I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$ \begin{align*} &\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\ & u(0)=u_0\in L_2 \end{align*} where the function $f$ is locally (but not globally!) Lipschitz in the first argument and is continuous in $t$, $x$. Assume also a certain dissipativity condition on $f$ to prevent blow-up (e.g. $(z_1-z_2)(f(z_1,t,x)-f(z_2,t,x))\le C-C|z_1-z_2|^2$.)

My question is how to prove that this equation has a unique classical solution. It should be very standard but I cannot find an appropriate reference.

I also would like to show that this solution is continuous on $(0,T)\times\mathbb{T}^d$.

Update: I guess uniqueness would follow by the standard argument (namely, by considering two solutions, taking their difference and applying the Gronwall lemma). Thus it remains to show global existence. This apparently was already asked at this forum but with no replies: Existence of solutions to a reaction-diffusion problem.

Update 2: the case when the initial condition is continuous or in $L_\infty$ follows from Lunardi's book, Proposition 7.3.1. Thus, the problem is that my initial data is in $L_2$. Is it possible to show that a limit of classical solutions corresponding to smooth initial conditions is also a classical solution?