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Evans' PDE book 2nd, (Theorem 7.1.5) says for parabolic PDEs, if the initial condition is in $H^1_0 \cap H^2$, and the source term satisfies the regularity $\mathbf { f }, \mathbf { f } ^ { \prime } \in L ^ { 2 } \left( 0 , T ; L ^ { 2 } ( U ) \right)$ then the solution satisfies $\mathbf { u } \in L ^ { \infty } \left( 0 , T ; H ^ { 2 } ( U ) \right) , \mathbf { u } ^ { \prime } \in L ^ { \infty } \left( 0 , T ; L ^ { 2 } ( U ) \right) \cap L ^ { 2 } \left( 0 , T ; H _ { 0 } ^ { 1 } ( U ) \right)$.

The preceding formal derivation of estimates says "Now $2 D u \cdot D u _ { t } = \frac { d } { d t } \left( | D u | ^ { 2 } \right)$". In the proof of Theorem 7.1.5, as I understand it the formal derivation is essentially done but with the Galerkin method, i.e., in the weak formulation and eigenfunction expansion.

Later in Section 7.4 Evans introduced the semigroup theory. My question is if an analogous estimate can be shown in the framework of the semigroup theory.

I have looked at Pazy's book Semigroup of Linear Operators and Applications to Partial Differential Equations. Section 6.3 is about semilinear equations with Analytic Semigroups. The setting seems to be very close, and I am expecting something like $$u'\in L ^ { \infty } \left( 0 , T ; X \right) \cap L ^ { 2 } \left( 0 , T ; D(A^{1/2}) \right)$$ Here, $u$ is the (strong?) solution of the equation $$\left\{ \begin{array} { l } { \frac { d u ( t ) } { d t } + A u ( t ) = f ( t , u ( t ) ) , \quad t > t _ { 0 } } \\ { u \left( t _ { 0 } \right) = x _ { 0 } } \end{array} \right.,$$ and $-A$ generates an analytic semigroup.

Pazy's book has several regularity estimates, and my attempt was to mimic the proofs: e.g., consider $ A^{1/2}(u(t)-u(s))/(t-s)=... $ and use properties such as $\|A^{1/2}T(t)\|\leq M t^{1/2}e^{-\delta t}$ for some $\delta$, but I do not know how to show an analogous result without assuming some spatial regularity of $f$.

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    $\begingroup$ "Without assuming some spatial regularity of $f$" won't work. Even if $A=0$, you still end up with a generic first-order differential equation in $X$. For every infinite-dimensional Banach space $X$ there exist a continuous function $f\colon\mathbb{R}\times X\to X$ and $x_0\in X$ such that the initial-value problem has no (local) solution. $\endgroup$ – MaoWao Feb 4 at 13:29
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    $\begingroup$ The situation in Evans' book is more restrictive in that $f$ only depends on $t$, not on $u$. In this case I think one might be able to obtain similar estimates in the abstract setting. $\endgroup$ – MaoWao Feb 4 at 13:32
  • $\begingroup$ Ah I see. Thank you! $\endgroup$ – user41467 Feb 4 at 14:06

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