METHOD 1:
The easiest way to install a newer kernel in Debian, is to install it from the backports. In order to install a kernel from the backports, we need to add the backports-repository for your Debian version to the apt-sources and update the list of available packages:
Add following line to /etc/apt/sources.list:
deb http://deb.debian.org/debian bullseye-backports main
sudo apt-get update
Now you can search for kernels:
sudo apt search linux-image
At this point, you can either install a specific version manually or choose to go for the latest release. To install the latest release, including necessary dependencies:
sudo apt -t bullseye-backports upgrade
After the upgrade, you can simply perform a reboot and the new kernel should be activated as the new default.
uname -r
METHOD 2:
sudo apt update && sudo apt upgrade -y
Add following to sources.list:
echo "deb http://deb.debian.org/debian unstable main contrib non-free" | sudo tee -a /etc/apt/sources.list
echo "deb-src http://deb.debian.org/debian unstable main contrib non-free" | sudo tee -a /etc/apt/sources.list
Next, an easy solution is to use apt-pinning to avoid having different version branches causing your system prompting for updates from the experimental repository. Open the following file using a text editor.
sudo nano /etc/apt/preferences
Next, add the following.
Package: * Pin: release a=bullseye Pin-Priority: 500 Package: linux-image-amd64 Pin: release a=unstable Pin-Priority: 1000 Package: * Pin: release a=unstable Pin-Priority: 100
Save it and get out (Ctrl-O and CTRL-X).
The order goes all updates are preferenced to Bullseye with a higher score (500) than unstable (100), so you are not prompted on various packages to be updated from the unstable repository.
However, to make it easy to keep the kernel up to date when you run the apt update command for your standard Bullseye packages, the example above has set linux-image-amd64 as a high priority (1000) using the unstable repository above any other source for that package only.
Next:
sudo apt update
sudo apt upgrade