When dealing with multi-line data, there are two basic possibilities: Either (1) read the entire file into memory (5000 lines is not that big, especially if the lines are only as short as you've shown) and apply a regex over the whole thing, or (2) read the file line-by-line and keep some kind of state, keeping in mind that you're only looking at one line at a time (Update: or, if the state you are keeping is a buffer, like a sliding window, a few lines). This second point is why your code isn't working: You're expecting your regex $beg to match two lines since you've included a \n in your regex, but if you're reading the file line-by-line, you'll only ever have one line in $_ at a time (or $line, that's unclear from your code).

The downside of the first approach is that it uses more RAM and will start being problematic if your files grow. Here's an example:

use warnings;
use strict;

my $data = do { open my $fh, '<', 'sample.txt' or die $!;
    local $/; <$fh> };  # slurp whole file into memory

my $regex = qr{ 
    ^ \s*APP>\sSTART_PARTU_8X8 \n
    ^ \s*APP>\sPARTITION_SPLIT \n
    (.*?)
    ^ \s*APP>\sEND_PARTU_8X8 $
    }msx;

if ($data=~$regex) {
    print $1;
}
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The downside of the second approach is that it makes the logic a bit more complex, but I tend to prefer it since it allows for any size input file, and it can be (IMO) more easily customized and extended for more complex state changes. Here's an example:

use warnings;
use strict;

my $beg1 = qr/^\s*APP>\sSTART_PARTU_8X8$/;
my $beg2 = qr/^\s*APP>\sPARTITION_SPLIT$/;
my $end  = qr/^\s*APP>\sEND_PARTU_8X8$/;

open my $fh, '<', 'sample.txt' or die $!;
my $prev_line;
my $state = 'not_in_block';
while ( my $line = <$fh> ) {
    if ( $state eq 'not_in_block' ) {
        if ( $prev_line && $prev_line=~$beg1 && $line=~$beg2 ) {
            $state = 'am_in_block';
        }
    }
    elsif ( $state eq 'am_in_block' ) {
        if ( $line=~$end ) {
            $state = 'not_in_block';
        }
        else {
            print $line;
        }
    }
} continue { $prev_line = $line }
close $fh;
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Using the sample input data from your post as sample.txt, both pieces of code output:

    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
    START_PARTU_4X4
    ........
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Of course, this being Perl, there are other ways to approach this, but these are two pretty common ones.