# VampNet: Music Generation via
Masked Acoustic Token Modeling

###### Abstract

We introduce VampNet, a masked acoustic token modeling approach to music
synthesis, compression, inpainting, and variation. We use a variable masking
schedule during training which allows us to sample coherent music from the
model by applying a variety of masking approaches (called prompts) during
inference. VampNet is non-autoregressive, leveraging a bidirectional
transformer architecture that attends to all tokens in a forward pass. With
just 36 sampling passes, VampNet can generate coherent high-fidelity musical
waveforms. We show that by prompting VampNet in various ways, we can apply it
to tasks like music compression, inpainting, outpainting, continuation, and
looping with variation (vamping). Appropriately prompted, VampNet is capable
of maintaining style, genre, instrumentation, and other high-level aspects of
the music. This flexible prompting capability makes VampNet a powerful music
co-creation tool. Code and audio samples are available online.

##  1 Introduction

Figure 1: VampNet overview. We first convert audio into a sequence of discrete
tokens using an audio tokenizer. Tokens are masked, and then passed to a
masked generative model, which predicts values for masked tokens via an
efficient iterative parallel decoding sampling procedure at two levels. We
then decode the result back to audio.

In recent years, advances in discrete acoustic token modeling have resulted in
significant leaps in autoregressive generation of speech [1, 2] and music [3].
Meanwhile, approaches that use non-autoregressive parallel iterative decoding
have been developed for efficient image synthesis [4, 5]. Parallel iterative
decoding promises to allow faster inference than autoregressive methods and is
more suited to tasks like infill, which require conditioning on both past and
future sequence elements.

In this work, we combine parallel iterative decoding with acoustic token
modeling, and apply them to music audio synthesis. To the best of our
knowledge, ours is the first 111While our work was under peer review, Google
released SoundStorm [6], which leverages a similar parallel iterative decoding
approach to ours. extension of parallel iterative decoding to neural audio
music generation. Our model, called VampNet, can be flexibly applied to a
variety of applications via token-based prompting. We show that we can guide
VampNet’s generation with selectively masked music token sequences, asking it
to fill in the blanks. The outputs of this procedure can range from a high-
quality audio compression technique to variations on the original input music
that match the original input music in terms of style, genre, beat and
instrumentation, while varying specifics of timbre and rhythm.

Unlike auto-regressive music models [2, 3], which can only perform music
continuations – using some prefix audio as a prompt, and having the model
generate music that could plausibly come after it – our approach allows the
prompts to be placed anywhere. We explore a variety of prompt designs,
including periodic, compression, and musically informed ones (e.g. masking on
the beat). We find that our model responds well to prompts to make loops and
variations, thus the name VampNet 222To vamp is to repeat a short passage of
music with variation.. We make our code open
source333https://github.com/hugofloresgarcia/vampnet and highly encourage
readers to listen to our audio samples444audio samples:
https://tinyurl.com/bdfj7rdx.

##  2 Background

Two-stage approaches to generative modeling have gained traction in image [5,
4, 7, 8] and audio [2, 3, 6, 9] synthesis, largely in part due to their
computational efficiency. In the first stage, a discrete vocabulary of
“tokens” is learned for the domain of interest. The input is put through an
encoder to obtain these tokens, which can be converted back into the input
domain via a corresponding decoder. In the second stage, a model is trained to
generate tokens, and is optionally given some conditioning (e.g. previous
tokens, a text description, a class label) to guide generation.

###  2.1 Stage 1: Tokenization

In images, visual tokenization has been leveraged for state-of-the-art
classification [10] and synthesis [4, 7, 11, 8]. The most popular approach is
to use vector quantization on a latent space. Similar approaches have been
explored for audio [12], but until recently such approaches have been
restricted to low sampling rates (e.g. 16khz), or have been restricted to
speech audio. The “sampling rate” of the latent space (the number of latent
vectors required every second to represent audio) is a critical aspect of the
tokenization scheme. The lower the sampling rate of the latent space, the
easier the next stage (generation) will be to accomplish. Recently, methods
based on residual vector quantization [13, 14] have been proposed for audio
tokenization at high compression rates with good reconstruction quality of
high-sample-rate audio.

The primary work we leverage for audio tokenization is the Descript Audio
Codec (DAC) [15]. With DAC, audio is encoded into a sequence of tokens via a
fully convolutional encoder. The output of this encoder is then quantized
using a hierarchical sequence of vector-quantizers [11]. Each quantizer
operates on the residual error of the quantizer before it. Because of this
residual vector quantization, DAC is able to reconstruct audio with very high
quality, at a high compression ratio. It, along with its predecessors [14,
13], are instrumental in enabling audio language models like AudioLM [2],
MusicLM [3], and VALL-E [1]. While we later briefly describe our tokenizer,
the key contributions of our work are applicable to the output of any audio
tokenizer and our specific audio tokenizer is not the focus of this work.

Figure 2: Training, sampling, and prompting VampNet. Training: we train
VampNet using Masked Acoustic Token Modeling, where we randomly mask a portion
of a set of input acoustic tokens and learn to predict the masked set of
tokens, using a variable masking schedule. Coarse model training masks coarse
tokens. Coarse-to-fine training only masks fine tokens. Sampling: we sample
new sequences of acoustic tokens from VampNet using parallel iterative
decoding, where we sample a subset of the most confident predicted tokens each
iteration. Prompting: VampNet can be prompted in a number of ways to generate
music. For example, it can be prompted periodically, where every $P$th
timestep in an input sequence is unmasked, or in a beat-driven fashion, where
the timesteps around beat markings in a song are unmasked.

###  2.2 Stage 2: Generation

Given audio encoded as tokens, the common approach is to use an autoregressive
model [16] for generation. State-of-the-art (SOTA) audio generation approaches
like AudioLM [2], MusicLM [3], and JukeBox [17] use this approach, generating
each acoustic token in the sequence in a step-by-step fashion using
transformer-based [18] decoder-only models. Autoregressive sampling is slow in
nature due to the high number of steps required at inference time [4].
Further, autoregressive models inherently restrict downstream applications, as
each generated token is only conditioned on the previous tokens. For an
autoregressive model to perform tasks like inpainting (“filling in the
middle”), one must re-arrange the data during training [19].

In language, masked modeling has been used extensively as a pre-training
procedure for high-quality semantic representations [20]. This procedure has
also been extended for representation learning in images [21] and audio [22].
Masked modeling for representation learning generally has a constant mask
probability. For example, in BERT [20], tokens are masked 15% of the time
during training. It has been shown that this approach is equivalent to a
single-step discrete diffusion model [23], that uses masking for its noising
procedure. Therefore, we can extend masked modeling to masked generative
modeling by varying the probability of masking a token during training. This
was done for image generation in MaskGIT [4], and in language [23]. Similar to
diffusion modeling [24, 25], which seeks to synthesize data starting from
random noise through a series of denoising steps, masked generative modeling
seeks to synthesize data starting from completely masked data through a series
of “unmasking” steps.

Key to the efficiency of MaskGIT and related approaches is a parallel
iterative decoding procedure. In parallel iterative decoding, the model
predicts every token in the output sequence in a single forward pass. However,
after just one forward pass of the model, the output often does not have high
quality. The output of the first sampling step is re-masked, with a lower
masking probability, and then put through the model again. In this way, masked
generative models can efficiently refine their output, resulting in high
quality generation.

In unconditional generation tasks, the model is asked to generate a realistic
sample from the target data distribution from scratch, without any guidance.
This is a difficult problem, as many target data distributions are highly
multimodal. Unconditional generative models are susceptible to mode collapse
[26], blurry samples, mode averaging, and other issues[27]. Therefore, some
conditioning is helpful as it provides some signal for the model to resolve
the multimodality. Conditioning is also a commonly used method to guide the
output of the system towards desired content.

Conditioning can take the form of a class label, a genre tag or lyrics [17],
or an associated text description [28, 8, 3]. Conditioning can also be applied
at every timestep, like the semantic tokens of AudioLM [2], or aligned text or
phonemes for text-to-speech generation [1].

In this work,we adopt a masked generative modeling approach with a parallel
iterative decoding procedure, inspired by work in vision such as MaskGIT [4]
and Paella [5], as illustrated in Figure 1. We do not apply any conditioning
beyond that provided by the unmasked tokens in our encoded audio. As we show
later, different approaches to masking, applied at inference time, can be used
to steer generation in useful and artistic ways.

In training, tokens are masked randomly throughout the sequence. The model is
then asked to predict the value of each of the masked tokens in a single
forward pass, but it is conditioned on all of the unmasked tokens, both in the
future as well as in the past. We vary the number of tokens that are masked
during training, allowing us to generate audio at inference time through a
sampling procedure. We now describe our method in more detail.

##  3 Method

We adapt the procedure of Masked Visual Token Modeling, proposed in MaskGIT
[4] to audio, accounting for several key differences between the vision and
audio domain. We call our approach Masked Acoustic Token Modeling.

###  3.1 Masked Acoustic Token Modeling

We first train an audio tokenizer based on the techniques described in DAC
[15]. Unlike the visual tokens of MaskGIT, our acoustic tokens are
hierarchical in nature due to residual vector quantization. As a first step,
the audio signal $x$ is encoded at each time step $t$ as a a $D$ dimensional
latent vector $Z$. We then quantize $Z$ using $N$ vector quantizers. Quantizer
1 produces $\hat{Z_{1}}$, a quantized approximation of $Z$ that has residual
error $R_{1}=Z-\hat{Z}_{1}$. Thereafter, the residual from each quantizer $i$
is passed to the next quantizer $i+1$, which produces a quantized
approximation of the remaining residual error: $R_{i}\approx\hat{Z_{i+1}}$.
Vector $Z$ is reconstructed by summing the output of the $N$ quantizers:
$Z=\sum_{i=1}^{N}{\hat{Z_{i}}}$.

Since the encoded signal is represented as a quantized vector of $N$ discrete
tokens at each timestep, we have $N$ tokens that can be masked or unmasked at
each timestep. Rather than attempt to generate all tokens at once, we instead
split the $N$ tokens into $N_{c}$ “coarse” tokens, and $N_{f}$ “fine” tokens,
as in AudioLM. We then train two generative models: one that generates the
fine tokens given the coarse tokens as conditioning, and one that generates
the coarse tokens given a sequence of coarse tokens. To generate a sample
(Figure 1), we chain the two models together. First, we apply the coarse model
to generate a sequence of coarse tokens. Then, we apply the coarse-to-fine
model to generate the fine tokens. We decode the tokens to a 44.1khz waveform
using the decoder of our audio tokenizer.

