# Contraction semigroup of nonlinear heat equations

Let $$(S(t))_{t\geq 0}$$ be a semigroup generated by $$\Delta$$ operator in $$L^{2}(\Omega)$$ for a bounded smooth domain $$\Omega \subset \mathbb{R}^{N}$$. Here we assume \begin{align} \begin{cases} D(B) = \{u \in H_{0}^{1}(\Omega)\,|\, \Delta u \in L^{2}(\Omega)\}\\ \forall u \in D(B), \, Bu=\Delta u \end{cases} \end{align}

Recall that a one-parameter family $$(S(t))_{t\geq 0}\subset \mathfrak{L}(L^{2}(\Omega))$$ (the space of bounded linear operator from $$L^{2}(\Omega)$$ to $$L^{2}(\Omega)$$) is a contraction semigroup in $$L^{2}(\Omega)$$ provided that:
1 $$\forall t \geq 0, ||S(t)||\leq 1$$
2 $$S(0) = I$$
3 $$\forall s,t\geq 0, \, S(t+s) = S(t)S(s)$$
4 $$\forall \phi \in L^{2}(\Omega)$$, the function $$t \mapsto S(t)\phi$$ belongs to $$C([0,\infty),L^{2}(\Omega))$$.

Now, my question is for this setting, what are the conditions such that I can say that for $$\phi \in L^{2}(\Omega)$$, it holds that $$\nabla S(t)\phi = S(t) \nabla \phi$$? I actually want to show that $$||(S(t)-I)\varphi||_{H_{0}^{1}(\Omega)}\to 0$$ as $$t\to0$$ for a given $$\varphi \in H_{0}^{1}(\Omega)$$.

Any help is much appreciated! Thank you very much!