Let $u$ solve the linear parabolic equation $$ u_t - \Delta u = f \text{ on } \Omega \times (0,T) $$ with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega \times (0,T)$. $\Omega$ is a bounded domain in $\mathbb R^d$ and $T>0$. Here, $u$ is meant to be a weak solution in $L^2(0,T;H^1_0(\Omega))$ with $u_t\in L^2(0,T;H^{-1}(\Omega))$.
I am interested in estimates of the type: $$ \|u\|_{L^p(\Omega\times(0,T))} \le c( \|f\|_{L^q(\Omega\times(0,T)} + \|u_0\|_{L^r(\Omega)}) $$ under suitable assumptions on $p,q,r$ and $d$. The proof of such estimate follows basic ideas: test the equation with $u|u|^{s-1}$ (for suitable $s$), get estimates for $|u|^{s/2}$ in $L^2(0,T;H^1_0(\Omega)) \cap L^\infty(0,T;L^2(\Omega))$, use Gagliardo-Nirenberg inequality and embedding theorems. No smoothness assumptions on the $\partial \Omega$ are necessary, also $-\Delta$ can be replaced by more general second-order elliptic operator.
Could you point me to a reference? All searches just turned up results for more complicated problems ($L^p$ estimates for derivatives of $u$). This result is so basic, it should be contained in some textbook. It is not in Evans, and not in Ladyshenskaya et al, though.