###  3.2 Training procedure

Let $\mathbf{Y}\in\mathbb{R}^{T\times N}$ be a matrix representing the output
of the encoder for some audio segment. Each element $y_{t,n}$ in $\mathbf{Y}$
is a token from the $n$th level codebook at timestep $t$. Let $\mathbf{Y}_{M}$
be the set of all masked tokens in $\mathbf{Y}$ and $\mathbf{Y}_{U}$ be the
set of all unmasked tokens in $\mathbf{Y}$. The model generates a probability
distribution over the set of possible codebook values for each token
$y\in\mathbf{Y}_{M}$, given the unmasked tokens and the model parameters
$\theta$. The training objective is to maximize the probability of the true
tokens. This corresponds to minimizing the negative log likelihood.

$\mathcal{L}=-\sum_{\forall y\in\mathbf{Y}_{M}}{\log
p(y|\mathbf{Y}_{U},\theta)\;\;\;\;}$ (1)

To predict the masked tokens, we use a multi-layer bidirectional transformer,
which predicts the probabilities of each possible token at every timestep, for
every quantizer. If each quantizer has a codebook size of $C$ possible values,
and there are $N$ quantizers, then the last layer of the network will be a
fully connected layer of shape $(E,CN)$, where $E$ is the dimensionality of
the output of the last layer. We then reshape this output into $(EN,C)$, and
compute the cross-entropy loss between the ground-truth one-hot token and the
predicted token. Because the transformer is bidirectional, it can attend to
all tokens in the input sequence to optimize the loss for each token.

For the coarse-to-fine generative model, the input sequence always contains
$N_{c}$ coarse tokens, and the masking operation is restricted to the $N_{f}$
fine tokens. The last layer of this network only predicts masked fine tokens.
Otherwise, the training procedure for both models is identical.

###  3.3 Sampling

We follow the same iterative confidence-based sampling approach used in
MaskGIT. More concretely, given $Y_{M}$ as the set of masked tokens and
$Y_{U}$ as the set of unmasked tokens, do:

  1. 1.

Estimate. For each masked token $y$ in $Y_{M}$, estimate the conditional
probability distribution over its vocabulary of codebook values $V$.

  2. 2.

Sample. For each masked token, sample from the distribution to generate an
associated token estimate $\hat{y}\in V$. We don’t use any sampling tricks in
this step, sampling from the categorical probability distribution for each
token as-is.

  3. 3.

Rank by Confidence. Compute a confidence measure for each of the sampled
tokens by taking their prediction log-probabilities and adding temperature-
annealed Gumbel noise to them:

$confidence(\hat{y}_{t})=log(p(\hat{y}_{t}))+temp\cdot g_{t}$ (2)

where $\hat{y}_{t}$ is a token estimate at timestep $t$, $g_{t}$ is an i.i.d
sample drawn from Gumbel(0,1) [29], and $temp$ is a hyperparameter that is
linearly annealed to 0 over the number of sampling iterations. Then, sort the
set of sampled token estimates by the confidence computed above. We find that
high temperature values (e.g. $>6.0$) result in higher quality samples.

  4. 4.

Select. Pick the number of tokens to mask at the next sampling iteration, $k$,
according to the masking schedule 555$k=\gamma(\frac{t}{t_{T}})D$, where $t$
is the current iteration, $t_{T}$ is the total number of iterations, and $D$
the total number of tokens in the sequence. The scheduling function $\gamma$
is a cosine schedule.. Take the $k$ lowest confidence estimates and toss them
out, re-masking their tokens. Place the remaining high-confidence token
estimates in $Y_{U}$, removing their tokens from $Y_{M}$.

  5. 5.

Repeat Return to step 1 until the number of iterations has been reached.

###  3.4 Prompting

Interactive music editing can be enabled by incorporating human guidance in
the sampling procedure through the conditioning prompt of unmasked tokens.
Because our approach isn’t conditioned on any signal other than the input
audio itself, we find that various types of prompts are useful for obtaining
coherent samples, as they lower the amount of multimodality when sampling from
the model. Like AudioLM, we can prompt our model with prefix audio of some
duration (usually between 1 and 4 seconds), and it will provide a continuation
of that audio. Unlike AudioLM, and other auto-regressive approaches, we can
also prompt our model with suffix audio, and it will generate audio that leads
up into that suffix. We can provide prefix and suffix audio, and the model
will generate the remaining audio, such that it is appropriate, giventhe
specified prefix and suffix.

We can also apply a “periodic” prompt, where all but every $P$th timestep are
masked.The lower $P$ is, the more the generated audio will sound like the
original, as the model is highly conditioned. For example if $P=2$, then the
model is essentially behaving like a upsampler, imputing the tokens for every
other timestep. As $P$ increases, the model shifts from behaving in a
compression mode to a generative mode, creating variations that match the
style of the original.

Another useful style of prompt are “compression” prompts, where all codebooks
other than the most coarse-grained are masked. This gives the model strong
conditioning on every timestep, so the model is likely to produce audio that
closely matches the original. We can combine this prompt with a periodic
prompt with low $P$ for even more extreme compression ratios. Given the
bitrate of the codec $B$ , which has number of codebooks $N$, a downsampling
rate $P$ for the periodic prompt, and a number of kept codebooks $N_{k}$, we
can achieve a bitrate of $B/P(N-N_{k})$.

Finally, we can design music-specific prompts, which exploit knowledge about
the structure of the music. More concretely, we explore beat-driven prompting,
where timesteps that fall on or around the beat are left unmasked. The model
is left to create music between these beats, resulting in interesting
variations on the original music. These prompts can all be combined to create
a very useful music creation tool. In concert with a well designed user
interface, VampNet shows promise as the basis for a next-generation music
editing and creation suite.

##  4 Experiments

Our experiments aim to evaluate VampNet’s capability to both compress and
generate music, given the various prompting strategies described in Section
3.4. For our objective audio quality measures, we use a multiscale mel
reconstruction error and the Fréchet Audio Distance (FAD). Mel-reconstruction
error is defined as the $L1$ distance between log-mel spectrograms at various
time-scales,

$\text{D}_{F,M}=||\hat{S}_{F,M}-S_{F,M}||_{1}$ (3)

where $F$ is the FFT size of each spectrogram, and $M$ is the number of mel-
frequency bins. We use $F\in[2048,512]$ and $M\in[150,80]$, with a hop size of
$\frac{1}{4}$ the FFT size. Mel-reconstruction is valuable as a metric for
compression quality, but not for generation quality, since it is likely that
models produce audio that does not match one to one with the original target
audio. For generation quality, we use FAD, which measures the overlap between
distributions of real and generated audio. Unlike mel-reconstruction, FAD is
geared more towards evaluating if sample quality falls within the data
distribution of the real audio, and can be used to evaluate generation
quality.

###  4.1 Dataset

Similar to JukeBox [17], we collect a large dataset of popular music
recordings. Our dataset consists of 797k tracks, with a sampling rate of 32
khz. These tracks are resampled to 44.1kHz to make compatible with our
tokenizer. Our dataset contains music from thousands of artists across genres
described in Echo Nest’s Every Noise at Once
666https://everynoise.com/engenremap.html.

We use a subset of 2k tracks for validation, and another subset of 2k tracks
for testing. We ensure that there is no artist overlap between train,
validation, and test tracks. In addition, we collect a set of music and non-
music data (speech, environmental sound), which we used to train our
tokenizer, using the datasets described in DAC [15]. All audio is normalized
to -24dbFS. We do not use any metadata about these files during training, as
our model is trained unconditionally.

Figure 3: Mel reconstruction error (top) and Fréchet Audio Distance (FAD,
bottom) for VampNet samples taken with varying numbers of sampling steps,
taken using a periodic prompt of $P=16$. The samples were generated by de-
compressing tokens at an extremely low bitrate (50 bps), effectively
generating variations of the input signals.

###  4.2 Network Architecture and Hyperparameters

The audio tokenizer model we use takes as input 44.1kHz audio, and compresses
it to a bitrate of 8kbps using 14 codebooks, with a downsampling rate of 768x.
The latent space therefore is at 57Hz, with 14 tokens to predict at every
timestep. We designate 4 of these tokens as the coarse tokens, and the
remaining 10 as the fine tokens. Refer to the Descript Audio Codec [15] for
details on the tokenizer architecture. We train the tokenizer for 250k steps.

The VampNet architecture (for both coarse and coarse-to-fine models) consists
of a bidirectional transformer [18] with relative attention [30] and an
embedding dimension of 1280 and 20 attention heads. The coarse model has 20
attention layers, while the coarse-to-fine model has 16. We train the coarse
and coarse-to-fine model for 1M and 500k steps, respectively. We train with
the AdamW optimizer [31] with $\beta_{1}$ and $\beta_{2}$ set to 0.9 and
0.999, respectively. We use the learning rate scheduler introduced by Vaswani
et al [18] with a target learning rate of 0.001 and 10k warmup steps. We use a
dropout of 0.1, and a batch size of 25, with a GPU memory budget of 72GB.

###  4.3 Efficiency of VampNet

We first validate that VampNet can generate realistic music audio in a low
number of steps. To do this, we run VampNet using one of our prompts (the
periodic prompt, with $P=16$) on our test set, on 10-second excerpts. We vary
the number of sampling steps in $[1,4,8,12,36,64,72]$, and report metrics for
each sampling step.

Figure 4:  Multiscale Mel-spectrogram error (top) and Fréchet Audio Distance
(FAD, bottom) for VampNet 10s samples taken with a different types of prompts.

###  4.4 Effect of prompts

We seek to understand how VampNet responds to different prompts, as discussed
in Section 3.4. The prompts range from “compression” prompts, which compress
music to a low bitrate, to more creative “generative” prompts. We examine
whether compression and generative prompts exist on a continuum, and whether
decompression from low bitrates results in generative behavior.

We draw 2000 10-second examples from our evaluation dataset, encode them into
token streams with our audio tokenizer, and manipulate the token streams in
four ways:

  1. 1.

