU) ML in 5G Challenge ITU (2021) requires ultra-
low latency for a different application, namely modulation classification in
communication networks, which displays the generalizability of real-time
scientific edge ML benchmarks. Benchmarking workloads of this nature can
ultimately be a catalyst in enabling a diverse range of new solutions and
applications.

##  3 Benchmarks

In this section, we first introduce our benchmark design philosophy to
establish a suite of real-time scientific benchmarks for edge machine
learning. Next, we introduce the tasks, models, datasets, and associated
metrics that define this initial suite. These benchmarks are in the initial
stages and the task list will continue to evolve over time.

###  3.1 Benchmark Design Philosophy

Our benchmarks tasks cover supervised learning, unsupervised learning, and
reinforcement learning and map generically onto other scientific applications.
The benchmarks are summarized in Table 2, which we explain in more detail in
the subsequent sections.

In each benchmark task, we are working at the edge, very close to the data
source. For each task, the input data has been digitized111In even more
extreme scenarios in the future, we may consider analog inputs., but the
analog signals from sensors are digitized via custom ADCs that have a custom
precision that is tailored for the sensor technology. Scientific instrument
sensors often may have a non-standard precision and further, may deal with
inputs of varying quantization. As is often the case in experimental design,
we lay out system constraints that will serve as guide rails for the various
benchmark tasks.

Solutions for the scientific ML edge benchmarks must satisfy the given
requirements, and then the algorithm performance is measured by a task-
specific metric. Each benchmark has specific system-level constraints. They
are all latency-bound, with latency requirements ranging from hundreds of
nanoseconds to milliseconds. In some cases, the algorithm latency and data
arrival frequency (pipeline interval) will not be the same. In one
application, we also have requirements for area and power.

In order to standardize measurement an implementation’s performance of a given
benchmark against a potentially diverse set of other hardware and
technologies, a number of related metrics should be captured and presented to
more accurately characterize a given aspect of a system. For example, when
measuring the latency of an implementation, both the end-to-end latency and
the initiation interval, or the time required before a system is ready to
accept an input sample after it’s just received one, should be measured and
presented as part of the results. To accurately support and measure various
comprehensive metrics, careful implementation of a “host” system, which is
responsible for administering a benchmark, must also taken into account.
Continuing the example of measuring latency of an implementation, the timing
and method which inputs are presented to the system under test must be capable
of strictly obeying the benchmark’s pipeline interval. A framework for
performing such measurements on varied hardware which we can adopt is based on
the MLPerf Tiny benchmark Banbury et al. (2021).

###  3.2 Supervised Learning: Jet Classification

At the LHC, proton bunches collide at an extreme frequency of 40 MHz, and data
rates at the two multipurpose high energy particle physics experiments, CMS
Chatrchyan et al. (2008) and ATLAS Airapetian et al. (1999), are of the order
of hundreds of terabytes per second. With such high data rates, the task of
real-time processing to filter events to reduce data rates to manageable
levels for offline processing is called triggering. The first level of the
trigger at CMS Sirunyan et al. (2020) and ATLAS Aad et al. (2020) is performed
in FPGAs in custom electronics platforms and have latency requirements at the
microsecond scale; to put this into proper perspective, it is worth noting
that the MLPerf inference tasks’ latency typically ranges between 10 to 100s
of milliseconds.

While this task is specific to particle physics, filtering data to find the
most interesting subset in real-time to reduce data transmission and volumes
in big data scientific experiments is a very common challenge. In this
scenario, we are looking for proton collisions with interesting energetic
radiation patterns called jets. In particular, we would like to identify rare
jet signatures originating from the W or Z boson or the top quark (t) in
contrast to more common signatures from the lighter quarks (q) or gluons (g).
Due to high fidelity simulations in particle physics experiments, this becomes
a supervised multi-classification task. A schematic illustration of the
different types of jets is shown in Fig. 2.

Figure 2: The five jet type classes—light quark (q), gluon (g), top quark (t),
W, and Z boson—for the supervised learning benchmark task. Figure adapted from
Moreno et al. (2020).

#### Dataset

We allow for two types of input features. Several previous related studies
used expert features in a dense, fully-connected network topology which we
will describe in the following section as our baseline model. There are 16
expert-designed features, traditionally standardized with a standard scalar
and represented with fixed-point precision with 16 total bits and six integer
bits. In realistic systems, the classification task would also require the
computation of those expert features, but we do not need that here. However,
to that end, we make available a more challenging set of point cloud inputs
which are the 100 most energetic particles in the jet (with zero padding if
there are less than 100 particles) Pierini et al. (2020). Each particle is
assumed to be massless and has a feature set size of 3 corresponding to the
momentum vector. Models built from these input particles can be more
performant than the expert features but could be more computationally
expensive. Given that the range of $p_{x}$, $p_{y}$, and $p_{z}$ for an
individual particle is approximately contained within $\pm$2 TeV, we adopt an
input integer quantization scheme of 16 total bits with a least significant
bit corresponding to 0.0625 GeV and a total range of $\pm$2048 GeV.

#### Performance Metrics

There are many metrics used in the literature for this benchmark model. The
two we will focus on are (1) classification accuracy and (2) FPR at TPR of 50%
for the signal being Z jet Hawks et al. (2021).

#### Real-time System Constraints

In the planned upgrade of the CMS trigger, the global correlator trigger would
contain jet tagging algorithms like the one described. This system features a
time-multiplexed design, such that information from 6 consecutive proton beam
crossings, which occur even 25 ns, is processed simultaneously. For this
system, a latency of no more than 1 $\mu$s and the ability to accept new
inputs every 150 ns is required CMS Collaboration (2020).

#### Baseline Model and Implementation

For our baseline model and performance, we consider the 16-input dense, fully-
connected architecture first presented in Ref. Duarte et al. (2018) and based
on expert inputs. Because we consider the model with expert inputs, a fully-
connected architecture is selected, but other neural network architectures may
be more optimal, particularly if particle level inputs are used.

The baseline model has an accuracy of 74.8% and the FPR at TPR of 50% metric
for the baseline model is 0.00129. The baseline model implementation with the
architecture consists of $16\rightarrow 64\rightarrow 32\rightarrow
32\rightarrow 5$ MLP with quantization-aware training (QAT) to a homogeneous
precision for the weights and biases of 6-bits as presented in Ref. Coelho et
al. (2021). This model is synthesized for a Virtex Ultrascale+ 9P FPGA with
the following resource usage: 124 DSPs, 39782 LUTs, and 8128 FFs. The latency
is 55 ns at a 5 ns clock with a pipeline interval of 1 clock cycle (5 ns).

###  3.3 Unsupervised Learning: Irregular Sensor Data Compression

Processing of LHC events requires two parallel streams of detector data to be
transmitted from on-detector readout chips. The first is complete event data
from all sensing elements, transmitted at the trigger accept rate of 100 kHz,
while the second is a compressed representation of the same data at the full
40 MHz collision rate. This second, lightweight representation is the basis
for trigger decisions, with a compression factor of $\mathcal{O}$(400),
ensuring parity between the bandwidth of each stream. The critical task is to
achieve this reduction in $\mathcal{O}$(100 ns) with minimal impact on
downstream physics algorithms.

This task is common to many detector systems and directly relevant to a host
of on-device compression tasks for complex sensor data. The scenario considers
the CMS high-granularity endcap calorimeter (HGCal), comprised of 6M channels,
each capturing a 5d (position, energy and time) image of showering high-energy
particles. HGCal is comprised of layers of hexagonal arrays, with a single
particle depositing energy into hundreds of individual sensors. Data from each
hexagonal array is compressed in an application-specific integrated circuit
(ASIC), with the encoded representation transmitted off-detector and
subsequently used to recover the initial detector image as in Figure 3. A
data-driven approach is required to tune the parameters of the compress and
decompress algorithms to minimize differences between the original and decoded
images.

Figure 3: Schematic of the compress-transmit-decompress pipeline for the HGCal
trigger data stream.

#### Dataset

Compression of the trigger data is accomplished in multiple steps. First,
nearest-neighbors within each hexagonal array are aggregated to form a set of
48 “trigger cells” with energies represented as a custom 7b float. These
48$\times$7b floating-point inputs are converted to 22b integers, summed, and
finally normalized to obtain 8b fixed-point trigger cell data. The sum is
preserved as a 9b float for transmission. The benchmark task begins from this
point, which must capture this $48\times 8=384$b representation in a budget of
either 144 or 48 bits, for the moderate—and extreme—compression variants,
respectively.

#### Real-time System Constraints

The encoder must accept new inputs at 40 MHz input rate and complete
processing within 100 ns. Furthermore, we must consider protection against
single-event effects due to the high-radiation environment of the on-detector
readout. As a rough first order guideline, on a low-power CMOS 65 nm
technology node, the algorithm area must not be greater than 4 mm2 while
drawing less than 100 mW.

#### Performance Metrics

Performance is assessed by directly comparing the individual energies of each
decoded array of hexagonal sensors (i.e. 48 trigger cells) to the original
image of normalized inputs. The energy mover’s distance Komiske et al. (2019)
(EMD) is used to compare the reconstructed radiation patterns, giving smaller
penalties for misreconstructed energies that are close-by to the original
deposit.

