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I'm facing following regularity issue and i wonder if anyone of you guys is able to help me. I'd like to show that the solution of a semilinear heat equation is classical, i.e. $C^2$ in space and $C^1$ in time:

\begin{align*} -\partial_t u(x,t) - \Delta u(x,t) +\frac{1}{2} \vert Du(x,t) \vert^2 = f(x,t)) \\ \end{align*} with the terminal condition $u(x,T) = g(x,T)$, where $f,g$ are Lipschitz continuous in space, and (1/2)-holdercontinuous in time $(\in C^{0,1/2}(\mathbb{R}^n \times [0,T])$, i.e. $$\vert f(x_1,t_1) -f(x_2,t_2) \vert \leq C ( \Vert x_1 - x_2\Vert_2 + \vert t_1 - t_2 \vert^{1/2})$$ and uniformly bounded by $C$.

Following statement is found in "linear and quasilinear equations of parabolic type (‎Ladyzhenska):

… under the above assumptions, there is a unique weak holdercontinuous solution $u \in C^{2,1/2}(\mathbb{R}^n \times [0,T])$ (holdercontinuity up to second derivative).

By comparison i was able to show that $\vert Du \vert \leq C$ uniformly in x and t, so that $u$ solves an equation of the form \begin{align*} -\partial_t u - \Delta u = A, \end{align*} where $A$ is uniformly bounded in $x$ and $t$. In the paper I'm reading the author conclude, that the solution $u$ is classical, but i've got no idea why this is true. I'm not very familiar with the regularity of heat equations and i can't found any hint for the above statement. Is there anyone who is? Thanks a lot!

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