Compression prompt: $C$ codebooks are left unmasked, starting from the
coarsest codebook. All other tokens are masked. We set $N_{k}=1$.

  2. 2.

Periodic prompt: every $P$th timestep is left unmasked. In an unmasked
timestep, tokens from every codebook are unmasked. All other tokens (e.g.
tokens in timesteps that do not correspond to the period $P$) are masked. We
set $P\in[8,16,32]$.

  3. 3.

Prefix and suffix (inpaint) prompts: a segment at the beginning and at the end
of the sequence is left unmasked. All other tokens are masked. This prompt is
parameterized by a context length in seconds. We set the context to be either
1 second or 2 seconds, which corresponds to 57 or 114 timesteps.

  4. 4.

Beat-driven prompt: we first process the audio waveform with a beat tracker
[32]. Then, around each detected beat, we unmask timesteps to the right of the
beat. We examine a 75ms unmasked section around each beat, which is about 4
timesteps per beat.

After manipulating the input token streams with our prompts, we generate new
musical signals from these masked token streams using VampNet, and compute FAD
and mel-reconstruction error between the generated signals and the input
signals from our music dataset. We include a noisy token stream baseline,
where a portion (as dictated by mask ratio $r$) of the tokens in the input
token stream are replaced with random tokens. We also include as baseline the
codec by itself, as well as the coarse-to-fine model.

Finally, we examine how these prompts can be combined - specifically the
compression and periodic prompts. By manipulating the hyperparameters of these
prompts ($C$ and $P$), we can shift the model behavior from compression to
generation. As more timesteps are masked, the model must generate plausible
musical excerpts that connect the unmasked timesteps, that may not match the
input music.

##  5 Results and discussion

Results for our experiment varying the number of sampling steps used to
generate samples with VampNet are shown on Figure 3. We find that VampNet
achieves the lowest FAD with 36 sampling steps, although 12 sampling steps
achieves comparable performance. In practice, we find that samples taken with
24 steps achieve a fair trade-off between generation quality and compute
speed, with 10-second samples taking around 6 seconds to sample on an NVIDIA
RTX3090. In contrast, to generate 10 seconds of audio with an autoregressive
model would require 574 steps, which would take around 1 min to generate 10
seconds of audio, given an autoregressive model with the same number of
parameters as ours, and the same tokenizer.

Results for our study on the effect of each prompt are shown in Figure 4.
First, we note that while the noisy token baseline has comparable mel
reconstruction to all prompts, it performs very poorly in terms of FAD. This
indicates that while our prompting strategies may result in audio that is not
a perfect match to the original input audio, it still falls inside the
distribution of plausible music.

Of our proposed prompts, we find that beat-driven prompts perform best,
achieving the lowest FAD of all prompts. A notable result here is that the
periodic prompt with $P=16$ (35 conditioning timesteps) performs on par with
inpainting with 1 second of context (57 conditioning timesteps). Therefore,
prompt techniques that spread out the conditioning tokens throughout the
sequence (periodic prompts) are able to use fewer conditioning timesteps to
generate samples of comparable quality to those generated by sampling
techniques that place all of the conditioning tokens at the start and end of
the sequences (inpainting).

Figure 5: Mel-spectrogram error (top) and Fréchet Audio Distance (FAD)
(bottom) for VampNet samples at varying bitrates. A baseline is provided by
replacing tokens in the input sequence with random tokens, per noise ratio
$r$.

Qualitatively, we also find that beat-driven prompts can keep a steadier tempo
than other prompts, though their outputs tend to resemble the original music
closer than periodic prompts. In practice, a mix of beat-driven, periodic, and
inpainting prompts can be employed to steer of VampNet in creative ways. To
illustrate, we highly encourage the reader to listen to the accompanying sound
samples 777audio samples: https://tinyurl.com/bdfj7rdx.

We then combined periodic and compression prompting to show how the model’s
behavior shifts between reconstruction and generation tasks, as more tokens
are masked away. Results for this experiment are shown in Figure 5. At higher
bitrates, (600 bps and above), VampNet is able to accurately reconstruct the
original music signal, achieving low mel-spectrogram error and FAD values with
respect to the evaluation music audio. At bitrates of 200bps and below,
VampNet has comparable reconstruction quality to the noisy token baselines,
indicating that the sampled VampNet signals no longer resemble the input audio
in terms of fine-grained spectral structure. However, the FAD for VampNet
samples at low bitrates is much lower than the FAD for noisy baselines. This
indicates that even though VampNet isn’t able to reconstruct the input music
signal at low bitrates, it is still able to generate coherent audio signals
with musical structure, that are closer to the distribution of “real music”
than our noisy baseline.

##  6 Conclusion

We introduced VampNet, a masked acoustic token modeling approach to music
generation. VampNet is bidirectional, and can be prompted a variety of ways
using an input audio file. Through different prompting techniques, VampNet can
operate in a continuum between music compression and generation, and is an
excellent tool for generating variations on a piece of music. With VampNet, a
musician could record a short loop, feed it into VampNet, and have VampNet
create musical variations on the recorded idea every time the looped region
repeats. In future work, we hope to investigate the interactive music co-
creation potential of VampNet and its prompting techniques, as well as explore
the representation learning capabilities of masked acoustic token modeling.

## References

  * [1] C. Wang, S. Chen, Y. Wu, Z. Zhang, L. Zhou, S. Liu, Z. Chen, Y. Liu, H. Wang, J. Li _et al._ , “Neural codec language models are zero-shot text to speech synthesizers,” _arXiv preprint arXiv:2301.02111_ , 2023.
  * [2] Z. Borsos, R. Marinier, D. Vincent, E. Kharitonov, O. Pietquin, M. Sharifi, O. Teboul, D. Grangier, M. Tagliasacchi, and N. Zeghidour, “Audiolm: a language modeling approach to audio generation,” _arXiv preprint arXiv:2209.03143_ , 2022.
  * [3] A. Agostinelli, T. I. Denk, Z. Borsos, J. Engel, M. Verzetti, A. Caillon, Q. Huang, A. Jansen, A. Roberts, M. Tagliasacchi _et al._ , “Musiclm: Generating music from text,” _arXiv preprint arXiv:2301.11325_ , 2023.
  * [4] H. Chang, H. Zhang, L. Jiang, C. Liu, and W. T. Freeman, “Maskgit: Masked generative image transformer,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 11 315–11 325.
  * [5] D. Rampas, P. Pernias, E. Zhong, and M. Aubreville, “Fast text-conditional discrete denoising on vector-quantized latent spaces,” _arXiv preprint arXiv:2211.07292_ , 2022.
  * [6] Z. Borsos, M. Sharifi, D. Vincent, E. Kharitonov, N. Zeghidour, and M. Tagliasacchi, “Soundstorm: Efficient parallel audio generation,” 2023.
  * [7] P. Esser, R. Rombach, and B. Ommer, “Taming transformers for high-resolution image synthesis,” in _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2021, pp. 12 873–12 883.
  * [8] R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer, “High-resolution image synthesis with latent diffusion models,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition._ , 2022.
  * [9] J. Copet, F. Kreuk, I. Gat, T. Remez, D. Kant, G. Synnaeve, Y. Adi, and A. Défossez, “Simple and controllable music generation,” 2023.
  * [10] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly _et al._ , “An image is worth 16x16 words: Transformers for image recognition at scale,” _arXiv preprint arXiv:2010.11929_ , 2020.
  * [11] A. Van Den Oord, O. Vinyals _et al._ , “Neural discrete representation learning,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [12] C. Gârbacea, A. van den Oord, Y. Li, F. S. Lim, A. Luebs, O. Vinyals, and T. C. Walters, “Low bit-rate speech coding with vq-vae and a wavenet decoder,” in _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 735–739.
  * [13] N. Zeghidour, A. Luebs, A. Omran, J. Skoglund, and M. Tagliasacchi, “Soundstream: An end-to-end neural audio codec,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 30, pp. 495–507, 2021.
  * [14] A. Défossez, J. Copet, G. Synnaeve, and Y. Adi, “High fidelity neural audio compression,” _arXiv preprint arXiv:2210.13438_ , 2022.
  * [15] R. Kumar, P. Seetharaman, A. Luebs, I. Kumar, and K. Kumar, “High-fidelity audio compression with improved rvqgan,” 2023.
  * [16] A. Radford, K. Narasimhan, T. Salimans, I. Sutskever _et al._ , “Improving language understanding by generative pre-training,” 2018.
  * [17] P. Dhariwal, H. Jun, C. Payne, J. W. Kim, A. Radford, and I. Sutskever, “Jukebox: A generative model for music,” _arXiv preprint arXiv:2005.00341_ , 2020.
  * [18] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [19] M. Bavarian, H. Jun, N. Tezak, J. Schulman, C. McLeavey, J. Tworek, and M. Chen, “Efficient training of language models to fill in the middle,” _arXiv preprint arXiv:2207.14255_ , 2022.
  * [20] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [21] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 16 000–16 009.
  * [22] Y.-A. Chung, Y. Zhang, W. Han, C.-C. Chiu, J. Qin, R. Pang, and Y. Wu, “W2v-bert: Combining contrastive learning and masked language modeling for self-supervised speech pre-training,” in _2021 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_. IEEE, 2021, pp. 244–250.
  * [23] J. Austin, D. D. Johnson, J. Ho, D. Tarlow, and R. van den Berg, “Structured denoising diffusion models in discrete state-spaces,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 17 981–17 993, 2021.
  * [24] Y. Song and S. Ermon, “Generative modeling by estimating gradients of the data distribution,” _Advances in neural information processing systems_ , vol. 32, 2019.
  * [25] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 6840–6851, 2020.
  * [26] A. Srivastava, L. Valkov, C. Russell, M. U. Gutmann, and C. Sutton, “Veegan: Reducing mode collapse in gans using implicit variational learning,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [27] T. Salimans, I. Goodfellow, W. Zaremba, V. Cheung, A. Radford, and X. Chen, “Improved techniques for training gans,” _Advances in neural information processing systems_ , vol. 29, 2016.
  * [28] H. Chang, H. Zhang, J. Barber, A. Maschinot, J. Lezama, L. Jiang, M.-H. Yang, K. Murphy, W. T. Freeman, M. Rubinstein _et al._ , “Muse: Text-to-image generation via masked generative transformers,” _arXiv preprint arXiv:2301.00704_ , 2023.
  * [29] E. J. Gumbel, “Statistical theory of extreme values and some practical applications; a series of lectures.” Washington, 1954.
  * [30] P. Shaw, J. Uszkoreit, and A. Vaswani, “Self-attention with relative position representations,” _arXiv preprint arXiv:1803.02155_ , 2018.
  * [31] I. Loshchilov and F. Hutter, “Fixing weight decay regularization in adam,” _CoRR_ , vol. abs/1711.05101, 2017. [Online]. Available: http://arxiv.org/abs/1711.05101
  * [32] C. J. Steinmetz and J. D. Reiss, “WaveBeat: End-to-end beat and downbeat tracking in the time domain,” in _151st AES Convention_ , 2021.


