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## A PRIORI ESTIMATES FOR FRACTIONAL HEAT EQUATION

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 $L^2$ norm in the time direction.  Let $0<\alpha<1$ and let $I^{\alpha}_t$ be the fractional integral

$I_t^{\alpha} f(t) = \int_0^t (t-s)^{\alpha-1} f(t)$

which is the inverse of the fractional derivative $D^{\alpha}_t$.  Consider

$D^{\alpha}_t u - \Delta u= f$ with initial data $u(\cdot,0)=g$.  Rigorous a priori estimates can use a finite dimensional approximation using eigenfunctions of Laplacian as a basis on a bounded domain in $\mathbf{R}^n$ 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 $|x|\rightarrow \infty$.

First, let $X=\mathbf{R}^n$ and

$\int_X f^2 dx = \int_X (D^{\alpha}_t u)^2 - 2\Delta u D^{\alpha}_t u + (\Delta u)^2 dx$

Now $D^{\alpha}_t (|Du|^2) = 2 Du DD^{\alpha}_t u$ since we are using the Caputo derivative.  So multiply $D^{\alpha}_t u - \Delta u = f$ with a test function and integrate by parts and set the test function to $u$:

$\frac{1}{2} D^{\alpha}_t \int_X u^2 dx + \int_X |Du|^2 = \int_X fu dx$ (1)

Now apply $I^{\alpha}_t$ first to the term on the right and use Cauchy-Schwartz:

$I^{\alpha}_t \int_X fu dx \le (I^{\alpha}_t \int_X f^2)^{1/2}(I^{\alpha}_t \int_X u^2)^{1/2}$

$\cdot \le \frac{1}{4 \epsilon} I^{\alpha}_t \int_X f^2 + \epsilon C(T) \int_X u^2 dx$

where $C(T) = I^{\alpha}_T(1) = \int_0^T (T-s)^{\alpha-1} ds$.  Choose $\epsilon C(T) = 1/4$ and plug in the estimate into (1).  Obtain

$\frac{1}{4}\sup_{[0,T]} \int_X u^2 + I^{\alpha}_t \int_X |Du|^2 \le C(I^{\alpha}_t\int_X f^2 dx + \frac{1}{2} \int_X g^2 dx)$

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.