#### Baseline Model

A convolutional neutral network (CNN) autoencoder architecture is used to
perform the compress-and-decompress task because the sensor data is image-
like, though the hexagonal geometry provides a unique challenge. Normalized
sensor data is re-arranged and fed to a CNN with one convolutional and one
dense layer with 6b weights, leading to a maximum of 16$\times$9b outputs,
saturating the 144b bandwidth Di Guglielmo et al. (2021). Data is decompressed
by a second CNN with inverted architecture, whose outputs are multiplied by
the energy sum to recover the original input image.

The compression logic is implemented in an “encoder ASIC” using a low-power
CMOS process with 65 nm feature size. The total latency for this circuit is 25
ns and is estimated to draw 60 mW in simulation.

Type | Benchmark | Input | Pipeline | Real-time | Misc. Req. | Baseline Model
---|---|---|---|---|---|---
Precision | Rate | Latency | Parameters
Supervised Learning | Jet Classification | 16b | 150 ns | 1 $\mu$s | - | 4,389
Unsupervised Learning | Sensor Data Compression | 9b | 25 ns | 100 ns | area, power (65 nm) | 2,288
Reinforcement Learning | Beam Control | 32b | 5 ms | 5 ms | - | 34,695

Table 2: Summary of constraints for three benchmark tasks and number of
parameters for the benchmark baseline models.

###  3.4 Reinforcement Learning: Beam Control

Intense and energetic particle beams are used for various applications, from
materials discovery to studying nuclear matter to fundamental particle
physics, and even cancer therapy. Controlling precise particle beams, such as
those at the Department of Energy User Facilities, requires intelligent
algorithms running at the edge in low-latency, real-time systems to steer
particles traversing miles of beamline at nearly the speed of light.

A dataset has been developed Kafkes & St John (2021) for studying how to
control the bending magnet ramping rate of power supplies St. John et al.
(2021) in the rapidly cycling Booster synchrotron ring Hubbard et al. (1973)
at the Fermilab Accelerator Complex. This is illustrated diagrammatically in
Figure 4. The power supply control signals are provided at 15 Hz. This beam
controls application can be framed as a reinforcement learning benchmark task.
Because an accurate and reliable simulation of the synchrotron is not possible
from first principles, a “virtual” accelerator complex surrogate model has
been developed to emulate the actual physical system. This surrogate model
will serve as the environment with which our reinforcement learning benchmark
interacts.

Figure 4: Synchrotron magnet power supply control system for the Fermilab
Booster Ring, adapted from St. John et al. (2021)

#### Dataset

A Booster synchrotron power supply regulation dataset provides cycle-by-cycle
time series of readings and settings from the most relevant devices available
in the Fermilab control system. This data was drawn from the time series of a
select subset of the roughly 200,000 entries that populate the device database
of the accelerator control network. Data was sampled at 15 Hz for 54 devices
pertaining to the system’s regulation. Because of how data is transmitted and
communicated, inputs are 32-bit floating-point numbers, but the sensor
source’s precision is, in many cases, less.

#### Real-time System Constraints

The Booster ramping cycle rate is 15 Hz, which sets the control loop’s time
scale. We define the algorithm latency requirement as 5 ms for this benchmark
due to data movement latency.

#### Performance Metrics

The primary performance metric in this reference benchmark is the reward, $R$,
defined as the negative of the error with respect to the reference expected
current in the Booster, $R=-|\Delta I_{\mathrm{min}}|$.

#### Baseline Model(s)

There are two models involved in this benchmark task: (1) the surrogate model
for the Booster accelerator and (2) the online agent, which is correcting the
reference magnet power supplies in real-time. The surrogate model is fixed in
this benchmark task and plays the role of the environment in this
reinforcement learning task. The long short-term memory (LSTM) Hochreiter &
Schmidhuber (1997) surrogate model inputs are the previous 150-time steps of
the top 5 causal variables—variables related to the synchrotron and downstream
accelerator currents and current errors concerning reference. The model has
approximately 1.5 million parameters.

The benchmark online agent running in the Arria10 system-on-chip (SoC) is a
multilayer perceptron taking the five input parameters, has three hidden
layers $(128,128,128)$ and approximately 35,000 parameters. The architecture
is a fully-connected neural network because the 5 inputs are already selected
expert variables. The deep Q-network Mnih et al. (2013; 2015) has 7 discrete
outputs and maximizes the reward, $R$, defined above. The reward metric is
measured as a function of RL episode and is presented in Fig. 7 (bottom) of
St. John et al. (2021).

The benchmark model weights and biases are quantized to 20 total bits in a
fixed-point representation in hardware. The lowest latency implementation of
the model is implemented for an Intel Arria10 SoC with a resource usage of 53
DSPs, 238 BRAMs, 672 MLABs, 43.3 kALMs, 92.6 kFFs. The algorithm has a latency
of 3.9 $\mu$s.

##  4 Discussion and Outlook

This position paper highlights both the need and challenges for developing
machine learning (ML) benchmarks for edge applications in science. Given the
demise of Moore’s law and Dennard scaling Dennard et al. (1974); Esmaeilzadeh
et al. (2011) and advances in scientific instrumentation resulting in rapidly
growing data rates, edge computing is becoming exceedingly crucial for
reducing and filtering scientific data in real-time to accelerate science
experimentation and enable more profound insights. There are challenges in
building well-constrained benchmark tasks with enough specification to be
generically applicable and accessible simultaneously. However, we can use
these edge applications in extreme data processing environments to advance
many scientific domains and enable the development of state-of-the-art tools.

In devising scientific ML edge benchmarks tasks, we aim to cover machine
learning and embedded system techniques. We choose three applications that
span supervised, unsupervised, and reinforcement learning. The system
constraints also span a wide range of latency requirements—from hundreds of
nanoseconds to milliseconds where technologies vary from ASIC to FPGA to more
relaxed constraints with the freedom to choose system architecture. Finally,
there is also a variation in input data representations from expert-level
inputs to image data, point cloud data, and time-series data. A summary of the
proposed scientific edge ML benchmarks are presented in Table 2 including the
input, latency, and pipeline interval constraints of the benchmark
applications. We also provide a prototype code repository for these benchmarks
Muhizi & Duarte (2022).

##  5 Conclusion

In this work, we present an initial set of scientific machine learning
benchmarks that are specifically geared towards real-time scientific edge
machine learning needs. Our goal is to first and foremost identify and provide
a suite of datasets and a code repository for these various edge ML tasks. We
will collect and document different solutions and, based on interest, extend
this work. In particular, there is potential to provide more domain
applications such as for astronomy, neuroscience, and microscopy that could
attract a broader set of use-cases.

## Acknowledgments

JD is supported by the US Department of Energy (DOE) Award Nos. DE-SC0021187,
DE-SC0021396 and the National Science Foundation (NSF) A3D3 Institute under
Cooperative Agreement OAC-2117997. BH, CH, JM, and NT are supported by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy (DOE), Office of Science, Office of High Energy Physics,
the DOE Early Career Research program under Award No. DE-0000247070, and the
DOE, Office of Science, Office of Advanced Scientific Computing Research under
Award No. DE-FOA-0002501.

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Backward and non-local problems for the Rayleigh-Stokes equation
Ravshan Ashurov1 and Nafosat Vaisova 2
1 Institute of Mathematics, Academy of Science of Uzbekistan

<EMAIL_ADDRESS>
2 Institute of Mathematics, Uzbekistan Academy of Science, Urgench

Abstract: The Fourier method is used to find conditions on the right-hand side
and on the initial data in the Rayleigh-Stokes problem, which ensure the
existence and uniqueness of the solution. Then, in the Rayleigh-Stokes
problem, instead of the initial condition, consider the non-local condition:
$u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta$ is either zero or one. It is
well known that if $\beta=0$, then the corresponding problem, called the
backward problem, is ill-posed in the sense of Hadamard, i.e. a small change
in $u(x,T)$ leads to large changes in the initial data. Nevertheless, we will
show that if we consider sufficiently smooth current information, then the
solution exists and it is unique and stable. It will also be shown that if
$\beta=1$, then the corresponding non-local problem is well-posed and coercive
type inequalities are valid.

Keywords: The Rayleigh-Stokes problem, the backward problem, non-local
problem, the Fourier method.

##  1\. Introduction

1. The main object of study in this paper is the following Rayleigh-Stokes problem for a generalized second-grade fluid with a time-fractional derivative model:

(1.1) $\left\\{\begin{aligned}
&\partial_{t}u(x,t)-(1+\gamma\,\partial_{t}^{\alpha})\Delta
u(x,t)=f(x,t),\quad x\in\Omega,\quad 0<t\leq T;\\\ &u(x,t)=0,\quad
x\in\partial\Omega,\quad 0<t\leq T;\\\ &u(x,0)=\varphi(x),\quad
x\in\Omega,\end{aligned}\right.$

where $\gamma>0$ is a fixed constant, $\varphi$ is the initial data,
$\partial_{t}=\partial/\partial t$, and $\partial_{t}^{\alpha}$ is the
Riemann-Liouville fractional derivative of the order $\alpha\in(0,1)$ defined
by (see, e.g. [1]):

(1.2)
$\partial_{t}^{\alpha}h(t)=\frac{d}{dt}\int\limits_{0}^{t}\omega_{1-\alpha}(t-s)h(s)ds,\quad\omega_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$

Here $\Gamma(\sigma)$ is Euler’s gamma function. Usually problem (1.1) is
considered in the domain $\Omega\subset R^{N}$, $N=1,2,3$, with a sufficiently
smooth boundary $\partial\Omega$.