# Decay rates for a variable-coefficient wave equation with nonlinear time-
dependent damping

Menglan Liao Corresponding author: Menglan Liao
Email addresses<EMAIL_ADDRESS>College of Science, Hohai University,
Nanjing, 210098, China

Abstract: In this paper, a class of variable-coefficient wave equations
equipped with time-dependent damping and the nonlinear source is considered.
We show that the total energy of the system decays to zero with an explicit
and precise decay rate estimate under different assumptions on the feedback
with the help of the method of weighted energy integral.

Keywords: Variable coefficient; Time-dependent damping; Energy estimates;
Nonlinear source.

MSC(2020): 35B35; 35L70; 35A01.

##  1 Introduction

In this paper, we are concerned with the following initial-boundary value
problem for a class of variable-coefficient wave equations with time-dependent
damping

$\begin{cases}u_{tt}-\mathcal{A}u+\gamma(t)g(u_{t})=f(u)&\quad
x\in\Omega,~{}t>0,\\\ u(x,t)=0&\quad x\in\partial\Omega,~{}t>0,\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)&\quad x\in\Omega,\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{3}$, for simplicity, is a bounded domain with
$C^{\infty}$ boundary. In particular, present results can be extended to
bounded domain in $\mathbb{R}^{n}$ by adopting the Sobolev embeddings, and by
adjusting some parameters imposed. Here $\gamma(t)g(u_{t})$ is a time-
dependent damping term, and $f\in C^{1}(\mathbb{R})$ such that
$|f^{\prime}(s)|\leq C(|s|^{p-1}+1)$ with $1\leq p<6$. To simplify
computations, we choose $f(u):=|u|^{p-1}u$ with $1\leq p<6$ in this paper. The
second-order differential operator $\mathcal{A}$ is defined by

$\mathcal{A}u=\mathrm{div}(A(x)\nabla
u)=\sum_{i,j=1}^{\infty}\frac{\partial}{\partial
x_{i}}\Big{(}a_{ij}(x)\frac{\partial u}{\partial x_{j}}\Big{)},$

where $A(x)=(a_{i,j}(x))$ is symmetric and positive definite matrices with
$a_{i,j}(x)\in C^{\infty}(\bar{\Omega})$ and satisfies the uniform ellipticity
conditions

$\sum_{i,j=1}^{n}a_{i,j}(x)\xi_{i}\xi_{j}\geq\omega\sum_{i=1}^{n}\xi_{i}^{2},\quad
x\in\bar{\Omega},~{}\omega>0.$ (1.2)

It is easily verified that the bilinear form
$a(\cdot,\cdot):H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)\to\mathbb{R}$
defined by

$a(u,v)=\sum_{i,j=1}^{n}\int_{\Omega}a_{i,j}(x)\frac{\partial u}{\partial
x_{j}}\frac{\partial u}{\partial x_{i}}dx=\int_{\Omega}A\nabla u\cdot\nabla
vdx$

is symmetric and continuous. Further, it follows from (1.2) that

$a(u,v)\geq\omega\|\nabla u\|_{2}^{2}.$ (1.3)

The problem of asymptotic stability of solutions of dissipative wave systems
equipped with time-dependent nonlinear damping forces has been given a lot of
attention. P. Pucci and J. Serrin [11] investigated the following nonlinear
damped wave system with Dirichlet data

$u_{tt}-\Delta u+Q(t,x,u,u_{t})+f(x,u)=0,$ (1.4)

where the function $Q$ represents a nonlinear damping satisfying
$(Q(t,x,u,v),v)\geq 0$, $f$ is a restoring force(that is $(f(x,u),u)\geq 0$).
Based on the a priori existence of a suitable auxiliary function, they proved
the natural energy $Eu(t)$ associated with solutions of the system satisfies
$\lim_{t\to+\infty}Eu(t)=0$. Problem (1.4) is well-known in a variety of
models in mathematical physics, for instance, elastic vibrations in a
dissipative medium, the telegraphic equation, and the damped Klein-Gordon
equation. P. Pucci and J. Serrin [13] studied again problem (1.4) not only for
potential energies which arise from restoring forces, but also for the effect
of _amplifying forces_. They pointed out that global asymptotic stability can
no longer be expected, and should be replaced by local stability. However,
whether an explicit and precise decay rate estimate of the total energy of the
system can be obtained was unknown. P. Martinez [10] considered the following
time-dependent dissipative systems

$u_{tt}-\Delta u+\rho(t,u_{t})=0$

with Dirichlet boundary, where
$\rho:\mathbb{R}^{+}\times\mathbb{R}\to\mathbb{R}$ is a continuous function
differentiable on $\mathbb{R}^{+}\times(-\infty,0)$. By generalizing the
method introduced to study autonomous wave equation damped with a boundary
nonlinear velocity feedback $\rho(u_{t})$ in [9], P. Martinez obtained that
the total energy of the system decays to zero with an explicit and precise
decay rate estimate under sharp assumptions on the feedback. M. Daoulatli [1]
studied problem (1.1) without $f(u)$, and they illustrated that if given
suitable conditions on the nonlinear terms, and the damping is modeled by a
continuous monotone function without any growth restrictions imposed at the
origin and infinity, the decay rate of the energy functional was obtained by
solving a nonlinear non-autonomous ODE. The Cauchy problem for second order
hyperbolic evolution equations with restoring force in a Hilbert space, under
the effect of nonlinear time-dependent damping has also been studied, the
interested readers can refer to papers [5, 6, 7, 8].

It is certainly beyond the scope of the present paper to give a comprehensive
review for dissipative wave systems equipped with time-dependent nonlinear
damping forces. However, the literature on energy decay estimates is rare for
the weak solution of the variable coefficient wave equations when it is
concerned with the interaction between time-dependent damping and source term.
Inspired by the paper [10], in this paper, the primary goal is to establish
the energy decay rates when the time-dependent damping satisfies different
assumptions. Because of the the interaction between time-dependent damping and
source term, some new difficulties need to be solved. The outline of the paper
is as follows. In Section 2, we shall give some assumptions, main results and
several remarks. In Section 3, we prove that the local (in time) existence of
the weak solution can be extended globally. Sections 4 and 5 are used to prove
the energy decay rates.

##  2 Preliminaries and main results

Throughout the paper, denote by $M$ the optimal embedding constant for the
embedding theorem $H_{0}^{1}(\Omega)\hookrightarrow L^{p+1}(\Omega).$ The
symbol $C$ is a generic positive constant, which may be different in various
positions. $C_{i}$ $(i=0,1,2,\cdots,33)$ represent some positive constants. We
impose the following assumptions:

$\mathbf{(H_{1})}$ $\gamma\in W_{loc}^{1,\infty}(\mathbb{R}^{+})$ is bounded
and nonnegative on $\mathbb{R}^{+}=[0,+\infty).$

$\mathbf{(H_{2})}$ $g$ is continuous and monotone increasing feedback with
$g(0)=0$. In addition, there exist positive constants $b_{1}$ and $b_{2}$ such
that

$b_{1}|s|^{m+1}\leq g(s)s\leq b_{2}|s|^{m+1},\quad\text{ where }m\geq 1\text{
and }|s|>1.$

$\mathbf{(H_{3})}$ $p\frac{m+1}{m}<6.$

$\mathbf{(H_{4})}$ $u_{0}(x)\in H_{0}^{1}(\Omega),~{}u_{1}(x)\in
L^{2}(\Omega).$

######  Definition 2.1 (Weak solution).

A function $u(t):=u(t,x)$ is said to a weak solution of problem (1.1) on
$[0,T]$ if $u\in C([0,T];H_{0}^{1}(\Omega))$ with $u_{t}\in
C([0,T];L^{2}(\Omega))\cap L^{m+1}(\Omega\times(0,T))$. In addition, for all
$t\in[0,T]$,

$\begin{split}&(u_{t}(t),\phi(t))-(u_{t}(0),\phi(0))+\int_{0}^{t}a(u(s),\phi(s))ds-\int_{0}^{t}\int_{\Omega}u_{t}(s)\phi_{t}(s)dxds\\\
&\quad+\int_{0}^{t}\int_{\Omega}\gamma(t)g(u_{t}(s))\phi_{t}(s)dxds=\int_{0}^{t}\int_{\Omega}f(u(s))\phi(s)dxds\end{split}$
(2.1)

for $\phi\in\\{\phi:\phi\in C([0,T];H_{0}^{1}(\Omega))\cap
L^{m+1}(\Omega\times(0,T))\text{ with }\phi_{t}\in C([0,T];L^{2}(\Omega))\\}.$

######  Theorem 2.2 (Local existence).

Let $\mathbf{(H_{1})}-\mathbf{(H_{4})}$ hold, then there exists a local (in
time) weak solutions $u(t)$ to problem (1.1) for $T>0$ depending on the
initial quadratic energy $\mathscr{E}(0)$. Moreover, the following identity
holds

$\mathscr{E}(t)+\int_{0}^{t}\int_{\Omega}\gamma(t)g(u_{t})u_{t}dxds=\mathscr{E}(0)+\int_{0}^{t}\int_{\Omega}f(u(s))u_{t}(s)dxds$
(2.2)

with the quadratic energy is defined by

$\mathscr{E}(t)=\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}a(u(t),u(t)).$
(2.3)

The following result illustrates that when the damping dominates the source,
then the solution is global in time.

######  Theorem 2.3 (Global existence I).

In addition to $\mathbf{(H_{1})}-\mathbf{(H_{4})}$, if $u_{0}(x)\in
L^{p+1}(\Omega)$ and $m\geq p$, then the weak solution of problem (1.1) is
global in time.

