Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in $L^p$-sense

We prove that multilevel Picard approximations are capable of approximating solutions of semilinear heat equations in $L^p$-sense, $p\in [2,\mathrm{\infty })$, in the case of gradient-dependent, Lipschitz-continuous nonlinearities, in the sense that the computational effort of the multilevel Picard approximations grow at most polynomially in both the dimension $d$ and the reciprocal $1\mathrm{/}\epsilon$ of the prescribed accuracy $\epsilon$.