The Rayleigh-Stokes problem (1.1) has received considerable attention in
recent years because of its practical importance (see, e.g. [2], [3]). This
non-stationary flow problem investigates the propagation of a vortex in a
half-space filled with a viscous in-compressible fluid that sets in motion
when an infinite flat plate suddenly acquires a constant velocity parallel to
itself from rest. The fractional derivative $\partial_{t}^{\alpha}$ in the
model is used to capture the viscoelastic behavior of the flow (see e.g. [4],
[5]).

In order to get an idea of the behavior of the solution of this model, a
number of authors have shown considerable interest in obtaining an exact
solution in some special cases; see e.g. [4], [5], [6]. For example, Shen et
al. [5] obtained an exact solution to the problem using the sine Fourier
transform and then applying the fractional Laplace transform. Zhao and Yang
[6] obtained exact solutions using the eigenfunction expansion in a
rectangular domain for the case of homogeneous initial and boundary
conditions. In this case, the eigenfunctions are explicitly written out. The
solutions obtained in these studies are of a formal nature, and the regularity
of the solution has not been particularly studied.

Questions of the regularity of the solution were studied in the fundamental
work Bazhlekova et al. [7]. The authors proved the Sobolev regularity of the
homogeneous problem both for smooth and non-smooth initial data $\varphi(x)$,
including $\varphi(x)\in L_{2}(\Omega)$.

The exact solutions obtained in the above studies involve infinite series and
special functions, such as generalized Mittag-Leffler functions, and are
therefore inconvenient for numerical imple-mentation. In addition, explicit
solutions are available only for a limited class of problem settings.
Therefore, it is extremely important to develop efficient and optimally
accurate numerical algorithms for problem (1.1). Numerous works of specialists
are devoted to this issue. An overview of the work in this direction is
contained in the above work [7].

The inverse problems of determining the right-hand side (the heat source
density) of the Rayleigh-Stokes equation have been considered by a number of
authors (see, e.g., [8], [9], [10] and the bibliography therein). However,
there is no general closed theory for the abstract case of the source function
$F(x,t)$. Known results deal with the separated source term $F(x,t)=h(t)f(x)$,
where $h(t)$ is a known function and $f(x)$ is an unknown one. Since this
inverse problem is an ill-posed problem in the sense of Hadamard, various
regularization methods are proposed in the above papers. Also in these papers,
the proposed regularized methods were tested by simple numerical experiments
to check the error estimate between the desired solution and the regularized
solution.

Note that such an inverse problem is ill-posed in the Hadamard sense also for
the subdiffusion equation (see, e.g. [11], [12], [13]).

2. In the case when the initial condition $u(x,0)=\varphi(x)$ in the problem (1.1) is replaced by $u(x,T)=\varphi(x)$, then the resulting problem is called _backward problem_. The backward problem for the Rayleigh-Stokes equation is of great importance and is aimed at determining the previous state of a physical field (for example, at t = 0) based on its current information (see, e.g., [14], [15] and the bibliography therein). However, this problem (as well as the inverse problem of finding the right-hand side of the equation) is ill-posed in the sense of Hadamard. In other words, a small change in $u(x,T)$ leads to a large change in the original data. The authors of the above works proposed various regularization methods and tested these methods using numerical experiments.

In the present paper we will pay special attention to the backward, since in
previous papers (see, e.g., [14], [15]) the authors considered only the case
$N\leq 3$. And this is connected with the method used in these works: if the
dimension of the space is less than four, then for the eigenvalues
$\lambda_{k}$ of the Laplace operator with the Direchlet condition, the series
$\sum_{k}\lambda_{k}^{-2}$ converges.

It is well known that in most models described by differential equations, an
initial condition is used to select a single solution. However, there are also
processes where we have to use non-local conditions, for example, the integral
over time intervals, or connection of solution values at different times, for
example, at the initial time and at the final time. It should be noted that
non-local conditions model some details of natural phenomena more accurately,
since they take into account additional information in the initial conditions.

In this work we consider problem (1.1) with a non-local time condition:

$u(x,T)=u(x,0)+\varphi(x),$

It turns out that the non-local problem is well-posed. In other words, a
solution to a non-local problem exists and is unique. Moreover, the solution
depends continuously on the function $\varphi(x)$ in the non-local condition.

Thus, the problem of determining the previous state of the physical field on
the basis of current information is an ill-posed problem. However, if, without
knowing the initial and present states, we want them to differ by the value
$\varphi(x)$, then such a problem turns out to be correct.

##  2\. Problem statements

The method of our research is the Fourier method. Therefore, only
eigenfunctions and eigenvalues are needed from the Laplace operator in the
Rayleigh-Stokes problem (1.1). With this in mind, instead of the Laplace
operator $(-\Delta)$ in problem (1.1), we consider an abstract positive
operator.

So in a Hilbert space $H$ with the scalar product $(\cdot,\cdot)$ and the norm
$||\cdot||$, consider an arbitrary unbounded positive selfadjoint operator
$A$. Let $A$ have a complete in $H$ system of orthonormal eigenfunctions
$\\{v_{k}\\}$ and a countable set of positive eigenvalues $\lambda_{k}:$
$0<\lambda_{1}\leq\lambda_{2}\cdot\cdot\cdot\rightarrow+\infty$. For a vector-
valued functions (or simply functions) $h:\mathbb{R}_{+}\rightarrow H$, we
define the Riemann-Liouville fractional derivative of order $0<\alpha<1$ in
the same way as (1.2) (see, e.g. [16]) Finally, let $C((a,b);H)$ stand for a
set of continuous functions $u(t)$ of $t\in(a,b)$ with values in $H$.

Consider the following Rayleigh-Stokes problem

(2.1) $\left\\{\begin{aligned}
&\partial_{t}u(t)+(1+\gamma\,\partial_{t}^{\alpha})Au(t)=f(t),\quad 0<t\leq
T;\\\ &u(T)=\beta u(0)+\varphi,\end{aligned}\right.$

where $\gamma>0$ is a fixed constant, $\varphi\in H$ and $f(t)\in C((0,T];H)$
and $\beta$ is equal to $0$ or $1$. If $\beta=0$ then this problem is called
the backward problem and if $\beta=1$, then _the non-local problem_. We will
consider both of these cases.

In this article, various initial problems are considered. The solution of
these problems is defined in a standard way: all vector functions and their
derivatives included in the equation and in the initial conditions must be
continuous in the variable $t$, and all the corresponding equalities should be
understood as the equality of the vectors from $H$ at each point $t$. As a
example, let us define a solution for the problem (2.1).

######  Definition 2.1.

A function $u(t)\in C([0,T];H)$ with the properties
$\partial_{t}u(t),\partial_{t}^{\alpha}Au(t)\in C((0,T);H)$ and satisfying
conditions (2.1) is called the solution of the nonlocal Rayleigh-Stokes
problem (2.1).

To solve problem (2.1) with $\beta=1$, we divide it into two auxiliary
problems:

(2.2) $\left\\{\begin{aligned}
&\partial_{t}v(t)+(1+\gamma\,\partial_{t}^{\alpha})Av(t)=f(t),\quad 0<t\leq
T;\\\ &v(0)=0\end{aligned}\right.$

and

(2.3) $\left\\{\begin{aligned}
&\partial_{t}w(t)+(1+\gamma\,\partial_{t}^{\alpha})Aw(t)=0,\quad 0<t\leq T;\\\
&w(T)=w(0)+\psi,\quad 0<\xi\leq T,\end{aligned}\right.$

where $\psi\in H$ is a given function.

Problem (2.2) and (2.3) are a special case of problem (2.1), and solutions to
problems (2.2) and (2.3) are defined similarly to Definition 2.1.

If $\psi=\varphi-v(\xi)$ and $v(t)$ and $w(t)$ are the corresponding
solutions, then it is easy to verify that function $u(t)=v(t)+w(t)$ is a
solution to problem (2.1). Therefore, it is sufficient to solve the auxiliary
problems.

######  Remark 2.2.

We note that, as operator $A$ one can take, for example, the Laplace operator
with the Dirichlet condition in an arbitrary $N$ (not only $\leq 3$) -
dimensional domain with a sufficiently smooth boundary. For our reasoning, it
is sufficient that the operator $A$ has the properties listed above.

In [20] and [21], a similar non-local problem with an arbitrary parameter
$\beta$ was studied in detail for subdiffusion equations with Riemann-
Liouville and Caputo derivatives. For the classical diffusion equation with
the parameter $\beta=1$, it was first considered in [22] and [23].