Theorem 2.2 can be proved directly by employing the theory of monotone
operators and nonlinear semigroups combined with energy methods. Theorem 2.3
also follows from a standard continuation argument from ODE theory. The
interested readers can follow from line to line as shown in [2](see also a
recent paper [3]) with slight differences to achieve it. The proof of Theorem
2.2 and Theorem 2.3 is not the point in question, so we omit it.

For $u\in L^{p+1}(\Omega)$, define the total energy $E(t)$ by

$E(t):=\mathscr{E}(t)-\frac{1}{p+1}\|u\|^{p+1}=\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}a(u(t),u(t))-\frac{1}{p+1}\|u\|_{p+1}^{p+1}.$
(2.4)

Then (2.2) can be transferred as

$E(t)+\int_{0}^{t}\int_{\Omega}\gamma(t)g(u_{t})u_{t}dxds=E(0).$ (2.5)

Further, we obtain the total energy $E(t)$ is monotone decreasing in time, and

$E^{\prime}(t)=-\int_{\Omega}\gamma(t)g(u_{t})u_{t}dx.$ (2.6)

######  Theorem 2.4 (Global existence II).

Let $1<p\leq 5$ and $\mathbf{(H_{1})}-\mathbf{(H_{4})}$ hold. Assume

$0<E(0)<\Big{(}\frac{1}{2}-\frac{1}{p+1}\Big{)}\Big{(}\frac{\omega^{\frac{p+1}{2}}}{M^{p+1}}\Big{)}^{\frac{2}{p-1}},\quad
a(u_{0},u_{0})<\Big{(}\frac{\omega^{\frac{p+1}{2}}}{M^{p+1}}\Big{)}^{\frac{2}{p-1}},$

then the weak solution of problem (1.1) is global.

We need to impose additional conditions to discuss the energy decay rates.

$\mathbf{(H_{5})}$ $\gamma:\mathbb{R}^{+}\to\mathbb{R}^{+}$ a nonincreasing
function of class $C^{1}$ and $\int_{0}^{\infty}\gamma(t)dt=\infty.$

$\mathbf{(H_{6})}$ There exists a strictly increasing and odd function $h\in
C^{1}[1,1]$ such that

$|h(s)s|\leq g(s)s\leq|h^{-1}(s)s|,\quad\text{ where }|s|\leq 1,$

where $h^{-1}$ is the inverse function of $h$.

######  Theorem 2.5 (Energy decay rates I).

In addition to all conditions of Theorem 2.4, $\mathbf{(H_{5})}$ and
$\mathbf{(H_{6})}$, we assume $1\leq m\leq 5$. Then for all $t\geq 0$

  1. $(1)$

if $h(s)$ is linear,

$E(t)\leq E(0)e^{1-C\int_{0}^{t}\gamma(s)ds}.$ (2.7)

  2. $(2)$

if $h(s)$ has polynomial growth,

$E(t)\leq
CE(0)\left(\frac{1}{1+\int_{0}^{t}\gamma(s)ds}\right)^{\frac{2}{m-1}}\text{with
}m>1.$ (2.8)

When $h(s)$ does not necessarily have polynomial growth, the energy decay
rates still can be obtained if we replace $\mathbf{(H_{5})}$ by the following

$\mathbf{(H_{5}^{\prime})}$ $\gamma(t)\geq\gamma_{0}>0$, where $\gamma_{0}$ is
constant.

######  Theorem 2.6 (Energy decay rates II).

In addition to all conditions of Theorem 2.4, $\mathbf{(H_{5}^{\prime})}$ and
$\mathbf{(H_{6})}$, we assume $1\leq m\leq 5$. Then for all $t\geq 1$

$E(t)\leq CE(0)\left(H^{-1}\Big{(}\frac{1}{t}\Big{)}\right)^{2}.$ (2.9)

Here $H(s):=h(s)s$.

######  Remark 2.7.

In the proof of Theorem 2.5, we only discuss the energy decay rates for $m\leq
5$, that is, the nonlinear damping is _subcritical_ and _critical_. The
restraint results from the embedding theorem $H_{0}^{1}(\Omega)\hookrightarrow
L^{m+1}(\Omega)$. A recent paper [4] investigated the energy decay estimates
for the automous wave equation with _supercritical_ nonlinear damping in the
absence of the driving source. Inspired by the paper [4], we reasonably
conjecture that if there exists $\kappa(t)$ satisfying some suitable
conditions, then

$E(t)\leq
CE(0)\left(\frac{1}{1+\int_{0}^{t}\kappa(t)\gamma(s)ds}\right)^{\frac{2}{m-1}},$

for $m>5$, that is the supercritical nonlinear damping.

######  Remark 2.8.

In this paper, we discuss the variable-coefficient wave equation with
nonlinear time-dependent damping and nonlinear source for standard growth
conditions. However, the energy decay rates for wave systems with nonstandard
growth condition can be of equal importance. By following this paper, it is
possible to discuss the energy decay rates to the initial-boundary value
problem

$\begin{cases}u_{tt}-\Delta
u+\gamma(t)|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u\quad&\text{ in
}\Omega\times(0,T),\\\ u(x,t)=0\quad&\text{ on }\partial\Omega\times(0,T),\\\
u(x,0)=u_{0}(x),~{}u_{t}(x,0)=u_{1}(x)\quad&\text{ in }\Omega.\end{cases}$

######  Remark 2.9.

The rest field $u(t,x)=0$ will be called _asymptotically stable in the mean_ ,
or simply _asymptotically stable_ , if and only if

$\lim_{t\to\infty}E(t)=0\text{ for all solutions }u(t):=u(t,x)\text{ of
problem }\eqref{1.1}.$

This concept was proposed first by P. Pucci and J. Serrin [12]. Obviously,
based on Theorem 2.3 or Theorem 2.4, by Lemma 3.2, we obtain
$\lim_{t\to\infty}E(t)=0.$ Hence, the rest field $u(x,t)=0$ is asymptotically
stable.

##  3 Proof of Theorem 2.4

Let us introduce a function $\mathcal{F}$ as follows

$\mathcal{F}(s)=\frac{1}{2}s-\frac{M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}s^{\frac{p+1}{2}}.$
(3.1)

By a direct computation, the function $\mathcal{F}$ satisfies that

  1. 1.

$\mathcal{F}(0)=0$;

  2. 2.

$\lim_{s\to+\infty}\mathcal{F}(s)=-\infty$;

  3. 3.

$\mathcal{F}$ is strictly increasing in $(0,s_{1})$, and is strictly
decreasing in $(s_{1},+\infty)$;

  4. 4.

$\mathcal{F}$ has a maximum at $s_{1}$ with the maximum value
$\mathcal{F}_{1}$. Here

$s_{1}=\Big{(}\frac{\omega^{\frac{p+1}{2}}}{M^{p+1}}\Big{)}^{\frac{2}{p-1}},\quad\mathcal{F}_{1}=\Big{(}\frac{1}{2}-\frac{1}{p+1}\Big{)}s_{1}.$

######  Lemma 3.1.

If $u(t)$ is a solution for problem $(\ref{1.1})$ and $E(0)<\mathcal{F}_{1},$
$a(u_{0},u_{0})<s_{1},$ then there exists a positive constant $s_{2}$
satisfying $0<s_{2}<s_{1}$ such that

$a(u(t),u(t))\leq s_{2}\quad\text{ for all }t\geq 0.$ (3.2)

###### Proof.

Using (1.3) and (2.1), the embedding theorem $H_{0}^{1}(\Omega)\hookrightarrow
L^{p+1}(\Omega)$, we obtain

$\begin{split}E(t)&\geq\frac{1}{2}a(u(t),u(t))-\frac{1}{p+1}\|u\|_{p+1}^{p+1}\geq\frac{1}{2}a(u(t),u(t))-\frac{M^{p+1}}{p+1}\|\nabla
u\|_{2}^{p+1}\\\
&\geq\frac{1}{2}a(u(t),u(t))-\frac{M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}[a(u(t),u(t))]^{\frac{p+1}{2}}:=\mathcal{F}(a(u(t),u(t))).\end{split}$
(3.3)

Since $E(0)<\mathcal{F}_{1}$, there exists a $s_{2}<s_{1}$ such that
$\mathcal{F}(s_{2})=E(0)$. It follows from (3.3) that
$\mathcal{F}(a(u_{0},u_{0}))\leq E(0)=\mathcal{F}(s_{2})$, which implies
$a(u_{0},u_{0})\leq s_{2}$ due to the given condition $a(u_{0},u_{0})<s_{1}$.
To complete the proof of (3.2), we suppose by contradiction that for some
$t^{0}>0$, $a(u(t^{0}),u(t^{0}))>s_{2}.$ The continuity of $a(u(t),u(t))$
illustrates that we may choose $t^{1}$ such that
$s_{1}>a(u(t^{1}),u(t^{1}))>s_{2}$, then we have
$E(0)=\mathcal{F}(s_{2})<\mathcal{F}(a(u(t^{1}),u(t^{1})))\leq E(t^{1}).$ This
is a contradiction for $E(t)$ is nonincreasing. ∎

######  Lemma 3.2.

Under all the conditions of Lemma $\ref{lem1},$ for all $t\geq 0,$ the
following holds

$0\leq\mathscr{E}(t)\leq C_{0}E(t)\leq C_{0}E(0).$ (3.4)

###### Proof.

Similar to (3.3), and then using (3.2) and $\eqref{2.4}$, then

$\begin{split}\frac{1}{p+1}\|u\|_{p+1}^{p+1}&\leq\frac{M^{p+1}}{p+1}\|\nabla
u\|_{2}^{p+1}\leq\frac{M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}[a(u(t),u(t))]^{\frac{p+1}{2}}\\\
&=\frac{M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}[a(u(t),u(t))]^{\frac{p-1}{2}}a(u(t),u(t))\\\
&\leq\frac{M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}s_{2}^{\frac{p-1}{2}}\Big{(}2E(t)+\frac{2}{p+1}\|u\|_{p+1}^{p+1}\Big{)},\end{split}$

which indicates

$\|u\|_{p+1}^{p+1}\leq\mathcal{M}E(t)\leq\mathcal{M}E(0),$ (3.5)

by recalling $E(t)$ is monotone decreasing. Here

$\mathcal{M}:=\frac{\frac{2M^{p+1}}{\omega^{\frac{p+1}{2}}}s_{2}^{\frac{p-1}{2}}}{1-\frac{2M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}s_{2}^{\frac{p-1}{2}}}<\frac{\frac{2M^{p+1}}{\omega^{\frac{p+1}{2}}}s_{1}^{\frac{p-1}{2}}}{1-\frac{2M^{p+1}}{(p+1)\omega^{\frac{p+1}{2}}}s_{1}^{\frac{p-1}{2}}}=\frac{2(p+1)}{p-1}.$

