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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?

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    $\begingroup$ Do you have some initial conditions as well? $\endgroup$
    – user35593
    Commented May 3, 2019 at 6:24
  • $\begingroup$ @user35593 Yes, $u_0$ should ideally be in $L_2$, but existence for continuous $u_0$ would also be fine for me. $\endgroup$
    – Oleg
    Commented May 3, 2019 at 9:42
  • $\begingroup$ @MichaelRenardy Thanks for your comment, but I also assumed that $f$ is continuous in $t$ in $x$. In this case if $f$ does not depend on $u$, then it is bounded on $[0,T]\times \mathbb{T}^d$ and there is no blow-up. $\endgroup$
    – Oleg
    Commented May 3, 2019 at 10:55
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    $\begingroup$ Your linked question is somewhat different. In the linked question $f$ is continuous but not Holder continuous, and the issue is local existence. In your case since $f$ is Lipschitz in $u$, you should be able to get local existence for $C^0$ initial data just using Picard iteration. Your dissipativity condition should give you also a priori, time-dependent upper bounds on $|u|$ using just the maximum principle, if I am not mistaken. This should give you most things you need for what you want. $\endgroup$
    – Willie Wong
    Commented May 3, 2019 at 15:16
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    $\begingroup$ @WillieWong Thank you for the comments! You are totally right: if $u_0$ is continuous, then this case is described in Lunardi's boook and local existence follows e.g. by the Picard iterations. But what to do if $u_0\in L^2$? $\endgroup$
    – Oleg
    Commented May 3, 2019 at 15:34

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