The remainder of this paper is composed of eight sections and the Conclusion.
In the next section, we introduce the Hilbert space associated with the degree
of operator A and recall some properties of function $B_{\alpha}(\lambda,t)$,
introduced in [7]. Section 4 is devoted to the study of the main problem (1.1)
with operator $A$ instead of the Laplace operator. Here, conditions are given
on the right-hand side of the equation and on the initial function, under
which the solution of problem (1.1) is found in the form of a Fourier series.
In Section 5, we study the backward problem, i.e. problem (2.1) with
$\beta=0$. This problem is not well-posed according to Hadamard: a small
change $u(T)$ leads to large changes in the solution of the problem.
Nevertheless, this section shows that if we consider $u(T)$ in the class of
smooth functions, then stability is preserved. Section 6 studies the
conditional stability of the backward problem. The next two sections are
devoted to the study of the non-local problem (2.1) with $\beta=1$.

##  3\. Preliminaries

In this section, we introduce the Hilbert space of $"$smooth$"$ functions
defined by the degree of operator $A$ and recall some properties of function
$B_{\alpha}(\lambda,t)$ obtained in [7] and [14] that we will use in what
follows.

Let $\tau$ be an arbitrary real number. We introduce the power of operator
$A$, acting in $H$ according to the rule (note, operator $A$ is positive and
therefore $\lambda_{k}>0$ for all $k$)

$A^{\tau}h=\sum\limits_{k=1}^{\infty}\lambda_{k}^{\tau}h_{k}v_{k}.$

Here and everywhere below, for the vector $h\in H$, the symbol $h_{k}$ will
denote the Fourier coefficients of this vector: $h_{k}=(h,v_{k})$. The domain
of definition of this operator is determined from condition $A^{\tau}h\in H$
and has the form:

$D(A^{\tau})=\\{h\in
H:\sum\limits_{k=1}^{\infty}\lambda_{k}^{2\tau}|h_{k}|^{2}<\infty\\}.$

For elements of $D(A^{\tau})$ we introduce the norm

$||h||^{2}_{\tau}=\sum\limits_{k=1}^{\infty}\lambda_{k}^{2\tau}|h_{k}|^{2}=||A^{\tau}h||^{2}.$

With this norm the lenear-vector space $D(A^{\tau})$ turns into a Hilbert
space.

Let $B_{\alpha}(\lambda,t)$ be the solution of the following Cauchy problem

(3.1) $Ly(t)\equiv
y^{\prime}(t)+\lambda(1+\gamma\partial_{t}^{\alpha})y(t)=0,\,\,t>0,\,\,\lambda>0,\,\,y(0)=1.$

The solution of such an equation is expressed in terms of the generalized
Wright function (see e.g. A.A. Kilbas et. al [1], Example 5.3, p. 289). But
function $B_{\alpha}(\lambda,t)$, the solution of this Cauchy problem, is
studied in detail in Bazhlekova, Jin, Lazarov, and Zhou [7]. See also Luc,
Tuan, Kirane, Thanh [14], where very important lower bounds are obtained. The
authors of [7], in particular, proved the following lemma.

######  Lemma 3.1.

Let $B_{\alpha}(\lambda,t)$ be a solution of the Cauchy problem (3.1). Then

  1. (1)

$B_{\alpha}(\lambda,0)=1,\,\,0<B_{\alpha}(\lambda,t)<1,\,\,t>0$,

  2. (2)

$\partial_{t}B_{\alpha}(\lambda,t)<0,\,\,t\geq 0$,

  3. (3)

$\lambda B_{\alpha}(\lambda,t)<C\min\\{t^{-1},t^{\alpha-1}\\},\,\,t>0\\}$,

  4. (4)

$\int\limits_{0}^{T}B_{\alpha}(\lambda,t)dt\leq\frac{1}{\lambda},\,\,T>0.$

It should only be noted that in paper [7], instead of assertion (2), a more
general proposition was proved. But the assertion (2) in our lemma easily
follows from the representation of the function $B_{\alpha}(\lambda,t)$, also
obtained in [7]:

(3.2)
$B_{\alpha}(\lambda,t)=\int\limits_{0}^{\infty}e^{-rt}b_{\alpha}(\lambda,r)dr,$

where

$b_{\alpha}(\lambda,r)=\frac{\gamma}{\pi}\frac{\lambda
r^{\alpha}\sin\alpha\pi}{(-r+\lambda\gamma
r^{\alpha}\cos\alpha\pi+\lambda)^{2}+(\lambda\gamma
r^{\alpha}\sin\alpha\pi)^{2}}.$

The following assertion is also implicitly contained in [7]. But in view of
its importance in our reasoning, we present a proof.

######  Lemma 3.2.

The solution of the Cauchy problem

(3.3)
$y^{\prime}(t)+\lambda(1+\gamma\partial_{t}^{\alpha})y(t)=f(t),\,\,t>0,\,\,\lambda>0,\,\,y(0)=0,$

has the form

(3.4) $y(t)=\int\limits_{0}^{t}B_{\alpha}(\lambda,t-\tau)f(\tau)d\tau.$

###### Proof.

Since $B_{\alpha}(\lambda,0)=1$, then

$\partial_{t}y(t)=f(t)+\int\limits_{0}^{t}\partial_{t}B_{\alpha}(\lambda,t-\tau)f(\tau)d\tau.$

On the other hand

$\partial_{t}^{\alpha}y(t)=\frac{1}{\Gamma(1-\alpha)}\partial_{t}\int\limits_{0}^{t}(t-\tau)^{-\alpha}\int\limits_{0}^{\tau}B_{\alpha}(\lambda,\tau-\xi)f(\xi)d\xi
d\tau=$

(change the order of integration)

$=\frac{1}{\Gamma(1-\alpha)}\partial_{t}\int\limits_{0}^{t}f(\xi)\int\limits_{\xi}^{t}(t-\tau)^{-\alpha}B_{\alpha}(\lambda,\tau-\xi)d\tau
d\xi=$

(change of variables: $\tau-\xi=\eta,\,\tau=\xi+\eta,\,d\tau=d\eta$)

$=\frac{1}{\Gamma(1-\alpha)}\partial_{t}\int\limits_{0}^{t}f(\xi)\int\limits_{0}^{t-\xi}(t-(\xi+\eta))^{-\alpha}B_{\alpha}(\lambda,\eta)d\eta
d\xi=$
$=\int\limits_{0}^{t}f(\xi)\frac{1}{\Gamma(1-\alpha)}\partial_{t}\int\limits_{0}^{t-\xi}(t-(\xi+\eta))^{-\alpha}B_{\alpha}(\lambda,\eta)d\eta
d\xi=\int\limits_{0}^{t}f(\xi)\partial_{t}^{\alpha}B_{\alpha}(\lambda,t-\xi)d\xi.$

Thus, given that $B_{\alpha}(\lambda,t)$ is a solution to the Cauchy problem
(3.1), we obtain

$Ly(t)=f(t)+\int\limits_{0}^{t}f(\xi)LB_{\alpha}(\lambda,\cdot-\xi)d\xi=f(t).$

∎

######  Corollary 3.3.

The solution of the Cauchy problem

(3.5)
$y^{\prime}(t)+\lambda(1+\gamma\partial_{t}^{\alpha})y(t)=f(t),\,\,t>0,\,\,\lambda>0,\,\,y(0)=y_{0},$

has the form

(3.6)
$y(t)=y_{0}B_{\alpha}(\lambda,t)+\int\limits_{0}^{t}B_{\alpha}(\lambda,t-\tau)f(\tau)d\tau.$

We also need the following properties of function $B_{\alpha}(\lambda,t)$.

######  Lemma 3.4.

There is a constant $C>0$, such that

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{C}{\lambda\,t^{2-\alpha}},\,\,t>0.$

###### Proof.

From (3.2) we have

$\partial_{t}B_{\alpha}(\lambda,t)=-\int\limits_{0}^{\infty}re^{-rt}b_{\alpha}(\lambda,r)dr.$

Therefore, by the definition of $b_{\alpha}$,

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{\gamma}{\pi}\int\limits_{0}^{\infty}\frac{\lambda
r^{\alpha}\sin\alpha\pi}{(\lambda\gamma
r^{\alpha}\sin\alpha\pi)^{2}}\,re^{-rt}dr=\frac{1}{\gamma\pi\lambda\sin\alpha\pi}\int\limits_{0}^{\infty}r^{1-\alpha}e^{-rt}dr=$

(change of variables: $\tau=rt,\,d\tau=tdr$)

$=\frac{t^{\alpha-2}}{\gamma\pi\lambda\sin\alpha\pi}\int\limits_{0}^{\infty}\tau^{1-\alpha}e^{-\tau}d\tau=\frac{t^{\alpha-2}}{\gamma\pi\lambda\sin\alpha\pi}\Gamma(2-\alpha)=\frac{C}{\lambda\,t^{2-\alpha}}.$

∎

One can also obtain an estimate for $\partial_{t}B_{\alpha}$ with a smaller
singularity at $t=0$.

