For the classical heat and parabolic equations, a priori estimates are extremely standard and can be found in Chapter 7 of L. C. Evans’ book and many others (evans-pde) For time-fractional heat equation this I believe is new but not very different. The essential difference is that one uses a weighted norm in the time direction. Let and let be the fractional integral
which is the inverse of the fractional derivative . Consider
with initial data . Rigorous a priori estimates can use a finite dimensional approximation using eigenfunctions of Laplacian as a basis on a bounded domain in as in Evans’ book but the key computation is the following where we assume that there is a smooth solution with sufficiently fast decay as .
First, let and
Now since we are using the Caputo derivative. So multiply with a test function and integrate by parts and set the test function to :
Now apply first to the term on the right and use Cauchy-Schwartz:
where . Choose and plug in the estimate into (1). Obtain
So what is happening here that is different from the standard case is that instead of the usual Lebesgue measure for time we have a weighted time measure; otherwise the argument is the same. Now the problem is that arguments involving the Gronwall lemma are a bit more complicated but there exists a Gronwall lemma proved for time-fractional functions in 2006 which could apply: fractionalGronwallInequality2006.