Combining (3.5) with (2.4), we directly obtain

$\mathscr{E}(t)=E(t)+\frac{1}{p+1}\|u\|^{p+1}\leq C_{0}E(t)\leq C_{0}E(0),$

which yields (3.4). ∎

It follows from Lemma 3.2 that the weak solution $u(t)$ of problem (1.1)
exists globally, that is $T=\infty.$

######  Remark 3.3.

By the well-known potential well theory, we can also prove the global
existence. In fact, $\mathcal{F}_{1}$ is equal to the potential well depth(the
mountain pass level) $d$ defined by

$d:=\inf_{u\in H_{0}^{1}(\Omega)\backslash\\{0\\}}\sup_{\lambda\geq
0}J(\lambda u),\text{ where
}J(u):=\frac{1}{2}a(u,u)-\frac{1}{p+1}\|u\|_{p+1}^{p+1}.$

##  4 Proof of Theorem 2.5

The proof of Theorem 2.5 relies on the following crucial lemma. We refer to
[9] for the detailed proof.

######  Lemma 4.1 ([9]).

Let $E:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a non-increasing function and
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a strictly increasing function of
class $C^{1}$ such that

$\psi(0)=0\text{ and }\psi(t)\to+\infty\text{ as }t\to+\infty.$

Assume that there exist $\sigma\geq 0$, $\sigma^{\prime}\geq 0$, $c\geq 0$ and
$\omega>0$ such that$:$

$\int_{t}^{+\infty}E^{1+\sigma}(s)\psi^{\prime}(s)ds\leq\frac{1}{\omega}E^{\sigma}(0)E(t)+\frac{c}{(1+\psi(s))^{\sigma^{\prime}}}E^{q}(0)E(s)$

then $E$ has the following decay property$:$

  1. $(1)$

if $\sigma=c=0,$ then $E(t)\leq E(0)e^{1-\omega\psi(t)}$ for all $t\geq 0;$

  2. $(2)$

if $\sigma>0,$ then there exists $C>0$ such that $E(t)\leq
CE(0)(1+\psi(t))^{-\frac{1+\sigma^{\prime}}{\sigma}}$ for all $t\geq 0$.

In what follows, let us prove Theorem 2.5. It dose make sense to multiply
(1.1) by $E^{\beta}(t)\phi^{\prime}(t)u(t)$ and to integrate over
$\Omega\times[S,T]$. Here $\phi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a concave
nondecreasing function of class $C^{2}$, such that $\phi^{\prime}$ is bounded,
and $\beta\geq 0$ is constant. Then we obtain

$\begin{split}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)[u_{tt}(t)-\mathcal{A}u(t)+\gamma(t)g(u_{t}(t))]dxdt=\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u(t)\|_{p+1}^{p+1}dt.\end{split}$
(4.1)

Integrating by parts yields

$\begin{split}&\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)u_{tt}(t)dxdt=E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)u_{t}(t)dx\Big{|}_{S}^{T}-\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u_{t}(t)\|_{2}^{2}\\\
&\quad-\int_{S}^{T}[\beta
E^{\beta-1}(t)E^{\prime}(t)\phi^{\prime}(t)+E^{\beta}(t)\phi^{\prime\prime}(t)]\int_{\Omega}u(t)u_{t}(t)dxdt.\end{split}$
(4.2)

Substituting (4.2) into (4.1), one obtains

$\begin{split}&E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)u_{t}(t)dx\Big{|}_{S}^{T}-\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\quad-\int_{S}^{T}[\beta
E^{\beta-1}(t)E^{\prime}(t)\phi^{\prime}(t)+E^{\beta}(t)\phi^{\prime\prime}(t)]\int_{\Omega}u(t)u_{t}(t)dxdt\\\
&\quad+\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)a(u(t),u(t))dt\\\
&\quad+\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\gamma(t)\int_{\Omega}u(t)g(u_{t}(t))dxdt=\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u(t)\|_{p+1}^{p+1}dt.\end{split}$
(4.3)

It follows from (2.4) that (4.3) can be written as

$\begin{split}&2\int_{S}^{T}E^{\beta+1}(t)\phi^{\prime}(t)dt=-E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)u_{t}(t)dx\Big{|}_{S}^{T}+2\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\quad+\int_{S}^{T}[\beta
E^{\beta-1}(t)E^{\prime}(t)\phi^{\prime}(t)+E^{\beta}(t)\phi^{\prime\prime}(t)]\int_{\Omega}u(t)u_{t}(t)dxdt\\\
&\quad+\frac{p-1}{p+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u(t)\|_{p+1}^{p+1}dt-\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\gamma(t)\int_{\Omega}u(t)g(u_{t}(t))dxdt\\\
&=\mathcal{J}_{1}+\mathcal{J}_{2}+\mathcal{J}_{3}+\mathcal{J}_{4}+\mathcal{J}_{5}.\end{split}$
(4.4)

In what follows, let us estimate every term on the right hand side above.

Using the embedding theorem $H_{0}^{1}(\Omega)\hookrightarrow L^{2}(\Omega)$,
(1.3) and (3.4) yields

$\begin{split}&\|u(t)\|_{2}\leq M\|\nabla u(t)\|_{2}\leq
M\Big{[}\frac{a(u(t),u(t))}{\omega}\Big{]}^{\frac{1}{2}}\leq
C_{1}E^{\frac{1}{2}}(t).\end{split}$ (4.5)

Applying Cauchy’s inequality, the boundedness of $\phi^{\prime}(t)$(we can
denote by $\theta$), (4.5) and (3.4), we arrive at

$\begin{split}|\mathcal{J}_{1}|&=\Big{|}-E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega}u(t)u_{t}(t)dx\Big{|}_{S}^{T}\Big{|}\\\
&\leq\theta
E^{\beta}(S)\big{[}\|u_{t}(T)\|_{2}\|u(T)\|_{2}+\|u_{t}(S)\|_{2}\|u(S)\|_{2}\big{]}\leq
C_{2}E^{\beta+1}(S),\end{split}$ (4.6)

moreover,

$\begin{split}|\mathcal{J}_{3}|&=\Big{|}\int_{S}^{T}[\beta
E^{\beta-1}(t)E^{\prime}(t)\phi^{\prime}(t)+E^{\beta}(t)\phi^{\prime\prime}(t)]\int_{\Omega}u(t)u_{t}(t)dxdt\Big{|}\\\
&\leq-\theta\beta\int_{S}^{T}E^{\beta}(t)E^{\prime}(t)dt+C_{3}\int_{S}^{T}E^{\beta+1}(t)(-\phi^{\prime\prime}(t))dt\\\
&\leq\frac{\theta\beta}{\beta+1}E^{\beta+1}(S)+C_{4}E^{\beta+1}(S)=C_{5}E^{\beta+1}(S).\end{split}$
(4.7)

It follows from (3.5) that

$\begin{split}|\mathcal{J}_{4}|=\frac{p-1}{p+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u(t)\|_{p+1}^{p+1}dt\leq\frac{p-1}{p+1}\mathcal{M}\int_{S}^{T}E^{\beta+1}(t)\phi^{\prime}(t)dt.\end{split}$
(4.8)

Using Young’s inequality with $\varepsilon_{1}>0$, (4.5), one has

$\begin{split}\int_{|u_{t}(t)|\leq
1}u(t)g(u_{t}(t))dx&\leq\frac{\varepsilon_{1}}{2}\|u(t)\|_{2}^{2}+\frac{1}{2\varepsilon_{1}}\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx\\\ &\leq
C_{6}\varepsilon_{1}E(t)+\frac{1}{2\varepsilon_{1}}\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx.\end{split}$

It follows from Young’s inequality with $\varepsilon_{1}>0$, the embedding
theorem $H_{0}^{1}(\Omega)\hookrightarrow L^{m+1}(\Omega)$, (1.3) and (3.4)
that

$\begin{split}&\int_{|u_{t}(t)|>1}u(t)g(u_{t}(t))dx\leq\frac{\varepsilon_{2}}{m+1}\|u(t)\|_{m+1}^{m+1}+\frac{m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dx\\\
&\leq\frac{M^{m+1}\varepsilon_{2}}{m+1}\|\nabla
u(t)\|_{2}^{m+1}+\frac{m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dx\\\
&\leq\frac{M^{m+1}\varepsilon_{2}}{m+1}\Big{[}\frac{a(u(t),u(t))}{\omega}\Big{]}^{\frac{m+1}{2}}+\frac{m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dx\\\
&\leq
C_{7}\varepsilon_{2}E(t)+\frac{m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dx.\end{split}$

Therefore, by recalling the boundedness of $\gamma(t)$(we can denote by
$c_{1}$), then

$\begin{split}|\mathcal{J}_{5}|=&\Big{|}-\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\gamma(t)\int_{\Omega}u(t)g(u_{t}(t))dxdt\Big{|}\\\
&\leq c_{1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\Big{|}\int_{|u_{t}(t)|\leq
1}u(t)g(u_{t}(t))dx+\int_{|u_{t}(t)|>1}u(t)g(u_{t}(t))dx\Big{|}dt\\\ &\leq
c_{1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\Big{[}C_{6}\varepsilon_{1}E(t)+\frac{1}{2\varepsilon_{1}}\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx\\\
&\quad+C_{7}\varepsilon_{2}E(t)+\frac{m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dx\Big{]}dt\\\
&\leq
C_{8}(\varepsilon_{1}+\varepsilon_{2})\int_{S}^{T}E^{\beta+1}\phi^{\prime}(t)dt+\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx\\\
&\quad+\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dxdt.\end{split}$
(4.9)

Inserting (4.6)-(4.9) into (4.4) indicates

$\begin{split}&\Big{[}2-\frac{p-1}{p+1}\mathcal{M}-C_{8}(\varepsilon_{1}+\varepsilon_{2})\Big{]}\int_{S}^{T}E^{\beta+1}(t)\phi^{\prime}(t)dt\\\
&\leq
C_{9}E^{\beta+1}(S)+2\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\quad+\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dxdt\\\
&\quad+\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dxdt.\end{split}$
(4.10)

Note that $2-\frac{p-1}{p+1}\mathcal{M}>0$, one gets

$2-\frac{p-1}{p+1}\mathcal{M}-C_{8}(\varepsilon_{1}+\varepsilon_{2})>0$

for sufficiently small positive constants $\varepsilon_{1}$ and
$\varepsilon_{2}$.