######  Lemma 3.5.

There is a constant $C>0$, such that

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq C\frac{\lambda}{t^{\alpha}},\,\,t>0.$

###### Proof.

Again by the definition of $b_{\alpha}$ one can obtain

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{\gamma}{\pi}\int\limits_{0}^{\infty}\frac{\lambda
r^{\alpha}\sin\alpha\pi}{r^{2}}\,re^{-rt}dr=\frac{\lambda\,\gamma\,\Gamma(\alpha)\,\sin\alpha\pi}{\pi
t^{\alpha}}.$

∎

Now, by combining these two estimates, we can obtain the following statement,
which we also apply further.

######  Lemma 3.6.

For any integer $n=2^{j},\,j=1,2,\cdots,$ there is a constant $C_{n}>0$, such
that

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{C_{n}\lambda^{\frac{1}{n}}}{t^{1-\frac{1-\alpha}{n}}},\,\,t>0.$

###### Proof.

Apply Lemmas 3.4 and 3.5 to get

(3.7) $|\partial_{t}B_{\alpha}(\lambda,t)|^{2}\leq C\frac{1}{\lambda
t^{2-\alpha}}\cdot\frac{\lambda}{t^{\alpha}}=\frac{C}{t^{2}},\,\,t>0.$

This estimate and Lemma 3.5 imply

$|\partial_{t}B_{\alpha}(\lambda,t)|^{2}\leq
C\frac{1}{t}\cdot\frac{\lambda}{t^{\alpha}}=\frac{C\lambda}{t^{1+\alpha}},\,\,t>0,$

or

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{C\lambda^{\frac{1}{2}}}{t^{1-\frac{1-\alpha}{2}}},\,\,t>0.$

Now, using this estimate and (3.7), we obtain

$|\partial_{t}B_{\alpha}(\lambda,t)|\leq\frac{C\lambda^{\frac{1}{4}}}{t^{1-\frac{1-\alpha}{4}}},\,\,t>0.$

Repeating this process gives the assertion of the lemma.

∎

In the future, instead of $\lambda$, we will have eigenvalues $\lambda_{k}$ of
operator $A$. The next lower bound for $B_{\alpha}(\lambda_{k},t)$ was
obtained in Luc, N.H., Tuan, N.H., Kirane, M., Thanh, D.D.X [14]. For the
convenience of the reader, we present this proof.

######  Lemma 3.7.

The following estimate holds for all $t\in[0,T]$ and $k\geq 1$:

$B_{\alpha}(\lambda_{k},t)\geq\frac{C(\alpha,\gamma,\lambda_{1})}{\lambda_{k}},$

where

$C(\alpha,\gamma,\lambda_{1})=\frac{\gamma\sin\alpha\pi}{4}\int\limits_{0}^{\infty}\frac{r^{\alpha}e^{-rT}}{\frac{r^{2}}{\lambda_{1}^{2}}+\gamma^{2}r^{2\alpha}+1}dr.$

###### Proof.

Since the modulo of trigonometric functions does not exceed one and
$(a+b+c)^{2}\leq 3(a^{2}+b^{2}+c^{2})$, then

$(-r+\lambda_{k}\gamma
r^{\alpha}\cos\alpha\pi+\lambda_{k})^{2}+(\lambda_{k}\gamma
r^{\alpha}\sin\alpha\pi)^{2}\leq
3(r^{2}+\lambda_{k}^{2}\gamma^{2}r^{2\alpha}+\lambda_{k}^{2})+\lambda_{k}^{2}\gamma^{2}r^{2\alpha}\leq$
$\leq
4\lambda_{k}^{2}(\frac{r^{2}}{\lambda_{1}^{2}}+\gamma^{2}r^{2\alpha}+1).$

Therefore, (3.2) implies

$B_{\alpha}(\lambda_{k},t)\geq\frac{\gamma\sin\alpha\pi}{4\lambda_{k}}\int\limits_{0}^{\infty}\frac{r^{\alpha}e^{-rT}}{\frac{r^{2}}{\lambda_{1}^{2}}+\gamma^{2}r^{2\alpha}+1}dr.$

It should be noted that the improper integral converges. Indeed,

$\int\limits_{0}^{\infty}\frac{r^{\alpha}e^{-rT}}{\frac{r^{2}}{\lambda_{1}^{2}}+\gamma^{2}r^{2\alpha}+1}dr\leq\int\limits_{0}^{\infty}r^{\alpha}e^{-rT}dr=T^{-(\alpha+1)}\int\limits_{0}^{\infty}\tau^{\alpha}e^{-\tau}d\tau=T^{-(\alpha+1)}\Gamma(\alpha+1).$

∎

##  4\. Forward problem

Consider the following Cauchy problem:

(4.1) $\left\\{\begin{aligned}
&\partial_{t}v(t)+(1+\gamma\,\partial_{t}^{\alpha})Av(t)=f(t),\quad 0<t\leq
T;\\\ &v(0)=\varphi,\end{aligned}\right.$

where $\varphi$ is a given vector of $H$. We also call this problem the
forward problem.

######  Theorem 4.1.

Let $\varphi\in H$ and $f(t)\in C([0,T];D(A^{\varepsilon}))$ for some
$\varepsilon\in(0,1)$. Then problem (4.1) has a unique solution and this
solution has the representation

(4.2)
$v(t)=\sum\limits_{k=1}^{\infty}B_{\alpha}(\lambda_{k},t)\varphi_{k}v_{k}+\sum\limits_{k=1}^{\infty}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}.$

There are constants $C$ and $C_{\varepsilon}>0$ such that the following
coercive type inequality holds:

(4.3) $||\partial_{t}v(t)||^{2}+||\partial_{t}^{\alpha}Av(t)||^{2}\leq
Ct^{-2}||\varphi||^{2}+C_{\varepsilon}\max\limits_{t\in[0,T]}||f||_{\varepsilon}^{2},\quad
0<t\leq T.$

Moreover, there is a constant $C_{T}$, depending on $T$, such that

(4.4) $||v(T)||_{1}^{2}\leq
C_{T}\big{(}||\varphi||^{2}+\max\limits_{t\in[0,T]}||f||^{2}\big{)}.$

As noted in the introduction, the Cauchy problem (4.1) in the case when $A$ is
the Laplace operator in the $N$-dimensional domain ($N=1,2,3$) has been
considered by many authors. So the authors of [7] establish the Sobolev
regularity of the homogeneous problem for both smooth and nonsmooth initial
data $\varphi$. In particular, the authors proved an estimate similar to (4.3)
(see estimate (2.16), note that in this paper $f(t)\equiv 0$).

Formula (4.2) for solving problem (4.1) is formally given in the same article
of Bazhlekova, Jin, Lazarov, and Zhou [7], but since this paper is devoted to
the study of a homogeneous problem, the question of for which $\varphi$ and
$f(t)$ it gives a solution to problem (4.1) is not considered. Note that
functions $\varphi$ and $f(t)$ must be such that, for example, the series
(4.2) allows term-by-term differentiation with respect to the variable $t$.

The formula is also contained in papers [10], equation (2.2), [14], equation
(2.6) and [15], equation (8). Again, since these works are devoted to the
study of other problems, the above issues were not considered.

###### Proof.

According to the Fourier method, we expand the functions $\varphi$ and $f(t)$
into a Fourier series in terms of the system of eigenfunctions $\\{v_{k}\\}$
with coefficients $\varphi_{k}$ and $f_{k}(t)=(f(t),v_{k})$ correspondingly.
The solution of problem (2.2) will be sought in the form of a series
$\sum_{k}T_{k}(t)v_{k}$ with unknown coefficients $T_{k}(t)$.

It is easy to verify that the functions $T_{k}(t)$ are solutions of the Cauchy
problem (3.5) with the right-hand side $f(t)=f_{k}(t)$, the initial condition
$y_{0}=\varphi_{k}$ and with the parameter $\lambda=\lambda_{k}$. Therefore
(see (3.4))

$T_{k}(t)=B_{\alpha}(\lambda_{k},t)\,\varphi_{k}+\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau.$

Thus, series (4.2) is a formal solution to problem (4.1). It remains to show
that the series itself and the series obtained after differentiation also
converge. Let us get started with this task.

