Estimates on mild solutions to the reaction-diffusion equation

Question

A mild solution to the IVP \begin{align} &u_t = \Delta u + F(u) \text{ in } [0,T] \times \mathbb{R^d} \\ &u(0,x) = u_0(x) \text{ in } \mathbb{R^d} \end{align} is given by $$ u(t,x) = (S(t)*u_0)(x) + \int_0^t(S(t-\tau)*F(u(\tau,\cdot)))(x)d\tau, $$ where $$ S(t) = \frac{1}{(4\pi t)^{d/2}}e^{-\frac{|x|^2}{4t}}. $$ If $u_0 \in L^p(\mathbb{R^d})$ and $u \in L^\infty(0,T; L^p(\mathbb{R^d}))$, is it true that $u \in L^2(0,T;W^{1,p}(\mathbb{R^d}))$? When I try to find an upper bound for $||\partial_{x_i}u(t,\cdot)||^2_{L^p(\mathbb{R^d})}$ I end up needing that $||\partial_{x_i}u_0||_{L^p(\mathbb{R^d})} < \infty$, that is, that $u_0 \in W^{1,p}(\mathbb{R^d})$. I've tried coming up with a counterexample (an initial condition that is in $L^p$ but with at least one of its first derivatives not in it), but I can't seem to get anywhere. Is my additional hypothesis actually necessary?