###  4.1 Case I: $h(s)$ is linear

Let us $\phi(t)=\int_{0}^{t}\gamma(s)ds$ in this section. From
$\mathbf{(H_{6})}$, there exists two positive constant $b_{3},~{}b_{4}$ such
that

$b_{3}s^{2}\leq g(s)s\leq b_{4}s^{2},\quad\text{ where }|s|\leq 1.$ (4.11)

Using (4.11), $\mathbf{(H_{2})}$ and (2.6) yields

$\begin{split}&2\int_{S}^{T}E^{\beta}(t)\gamma(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\leq 2\int_{S}^{T}E^{\beta}(t)\gamma(t)\Big{[}\int_{|u_{t}(t)|\leq
1}|u_{t}(t)|^{2}dx+\int_{|u_{t}(t)|>1}|u_{t}(t)|^{2}dx\Big{]}dt\\\
&\leq\frac{2}{b_{3}}\int_{S}^{T}E^{\beta}(t)\int_{\Omega}\gamma(t)g(u_{t}(t))u_{t}(t)dxdt+2\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|>1}|u_{t}(t)|^{m+1}dxdt\\\
&\leq\frac{2}{b_{3}}\int_{S}^{T}E^{\beta}(t)\int_{\Omega}\gamma(t)g(u_{t}(t))u_{t}(t)dxdt+\frac{2}{b_{1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{\Omega}g(u_{t}(t))u_{t}(t)dxdt\\\
&\leq-\Big{[}\frac{2}{b_{3}}+\frac{2}{b_{1}}\Big{]}\int_{S}^{T}E^{\beta}(t)E^{\prime}(t)dt\leq
C_{10}E^{\beta+1}(S).\end{split}$ (4.12)

By using (4.11) and (2.6), we gets

$\begin{split}&\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx\\\
&\leq\frac{c_{1}b_{4}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|\leq
1}g(u_{t}(t))u_{t}(t)dx\\\
&\leq-\frac{c_{1}b_{4}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)E^{\prime}(t)dt\leq
C_{11}E^{\beta+1}(S).\end{split}$ (4.13)

It follows from $\mathbf{(H_{2})}$ and (2.6) that

$\begin{split}&\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dxdt\\\
&=\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{1}{m}}|g(u_{t}(t))|dxdt\\\
&\leq\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}b_{2}^{\frac{1}{m}}}{(m+1)}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|>1}g(u_{t}(t))u_{t}(t)dxdt\leq
C_{12}E^{\beta+1}(S).\end{split}$ (4.14)

Inserting (4.12)-(4.14) into (4.10), we easily deduce

$\int_{S}^{T}E^{\beta+1}(t)\gamma(t)dt\leq\frac{1}{C_{13}}E^{\beta}(0)E(S).$
(4.15)

When $T$ goes to $+\infty$, choose $\beta=0$,
$\psi(t)=\int_{0}^{t}\gamma(s)ds$ in Lemma 4.1, then we derive (2.7).

###  4.2 Case II: $h(s)$ has polynomial growth

In this subsection, we still choose $\phi(t)=\int_{0}^{t}\gamma(s)ds$. To
simplicity, denote $h(s)=b_{5}|s|^{m-1}s$ with $m>1,~{}|s|\leq 1,~{}b_{5}>0$.
From $\mathbf{(H_{6})}$, there exists two positive constant $b_{6},~{}b_{7}$
such that

$b_{6}s^{m+1}\leq g(s)s\leq b_{7}|s|^{\frac{m+1}{m}},\quad\text{ where
}|s|\leq 1.$ (4.16)

It follows from Hölder’s inequality, (4.16), Young’s inequality with
$\varepsilon_{3}>0$, $\mathbf{(H_{2})}$ and (2.6) yields

$\begin{split}&2\int_{S}^{T}E^{\beta}(t)\gamma(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\leq 2\int_{S}^{T}E^{\beta}(t)\gamma(t)\Big{[}\int_{|u_{t}(t)|\leq
1}|u_{t}(t)|^{2}dx+\int_{|u_{t}(t)|>1}|u_{t}(t)|^{2}dx\Big{]}dt\\\ &\leq
2|\Omega|^{\frac{m-1}{m+1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\Big{(}\int_{|u_{t}(t)|\leq
1}|u_{t}(t)|^{m+1}dx\Big{)}^{\frac{2}{m+1}}dt+\frac{2}{b_{1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{\Omega}g(u_{t}(t))u_{t}(t)dxdt\\\
&\leq
2|\Omega|^{\frac{m-1}{m+1}}\frac{1}{b_{6}^{\frac{2}{m+1}}}\int_{S}^{T}E^{\beta}(t)\gamma^{\frac{m-1}{m+1}}(t)\Big{(}\int_{\Omega}\gamma(t)u_{t}(t)g(u_{t}(t))dx\Big{)}^{\frac{2}{m+1}}dt-\frac{2}{b_{1}}\int_{S}^{T}E^{\beta}(t)E^{\prime}(t)dxdt\\\
&\leq
C_{14}\varepsilon_{3}\int_{S}^{T}E^{\frac{\beta(m+1)}{m-1}}(t)\gamma(t)dt+C_{15}\varepsilon_{3}^{-\frac{m-1}{2}}\int_{S}^{T}(-E^{\prime}(t))dt+C_{16}E^{\beta+1}(S)\\\
&\leq
C_{14}\varepsilon_{3}\int_{S}^{T}E^{\frac{\beta(m+1)}{m-1}}(t)\gamma(t)dt+C_{17}E(S)+C_{16}E^{\beta+1}(S).\end{split}$
(4.17)

Similar to (4.17), using the second inequality in (4.16), we obtain

$\begin{split}&\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{2}dx\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}|\Omega|^{\frac{m-1}{m+1}}\int_{S}^{T}E^{\beta}(t)\gamma(t)\Big{(}\int_{|u_{t}(t)|\leq
1}|g(u_{t}(t))|^{m+1}dx\Big{)}^{\frac{2}{m+1}}dt\\\
&\leq\int_{S}^{T}E^{\beta}(t)\gamma^{\frac{m-1}{m+1}}(t)\cdot\frac{c_{1}}{2\varepsilon_{1}}|\Omega|^{\frac{m-1}{m+1}}b_{7}^{\frac{2m}{m+1}}\Big{(}\int_{\Omega}\gamma(t)u_{t}(t)g(u_{t}(t))dx\Big{)}^{\frac{2}{m+1}}dt\\\
&\leq
C_{18}\varepsilon_{4}\int_{S}^{T}E^{\frac{\beta(m+1)}{m-1}}(t)\gamma(t)dt+C_{19}\varepsilon_{4}^{-\frac{m-1}{2}}\int_{S}^{T}(-E^{\prime}(t))dt\\\
&\leq
C_{18}\varepsilon_{4}\int_{S}^{T}E^{\frac{\beta(m+1)}{m-1}}(t)\gamma(t)dt+C_{20}E(S).\end{split}$
(4.18)

Moreover, (4.14) still holds in this case.

Inserting (4.14), (4.17) and (4.18) into (4.10), and choosing
$\beta=\frac{m-1}{2}$, we easily deduce

$\begin{split}\Big{[}2-\frac{p-1}{p+1}\mathcal{M}-C_{8}(\varepsilon_{1}+\varepsilon_{2})-C_{21}(\varepsilon_{3}+\varepsilon_{4})\Big{]}\int_{S}^{T}E^{\beta+1}(t)\gamma(t)dt\leq
C_{22}E^{\beta+1}(0)E(S).\end{split}$ (4.19)

We also have

$2-\frac{p-1}{p+1}\mathcal{M}-C_{8}(\varepsilon_{1}+\varepsilon_{2})-C_{21}(\varepsilon_{3}+\varepsilon_{4})>0$

for sufficiently small positive constants $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$ and $\varepsilon_{4}$.

Therefore, we get

$\int_{S}^{T}E^{\beta+1}(t)\gamma(t)dt\leq\frac{1}{C_{23}}E^{\beta}(0)E(S).$
(4.20)

When $T$ goes to $+\infty$, note that $\beta=\frac{m-1}{2}$,
$\psi(t)=\int_{0}^{t}\gamma(s)ds$ in Lemma 4.1 then we derive (2.8).

##  5 Proof of Theorem 2.6

To prove Theorem 2.6, we need to the following lemma included in [9].

######  Lemma 5.1.

The function $\phi:[1,+\infty)\to[1,\infty)$ defined by

$\phi(t):=\tilde{\psi}^{-1}(t)$ (5.1)

is a strictly increasing concave function of class $C^{2}$ and satisfies

$\lim_{t\to+\infty}\phi(t)=+\infty,~{}\lim_{t\to+\infty}\phi^{\prime}(t)=0,$

and $\int_{1}^{\infty}\phi^{\prime}(t)|h^{-1}(\phi^{\prime}(t))|^{2}dt$
converges. Here

$\tilde{\psi}(t):=1+\int_{1}^{t}\frac{1}{h(1/s)}ds,$ (5.2)

and the function $h$ satisfies $\mathbf{(H_{6})}$.

In this section, define

$\chi(t):=\frac{1}{\phi(t)}=\frac{1}{\tilde{\psi}^{-1}(t)}.$ (5.3)

Now we are in a position to estimate (4.10).