By Parseval’s equality, the first assertion of the Lemma 3.1, and Holder’s
inequality, we have

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}^{2}=\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]^{2}\leq$
$\leq\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}|f_{k}(\tau)|d\tau\right]^{2}\leq
C\max_{0\leq t\leq T}||f(t)||^{2}.$

Same way,

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}B_{\alpha}(\lambda_{k},t)\varphi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq
C||\varphi||^{2}.$

Now consider the series after differentiation. Apply Parseval’s equality to
obtain

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\left[\partial_{t}\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}^{2}=\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\left[f_{k}(t)-\int\limits_{0}^{t}\partial_{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}^{2}\leq$
$\leq 2\max_{0\leq t\leq
T}||f(t)||^{2}+2\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}\partial_{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]^{2}.$

Generalized Minkowski inequality and Lemma 3.6 imply

$\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}\partial_{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]^{2}\leq
C_{n}\bigg{(}\int\limits_{0}^{t}\tau^{\frac{1-\alpha}{n}-1}\bigg{(}\sum\limits_{k=1}^{j}|\lambda_{k}^{\frac{1}{n}}f_{k}(\tau)|^{2}\bigg{)}^{\frac{1}{2}}d\tau\bigg{)}^{2}\leq$

(choose $n$ such that $\frac{1}{n}<\varepsilon$)

$\leq C_{\varepsilon}\max_{t\in[0,T]}||f(t)||_{\varepsilon}.$

Similarly, apply Parseval’s equality and (3.7) to get

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\partial_{t}B_{\alpha}(\lambda_{k},t)\varphi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq\frac{C}{t^{2}}||\varphi||^{2}.$

Equation (2.2) implies

$||(1+\gamma\partial_{t}^{\alpha})Av||^{2}\leq||\partial_{t}v||^{2}+||f(t)||^{2}.$

Therefore, it follows from the above

$\bigg{|}\bigg{|}(1+\gamma\partial_{t}^{\alpha})A\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}^{2}\leq
C_{\varepsilon}\max_{t\in[0,T]}||f(t)||_{\varepsilon},$

and

$\bigg{|}\bigg{|}(1+\gamma\partial_{t}^{\alpha})A\sum\limits_{k=1}^{j}B_{\alpha}(\lambda_{k},t)\varphi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq
C(\frac{1}{t^{2}}||\varphi||+||f(t)||).$

Note that $||f(t)||\leq||f(t)||_{\varepsilon}$, $\varepsilon>0$.

Thus, it has been shown that the function defined by the series (4.2) is
indeed a solution to problem (2.2).

Obviously, (4.3) follows from the established estimates.

The uniqueness of the solution to problem (2.2) can be proved by the standard
technique based on completeness of the set of eigenfunctions $\\{v_{k}\\}$ in
$H$. Indeed, suppose the problem has two solutions $v^{1}$ and $v^{2}$. Then
the difference $v=v^{1}-v^{2}$ is a solution of the homogeneous problem:

$\left\\{\begin{aligned}
&\partial_{t}v(t)+(1+\gamma\,\partial_{t}^{\alpha})Av(t)=0,\quad 0<t\leq T;\\\
&v(0)=0.\end{aligned}\right.$

Let $v(t)$ be any soltion of this problem. Consider the Fourier coefficients
$T_{k}(t)=(v(t),v_{k})$. It is not hard to see, that $T_{k}$ is a solution of
the Cauchy problem

$\partial_{t}T_{k}(t)+\lambda_{k}(1+\gamma\,\partial_{t}^{\alpha})T_{k}(t)=0,\quad
0<t\leq T,\,\,T_{k}(0)=0.$

Corollary 3.6 implies that $T_{k}(t)\equiv 0$ for all $k$. Since the set of
eigenfunctions $\\{v_{k}\\}$ complete in $H$, then $v(t)\equiv 0$.

It remains to prove the estimate (4.4). Let $S_{j}(t)$ be the partial sum of
the series (4.2). Then

$||AS_{j}(T)||\leq\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\lambda_{k}B_{\alpha}(\lambda_{k},T)\varphi_{k}v_{k}\bigg{|}\bigg{|}+\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{T}\lambda_{k}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}.$

Apply Parseval’s equality and estimate (3) of Lemma 3.1 to obtain

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\lambda_{k}B_{\alpha}(\lambda_{k},T)\varphi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq
C_{T}\sum\limits_{k=1}^{j}|\varphi|^{2}\leq C_{T}||\varphi||^{2}.$

Similarly,

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{T}\lambda_{k}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}\bigg{|}\bigg{|}^{2}\leq
C_{T}\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{T}\tau^{-\alpha}f_{k}(\tau)d\tau\right]^{2}\leq$

(apply generalized Minkowski inequality)

(4.5) $\leq
C_{T}\bigg{(}\int\limits_{0}^{T}\tau^{-\alpha}\bigg{(}\sum\limits_{k=1}^{j}|f_{k}(\tau)|^{2}\bigg{)}^{1/2}d\tau\bigg{)}^{2}\leq
C_{T}\max_{t\in[0,T]}||f(t)||^{2}.$

The last two estimates imply (4.4).

The theorem is completely proven.

∎

##  5\.  Backward problem

In the present paragraph we consider the following backward problem:

(5.1) $\left\\{\begin{aligned}
&\partial_{t}\omega(t)+(1+\gamma\,\partial_{t}^{\alpha})A\omega(t)=f(t),\quad
0<t<T;\\\ &\omega(T)=\psi,\end{aligned}\right.$

where $\psi$ is a given vector of $H$. It should be specially emphasized that
for the backward problem the parameter $\gamma$ in the Rayleigh-Stokes
equation plays an important role. If this parameter is equal to zero, then we
get the classical diffusion equation, for which the backward problem is
strongly ill-posed: not for any $\psi\in H$ (even not for any $\psi\in
D(A^{k}),\,k\geq 1$) there is a solution, and if it exists, then a small
change in $\psi$ leads to a very strong change in the solution (see e.g.
Chapter 8.2 of [17]). In the next Theorem 5.1, we show that the parameter
$\gamma\neq 0$ $"$ennobles$"$ the backward problem for the Rayleigh-Stokes
equation and its solution already exists for all $\psi\in D(A)$.

Nevertheless problem (5.1) is also ill-posed in the sense of Hadamard because
of the same reason as the classical one ($\gamma=0$): a small variation of
$u(T)$ in the norm of space $H$ may cause arbitrarily large variations in the
initial data. Indeed, let $f(t)\equiv 0$ and take

$\omega(0)=\lambda^{-1+\varepsilon}_{k}\cdot\frac{1}{B_{\alpha}(\lambda_{k},T)}\cdot
v_{k},\,\,\varepsilon>0,$

as a initial condition in problem (4.1). Then the unique solution of the
problem (4.1) is (see Theorem 4.1)

$\omega(t)=\lambda^{-1+\varepsilon}_{k}\cdot\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)}\cdot
v_{k}.$

This function is a unique solution of problem (5.1) with
$u(T)=\lambda^{-1+\varepsilon}_{k}v_{k}$.

Therefore, on the one hand, $||u(T)||=\lambda^{-1+\varepsilon}_{k}$ and it
tends to zero as $k\rightarrow\infty$ (even $||u(T)||_{a}\rightarrow 0$ for
any $a<1-\varepsilon$), and on the other, according to Lemma 3.7,

$||\omega(0)||\geq\lambda^{-1+\varepsilon}_{k}\cdot\frac{\lambda_{k}}{C(\alpha,\gamma,\lambda_{1})}\rightarrow\infty\quad\text{when}\quad
k\rightarrow\infty.$

However, if we consider the norm of $u(T)$ in space $D(A)$, then the situation
will change completely; note the norm $||u(T)||_{1}=\lambda_{k}^{\varepsilon}$
in this example is unbounded as $k\rightarrow\infty$.

######  Theorem 5.1.

Let $f(t)\in C([0,T];H)$. Then for any $\psi\in D(A)$ problem (4.1) has a
unique solution. Moreover there exists a constant
$C=C(\alpha,\gamma,\lambda_{1},T)>0$, depending on $\alpha,\gamma,\lambda_{1}$
and $T$, such that

(5.2) $||\omega(t)||\leq
C\,\big{(}||\omega(T)||_{1}+\max\limits_{t\in[0,T]}|f(t)||\big{)}.$

Let $f(t)\equiv 0$. Then there exist constants $C_{1},C_{2}>0$, such that

(5.3) $C_{1}||\omega(0)||\leq||\omega(T)||_{1}\leq C_{2}||\omega(0)||.$

Note that in work [18] the baclward problem for the subdiffusion equation
$\partial_{t}^{\alpha}u+Au=f$ was studied. It is shown that an estimate
similar to (5.3) is valid only with the norm $||\omega(T)||_{2}$. However, if
we take the derivative in the sense of Caputo in the subdiffusion equation,
then the estimate (5.3) with the norm $||\omega(T)||_{1}$ is valid (see
([19])).

###### Proof.

Let us take a solution (4.2) with an unknown initial function $\varphi$ and
use condition $\omega(T)=\psi$ to determine this unknown function. Then the
Fourier coefficients of $\varphi$ has the form

(5.4)
$\varphi_{k}=\frac{1}{B_{\alpha}(\lambda_{k},T)}\left(\psi_{k}-\int\limits_{0}^{T}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right),\,\,k\geq
1,$

and the unique formal solution of the backward problem can be written as

$\omega(t)=\sum\limits_{k=1}^{\infty}\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)}\left(\psi_{k}-\int\limits_{0}^{T}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right)v_{k}+\sum\limits_{k=1}^{\infty}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}.$

According to Theorem 4.1, this formal solution will indeed be a unique
solution to the backward problem if the condition: $\varphi\in H$ is
satisfied. Let us check this condition.