For $t\geq 1$, set

$\begin{split}\Omega^{1}&=\\{x\in\Omega:~{}|u_{t}(t)|\leq\chi(t)\\};\\\
\Omega^{2}&=\\{x\in\Omega:~{}\chi(t)<|u_{t}(t)|\leq 1\\};\\\
\Omega^{3}&=\\{x\in\Omega:~{}|u_{t}(t)|>1\\}.\end{split}$

Using Lemma 5.1, $\mathbf{(H_{6})}$, $\mathbf{(H_{5}^{\prime})}$ and (2.6), we
gets

$\begin{split}&\int_{S}^{T}E^{\beta}(t)\int_{\Omega^{2}}\phi^{\prime}(t)|u_{t}(t)|^{2}dxdt\\\
&=\int_{S}^{T}E^{\beta}(t)\int_{\Omega^{2}}h(\chi(t))|u_{t}(t)|^{2}dxdt\leq\int_{S}^{T}E^{\beta}(t)\int_{\Omega^{2}}h(u_{t}(t))|u_{t}(t)|^{2}dxdt\\\
&\leq\int_{S}^{T}E^{\beta}(t)\frac{1}{\gamma(t)}\int_{\Omega^{2}}\gamma(t)g(u_{t}(t))u_{t}(t)dxdt\leq\frac{1}{\gamma_{0}}E^{\beta+1}(S).\end{split}$
(5.4)

It follows from (2.6), (5.3) and Lemma 5.1 that

$\begin{split}&\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega^{1}}|u_{t}(t)|^{2}dxdt\leq|\Omega|\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\chi^{2}(t)dt\\\
&\leq|\Omega|E^{\beta}(S)\int_{S}^{T}\phi^{\prime}(t)\chi^{2}(t)dt\leq|\Omega|\frac{E^{\beta}(S)}{\phi(S)}.\end{split}$
(5.5)

Recalling (4.17), and using (5.4) and (5.5), one gets

$\begin{split}&2\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\|u_{t}(t)\|_{2}^{2}dt\\\
&\leq 2\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|\leq
1}|u_{t}(t)|^{2}dxdt+\frac{2}{b_{1}}\int_{S}^{T}E^{\beta}(t)\frac{\phi^{\prime}(t)}{\gamma(t)}\int_{\Omega^{3}}\gamma(t)g(u_{t}(t))u_{t}(t)dxdt\\\
&\leq
2\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega^{1}\cup\Omega^{2}}|u_{t}(t)|^{2}dxdt+C_{24}E^{\beta+1}(S)\\\
&\leq C_{25}E^{\beta+1}(S)+|\Omega|\frac{E^{\beta}(S)}{\phi(S)}.\end{split}$
(5.6)

Next we continue to estimate the remaining terms in (4.10).
For $t\geq 1$ and $\phi^{\prime}(t)\leq 1$, set

$\begin{split}\Omega_{1}&=\\{x\in\Omega:~{}|u_{t}(t)|\leq\phi^{\prime}(t)\\};\\\
\Omega_{2}&=\\{x\in\Omega:~{}\phi^{\prime}(t)<|u_{t}(t)|\leq 1\\};\\\
\Omega_{3}&=\\{x\in\Omega:~{}|u_{t}(t)|>1\\}.\end{split}$

For $t\geq 1$ and $\phi^{\prime}(t)>1$, set

$\begin{split}\Omega_{4}&=\\{x\in\Omega:~{}|u_{t}(t)|\leq
1<\phi^{\prime}(t)\\};\\\
\Omega_{5}&=\\{x\in\Omega:~{}1<|u_{t}(t)|\leq\phi^{\prime}(t)\\};\\\
\Omega_{6}&=\\{x\in\Omega:~{}|u_{t}(t)|>\phi^{\prime}(t)>1\\}.\end{split}$

Hence

$\\{x\in\Omega:~{}|u_{t}(t)|\leq 1\\}=\Omega_{1}\cup\Omega_{2}(\text{or
}\Omega_{4}),~{}\\{x\in\Omega:~{}|u_{t}(t)|>1\\}=\Omega_{3}(\text{or
}\Omega_{5}\cup\Omega_{6}).$

Similar to (4.14),

$\begin{split}\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega^{i}}|g(u_{t}(t))|^{\frac{m+1}{m}}dxdt\leq
C_{26}E^{\beta+1}(S),\quad\text{i=3,5,6}.\end{split}$

Thus

$\begin{split}\frac{c_{1}m\varepsilon_{2}^{-\frac{1}{m}}}{m+1}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{|u_{t}(t)|>1}|g(u_{t}(t))|^{\frac{m+1}{m}}dxdt\leq
C_{27}E^{\beta+1}(S).\end{split}$ (5.7)

Using $\mathbf{(H_{6})}$ and $\mathbf{(H_{5}^{\prime})}$, then

$\begin{split}&\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega_{2}}|g(u_{t}(t))|^{2}dxdt\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\int_{\Omega_{2}}|u_{t}(t)||g(u_{t}(t))|^{2}dxdt\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\int_{\Omega_{2}}u_{t}(t)g(u_{t}(t))|h^{-1}(u_{t}(t))|dxdt\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}h^{-1}(1)\int_{S}^{T}E^{\beta}(t)\frac{1}{\gamma(t)}\int_{\Omega_{2}}\gamma(t)u_{t}(t)g(u_{t}(t))dxdt\leq
C_{28}E^{\beta+1}(S),\end{split}$ (5.8)

and

$\begin{split}&\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega_{i}}|g(u_{t}(t))|^{2}dxdt\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega_{i}}|h^{-1}(u_{t}(t))|^{2}dxdt\\\
&\leq\frac{c_{1}}{2\varepsilon_{1}}\int_{S}^{T}E^{\beta}(t)\phi^{\prime}(t)\int_{\Omega_{i}}|h^{-1}(\phi^{\prime}(t))|^{2}dxdt\\\
&\leq
C_{29}E^{\beta}(S)\int_{S}^{T}\phi^{\prime}(t)|h^{-1}(\phi^{\prime}(t))|^{2}dt,\quad\text{i=1,4}.\end{split}$
(5.9)

Substubiting (5.6)-(5.9) into (4.10), we get

$\begin{split}&\Big{[}2-\frac{p-1}{p+1}\mathcal{M}-C_{8}(\varepsilon_{1}+\varepsilon_{2})\Big{]}\int_{S}^{T}E^{\beta+1}(t)\phi^{\prime}(t)dt\\\
&\leq
C_{30}E^{\beta+1}(S)+|\Omega|\frac{E^{\beta}(S)}{\phi(S)}+C_{31}E^{\beta}(S)\int_{S}^{T}\phi^{\prime}(t)|h^{-1}(\phi^{\prime}(t))|^{2}dt.\end{split}$
(5.10)

Since $\int_{1}^{\infty}\phi^{\prime}(t)|h^{-1}(\phi^{\prime}(t))|^{2}dt$
converges in Lemma 5.1, then

$\int_{S}^{T}E^{\beta+1}(t)\phi^{\prime}(t)dt\leq\frac{1}{C_{32}}E^{\beta}(0)E(S)+\frac{C_{33}}{\phi(S)}E^{\beta}(0)E(S).$
(5.11)

When $T$ goes to $+\infty$, choose $\beta=1$, and choose $\psi(t)=\phi(t)-1$
in Lemma 4.1, then we derive

$E(t)\leq\frac{CE(0)}{\phi^{2}(t)}.$ (5.12)

Let us choose $s_{0}$ such that $h(\frac{1}{s_{0}})\leq 1$, then by (5.2) and
$H(s):=h(s)s$, for $s\geq s_{0}$

$\tilde{\psi}(s)\leq 1+(s-1)\frac{1}{h\Big{(}\frac{1}{s}\Big{)}}\leq
s\frac{1}{h\Big{(}\frac{1}{s}\Big{)}}=\frac{1}{H\Big{(}\frac{1}{s}\Big{)}}.$

Hence, by (5.1), for $s\geq s_{0}$

$s\leq\phi\left(\frac{1}{H\Big{(}\frac{1}{s}\Big{)}}\right)=\phi(t)\quad\text{
with }t=\frac{1}{H\Big{(}\frac{1}{s}\Big{)}}.$

Further,

$\frac{1}{\phi(t)}\leq\frac{1}{s}=H^{-1}\Big{(}\frac{1}{t}\Big{)}.$

which together with (5.12) implies (2.9).

## Acknowledgements

This work is supported by the Fundamental Research Funds for Central
Universities(B230201033).

## Competing Interests

The authors declare that they have no competing interests.

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## References

  * [1] M. Daoulatli, _Rates of decay for the wave systems with time-dependent damping_ , Discrete Contin. Dyn. Syst., 31 (2011) 407–443.
  * [2] Y.Q. Guo, M.A. Rammaha, et al., _Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping_ , J. Differential Equations, 257 (2014), 3778–3812.
  * [3] T.G. Ha, _Global solutions and blow-up for the wave equation with variable coefficients: I. Interior supercritical source._ Appl. Math. Optim., 84 (2021), 767–803.
  * [4] A. Haraux, L. Tebou, Energy decay estimates for the wave equation with supercritical nonlinear damping, arXiv preprint arXiv:2204.11494.
  * [5] J.R. Luo, T.J. Xiao, _Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping_ , Evol. Equ. Control Theory, 9 (2020), 359–373.
  * [6] J.R. Luo, T.J. Xiao, _Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions_ , Math. Methods Appl. Sci., 44 (2021), 303–314.
  * [7] J.R. Luo, T.J. Xiao, _Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping_ , ESAIM: COCV, 27 (2021) 59.
  * [8] J.R. Luo, T.J. Xiao, _Optimal decay rates for semi-linear non-autonomous evolution equations with vanishing damping_ , Nonlinear Anal., 230 (2023) 113247.
  * [9] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control Optim. Calc. Var., 4 (1999), 419–444.
  * [10] P. Martinez, _Precise decay rate estimates for time-dependent dissipative systems_ , Israel J. Math., 119 (2000), 291–324.
  * [11] P. Pucci and J. Serrin, _Asymptotic stability for non–autonomous dissipative wave systems_ , Comm. Pure Appl. Math., XLIX (1996), 177–216.
  * [12] P. Pucci and J. Serrin, _Asympptotic stablility for nonlinear parabolic systems_ //S.N. Antontsev, J.I. Diaz, S.I. Shmarev. Energy Methods in Continuum Mechanics. Dordrecht: Kluwer Acad Publ, 1996: 66–74.
  * [13] P. Pucci and J. Serrin, _Local asymptotic stability for dissipative wave systems_ , Israel J. Math., 104 (1998), 29–50.