Apply Lemma 3.7 to obtain

$\sum\limits_{k=1}^{j}\bigg{|}\frac{\psi_{k}}{B_{\alpha}(\lambda_{k},T)}\bigg{|}^{2}\leq\frac{1}{C(\alpha,\gamma,\lambda_{1})}\sum\limits_{k=1}^{j}|\lambda_{k}\psi_{k}|^{2}\leq\frac{1}{C(\alpha,\gamma,\lambda_{1})}||\psi||_{1}^{2}.$

Again this lemma and estimate (3) of Lemma 3.1 imply (see (4.5))

$\sum\limits_{k=1}^{j}\bigg{|}\frac{1}{B_{\alpha}(\lambda_{k},T)}\bigg{|}^{2}\left(\int\limits_{0}^{T}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right)^{2}\leq\frac{C_{T}}{C(\alpha,\gamma,\lambda_{1})}\max_{t\in[0,T]}||f(t)||^{2}.$

Thus $\varphi=\sum_{k}\varphi_{k}v_{k}\in H$ and therefore, function
$\omega(t)$ defined above is indeed a solution to the backward problem ((4.1).

Let us pass to the proof of estimate (5.2). We note right away that in order
to estimate the norm in $H$, by virtue of the Parseval equality, it is
sufficient to estimate a series of Fourier coefficients.

Application of Lemmas 3.1 and 3.7 gives

$\sum\limits_{k=1}^{j}\left|\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)}\psi_{k}\right|^{2}\leq\frac{1}{C(\alpha,\gamma,\lambda_{1})}\sum\limits_{k=1}^{j}|\lambda_{k}\psi_{k}|^{2}\leq\frac{1}{C(\alpha,\gamma,\lambda_{1})}||\psi||^{2}.$

Apply Lemma 3.7 for $B_{\alpha}$ in the denominator, estimate (1) of Lemma 3.1
for $B_{\alpha}$ in the numerator, and finally estimate (3) of Lemma 3.1 for
$B_{\alpha}$ under the integral sign (see (4.5)). Then

$\sum\limits_{k=1}^{j}\bigg{|}\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)}\bigg{|}^{2}\left(\int\limits_{0}^{T}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right)^{2}\leq\frac{C_{T}}{C(\alpha,\gamma,\lambda_{1})}\max_{t\in[0,T]}||f(t)||^{2}.$

Similarly, by virtue of estimate (3) of Lemma 3.1 we obtain

$\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]^{2}\leq
C_{T}\max_{t\in[0,T]}||f(t)||^{2}.$

As for estimate (5.3), its left part follows from the just proved estimate
(5.2), while the right part is contained in (4.4).

∎

As noted in the introduction, statements similar to Theorem 5.1 were known
earlier (see, e.g., [14], [15] and the bibliography therein). In these papers,
operator $A$ is the Laplace operator with the Dirichlet condition in the
domain $\Omega\subset R^{N}$, $N\leq 3$.

Let us make a few remarks:

1\. The left side of the estimate (5.3) means that a small change of
$\omega(T)$ in the norm $||\cdot||_{1}$ entails a small change in the norm
$||\cdot||$ of the initial data.

2\. The right side of this estimate asserts the unimprovability of the left
side, i.e. it is impossible to replace the norm $||\cdot||_{1}$ with the norm
$||\cdot||_{1-\varepsilon}$ with any $\varepsilon>0$.

3\. For the classical diffusion equation (i.e. $\gamma=0$), the statement of
Theorem 5.1 naturally does not hold, since $C(\alpha,0,\lambda_{1})=0$ (see
the definition of this constant in Lemma 3.7). It should also be noted, that
for the backward problem for the classical diffusion equation (i.e.
$\gamma=0$) estimates of the type (5.3) on the scales of spaces $D(A^{a})$ are
generally impossible (see, for example, Chapter 8.2 of [17]).

##  6\. Conditional stability

Following work Luc, Huynh, O’Regan and Can [15], in this section we consider
the problem of conditional stability of the backward problem. In other words,
we impose the following a priori bound condition on the initial data
$\omega(0)=\varphi$:

(6.1)
$||\varphi||^{2}_{\varepsilon}=\sum_{k=1}^{\infty}\lambda_{k}^{2\varepsilon}|\varphi_{k}|^{2}\leq\Phi^{2}_{0},$

where $\varepsilon$ and $\Phi_{0}$ are positive constants, and consider a
class of functions that satisfy this condition. The following statement is
true:

######  Theorem 6.1.

Let $\varphi\in D(A^{\varepsilon})$ satisfy condition (6.1). Then there is a
constant $C$ depending on $\alpha,\gamma,\lambda_{1}$ and $T$ such, that

(6.2) $||\varphi||\leq C\big{[}||\psi||+\max_{0\leq t\leq
T}||f(t)||\big{]}^{\frac{\varepsilon}{1+\varepsilon}}\Phi_{0}^{\frac{1}{1+\varepsilon}}.$

This theorem for the case of the Laplace operator with the Dirichlet condition
in $N$, $N\leq 3$, - dimensional domain was proved in the above paper [15]
(note that the condition of the theorem in [15] is slightly different). Note
that in this case the estimates $\lambda_{k}\geq Ck^{\frac{2}{N}}$ hold for
the eigenvalues of the Laplace operator, and the proof in [15] is based on the
convergence of the series

$\sum_{k=1}^{\infty}\frac{1}{\lambda^{2}_{k}}\leq
C\sum_{k=1}^{\infty}\frac{1}{k^{\frac{4}{N}}}<\infty.$

The proof of Theorem 6.1 repeats the proof of [15] with a slight change. For
the convenience of the reader, we present this proof.

We emphasize that the assertion of Theorem 6.1 is valid without any conditions
on the spectrum of operator $A$.

###### Proof.

Set (see (5.4))

$\Phi_{k}=\psi_{k}-\int\limits_{0}^{T}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau,\,k\geq
1.$

Then

$||\varphi||^{2}=\sum_{k=1}^{\infty}\left|\frac{\Phi_{k}}{B_{\alpha}(\lambda_{k},t)}\right|^{2}=\sum_{k=1}^{\infty}\frac{|\Phi_{k}|^{\frac{2\varepsilon}{1+\varepsilon}}|\Phi_{k}|^{\frac{2}{1+\varepsilon}}}{|B_{\alpha}(\lambda_{k},t)|^{2}}.$

Apply the Holder inequality with parametrs
$p=\frac{1+\varepsilon}{\varepsilon}$ and $q=1+\varepsilon$ to obtain

$||\varphi||^{2}\leq\left(\sum_{k=1}^{\infty}|\Phi_{k}|^{2}\right)^{\frac{\varepsilon}{1+\varepsilon}}\left(\sum_{k=1}^{\infty}\frac{1}{|B_{\alpha}(\lambda_{k},t)|^{2\varepsilon}}\left|\frac{\Phi_{k}}{B_{\alpha}(\lambda_{k},t)}\right|^{2}\right)^{\frac{1}{1+\varepsilon}}.$

We use Lemma 3.7 to get

$\sum_{k=1}^{\infty}\frac{1}{|B_{\alpha}(\lambda_{k},t)|^{2\varepsilon}}\left|\frac{\Phi_{k}}{B_{\alpha}(\lambda_{k},t)}\right|^{2}\leq\frac{1}{C^{2\varepsilon}(\alpha,\gamma,\lambda_{1})}\sum_{k=1}^{\infty}|\lambda_{k}^{\varepsilon}\,\varphi_{k}|^{2}\leq\frac{\Phi_{0}^{2}}{C^{2\varepsilon}(\alpha,\gamma,\lambda_{1})}.$

On the other hand estimate (1) of Lemma 3.1 and the Holder inequality give

$\sum_{k=1}^{\infty}|\Phi_{k}|^{2}\leq
2||\psi||^{2}+2\sum_{k=1}^{\infty}\left[\int\limits_{0}^{T}|f_{k}(\tau)|d\tau\right]^{2}\leq
2||\psi||^{2}+2T\int\limits_{0}^{T}\sum_{k=1}^{\infty}|f_{k}(\tau)|^{2}d\tau\leq$
$\leq 2||\psi||^{2}+2T\max_{0\leq t\leq T}||f(t)||^{2}.$

The last two estimates imply estimate (6.1). ∎

##  7\. Auxiliary problem (2.2)

If we set $\varphi=0$, then from Theorem 4.1 we have the following result for
our auxiliary problem (2.2).

######  Theorem 7.1.

Let $f(t)\in C([0,T];D(A^{\varepsilon}))$ for some $\varepsilon\in(0,1)$. Then
problem (2.2) has a unique solution and this solution has the representation

(7.1)
$v(t)=\sum\limits_{k=1}^{\infty}\left[\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau\right]v_{k}.$

Moreover, there is a constant $C_{\varepsilon}>0$ such that the following
coercive type inequality holds:

(7.2) $||\partial_{t}v(t)||^{2}+||\partial_{t}^{\alpha}Av(t)||^{2}\leq
C_{\varepsilon}\max\limits_{t\in[0,T]}||f||_{\varepsilon}^{2},\quad 0\leq
t\leq T.$

If $f(t)$ does not depend on $t$, then the statement of Theorem 2.2 is true
for all $f\in H$.

######  Corollary 7.2.

Let $f\in H$. Then problem (2.2) has a unique solution and this solution has
the representation

(7.3)
$v(t)=\sum\limits_{k=1}^{\infty}f_{k}v_{k}\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},\tau)d\tau.$

Moreover, there is a constant $C>0$ such that the following coercive type
inequality holds:

(7.4) $||\partial_{t}v(t)||^{2}+||\partial_{t}^{\alpha}Av(t)||^{2}\leq
C||f||^{2},\quad 0<t\leq T.$

###### Proof.

Estimate (1) of Lemma 3.1 implies

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}f_{k}v_{k}\partial_{t}\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},\tau)d\tau\bigg{|}\bigg{|}^{2}=\sum\limits_{k=1}^{j}|f_{k}|^{2}|B_{\alpha}(\lambda_{k},t)|^{2}\leq||f||.$

Using this estimate and repeating the arguments similar to the proof of
Theorem 2.2, it is easy to check that (7.3) is indeed a unique solution to
problem (2.2) and estimate (7.4) holds true.

∎

Note that the integral in (7.3) can be rewritten in the form

$\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},\tau)d\tau=\frac{1}{\lambda_{k}}(1-A_{\alpha}(\lambda_{k},t)),$

where function $A_{\alpha}(\lambda_{k},t))$ is also studied in [7]. In
particular, it is shown that the Laplace transform of this function is

$\mathcal{L}\\{A_{\alpha}(\lambda_{k},\cdot)\\}(z)=\frac{1+\gamma\lambda_{k}z^{\alpha-1}}{z+\gamma\lambda_{k}z^{\alpha}+\lambda_{k}},$

and the estimates

$0<A_{\alpha}(\lambda_{k},t)\leq 1$

hold.

##  8\. The second auxiliary and non-local problems

In this section we first consider auxiliary problem (2.3) and then we obtain
the result for the non-local problem (2.1).

######  Theorem 8.1.

Let $\psi\in H$. Then problem (2.3) has a unique solution and this solution
has the representation

(8.1)
$w(t)=\sum\limits_{k=1}^{\infty}\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)-1}\psi_{k}v_{k}.$

Moreover, there is a constant $C>0$ such that the following coercive type
inequality holds:

(8.2) $||\partial_{t}w(t)||^{2}+||\partial_{t}^{\alpha}Aw(t)||^{2}\leq
C\frac{1}{t^{2}}||\psi||^{2},\quad 0<t\leq T.$

###### Proof.

The solution to problem (2.3) will be sought in the form of a Fourier series

$w(t)=\sum_{k=1}^{\infty}T_{k}(t)v_{k},$

where $T_{k}(t)$ is the solution to the non-local problem

(8.3) $\left\\{\begin{aligned}
&\partial_{t}T(t)+\lambda_{k}(1+\gamma\,\partial_{t}^{\alpha})T_{k}(t)=0,\quad
0<t<T;\\\ &T_{k}(T)=T_{k}(0)+\psi_{k},\end{aligned}\right.$

Assuming $T_{k}(0)=h_{k}$ to be known, we write a solution to equation (8.3)
with this initial condition (see (3.3))

$T_{k}(t)=h_{k}B_{\alpha}(\lambda_{k},t).$

Now, using the non-local condition in (8.3), we obtain an equation for finding
the unknowns $h_{k}$:

$h_{k}(B_{\alpha}(\lambda_{k},T)-1)=\psi_{k}.$

Since $T>0$ and $\lambda_{k}>0$, then $0<B_{\alpha}(\lambda_{k},T)<1$ (see
Lemma 3.1). Therefore

$h_{k}=\frac{\psi_{k}}{B_{\alpha}(\lambda_{k},T)-1},\,\,|h_{k}|\leq
C|\psi_{k}|,\,\,k\geq 1,$

and (8.1) is a formal solution of problem (2.3).

The fact that the series (8.1) converges is beyond doubt. It remains to show
that this series can be term-by-term differentiated.

First apply Parseval’s equality and (3.7) to get

$\bigg{|}\bigg{|}\sum\limits_{k=1}^{j}\frac{\partial_{t}B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)-1}\psi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq\frac{C}{t^{2}}||\psi||^{2},\,\,t>0.$

On the other hand equation (2.3) implies

$||(1+\gamma\partial_{t}^{\alpha})Aw||^{2}\leq||\partial_{t}w||^{2}.$

Therefore, it follows from the above

$\bigg{|}\bigg{|}(1+\gamma\partial_{t}^{\alpha})A\sum\limits_{k=1}^{j}\frac{B_{\alpha}(\lambda_{k},t)}{B_{\alpha}(\lambda_{k},T)-1}\psi_{k}v_{k}\bigg{|}\bigg{|}^{2}\leq
C\frac{1}{t^{2}}||\psi||,\,\,t>0.$

The uniqueness of the problem solution (2.3) is based on the completeness of
the set of eigenfunctions $\\{v_{k}\\}$ in $H$ and the estimate
$B_{\alpha}(\lambda_{k},T)<1$. Indeed, if we assume that there are two
solutions $w^{1}$ and $w^{2}$, then the difference $w=w^{1}-w^{2}$ is a
solution to the homogeneous problem:

$\left\\{\begin{aligned}
&\partial_{t}w(t)+(1+\gamma\,\partial_{t}^{\alpha})Aw(t)=0,\quad 0<t\leq T;\\\
&w(T)=w(0).\end{aligned}\right.$

Let $w(t)$ be any soltion of this problem. Consider the Fourier coefficients
$T_{k}(t)=(w(t),v_{k})$. It is not hard to see, that $T_{k}$ is a solution of
the problem

$\partial_{t}T_{k}(t)+\lambda_{k}(1+\gamma\,\partial_{t}^{\alpha})T_{k}(t)=0,\quad
0<t\leq T,\,\,T_{k}(T)=T_{k}(0).$

The solution of the Cauchy problem with the initial data $T_{k}(0)=h_{k}$ has
the form $T_{k}(t)=h_{k}B_{\alpha}(\lambda_{k},t)$ (see Corollary 3.6). The
non-local condition implies $h_{k}B_{\alpha}(\lambda_{k},T)=h_{k}$. Therefore
$T_{k}(t)\equiv 0$ for all $k$. Since the set of eigenfunctions $\\{v_{k}\\}$
complete in $H$, then $w(t)\equiv 0$.

∎

Combining the statements of the last two theorems, we obtain the following
assertion for the non-local problem (2.1).

######  Theorem 8.2.

Let $\varphi\in H$ and $f(t)\in C([0,T];D(A^{\varepsilon}))$ for some
$\varepsilon\in(0,1)$. Then problem (2.1) has a unique solution and this
solution has the form

(8.4)
$u(t)=\sum\limits_{k=1}^{\infty}\left[\frac{\varphi_{k}-y_{k}(T)}{B_{\alpha}(\lambda_{k},T)-1}B_{\alpha}(\lambda_{k},t)+y_{k}(t)\right]v_{k},$

where

$y_{k}(t)=\int\limits_{0}^{t}B_{\alpha}(\lambda_{k},t-\tau)f_{k}(\tau)d\tau$

Moreover, there are positive constants $C$ and $C_{\varepsilon}$ such that the
following coercive type inequality holds:

(8.5) $||\partial_{t}u(t)||^{2}+||\partial_{t}^{\alpha}Au(t)||^{2}\leq
C\frac{1}{t^{2}}||\varphi||+C_{\varepsilon}\max\limits_{t\in[0,T]}||f||_{\varepsilon}^{2},\quad
0<t<T.$

###### Proof.

Let $\varphi\in H$ and $f(t)\in C([0,T];D(A^{\varepsilon}))$ for some
$\varepsilon\in(0,1)$. As noted above, if we put $\psi=\varphi-v(T)\in H$ and
$v(t)$ and $w(t)$ are the corresponding solutions of problems (2.2) and (2.3),
then function $u(t)=v(t)+w(t)$ is a solution to problem (2.1). Therefore, the
solution of problem (2.1) has the form (8.1).

Estimate (8.5) follows from estimates (7.2) and (8.2).

∎

##  9\. Conclusion

In many papers, explicit solutions of the simplest Rayleigh-Stokes problems
are constructed. In [7], the regularity of the solution was studied for the
first time and a formal formula for solving an inhomogeneous problem in a
three-dimensional domain was given. The backward problem has also been studied
by the authors in domains with dimensions less than four. Some proofs of these
results essentially use the fact that the dimension of the domain under
consideration is $N\leq 3$.

In this paper, all these questions are investigated for the case of an
abstract operator instead of the Laplace operator in the Rayleigh-Stokes
problem. In addition, a new non-local problem for equation Rayleigh-Stokes is
considered and it is shown that, unlike the backward problem, it is well-
posed. As the operator $A$, we can take the Laplace operator with the
Dirichlet condition in an arbitrary $N$-dimensional domain with a sufficiently
smooth boundary.

##  10\. Acknowledgement

The authors are grateful to Sh. A. Alimov for discussions of these results.

The authors acknowledge financial support from the Ministry of Innovative
Development of the Republic of Uzbekistan, Grant No F-FA-2021-424.

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