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Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$ , $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE $$ \partial_t\Phi(t,z) - \phi(0,z) - \sum_i \mu_i \partial_{z_i}\Phi(t,z) - \frac{\sum_{i,j} \rho_{i,j}\partial_{z_i}\Phi(t,z)\partial_{z_j}\Phi(t,z) }{2} = \frac{\sum_{i,j} \rho_{i,j}\partial_{z_i,z_j}^2\Phi(t,z) }{2} $$ where $\Phi(t,z)\triangleq \int_0^t \phi(s,z)\,ds$.

If $\mu_i$ and $\rho_{i,j}$ are given is it possible to solve the given PDE for $\Phi$? It not can we calculate or at-least estimate the $L^{\infty}$ norm of $\phi$?


Potential Extra Assumption: If need be I may be able to assume (but would prefer not to) that the term on the Right-hand-side is 0.

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  • $\begingroup$ You might want to explain what $\partial_x\Phi(t,z)$ is. $\endgroup$ Aug 24, 2016 at 14:14
  • $\begingroup$ $\partial_t\Phi(t,z)$ is the partial derivative with respect to $t$. $\endgroup$
    – ABIM
    Aug 24, 2016 at 14:36
  • $\begingroup$ Pick $\phi(0,z)$ arbitrarily. Once you have done so, your equation becomes a semilinear parabolic equation for $\Phi$. There is plenty of literature on such equations. $\endgroup$ Aug 24, 2016 at 15:31
  • $\begingroup$ Hmm I actually do know $\phi(0,z)$ can you point me to a reference which gives explicit solutions to this type of PDE? $\endgroup$
    – ABIM
    Aug 24, 2016 at 18:18
  • $\begingroup$ Explicit solutions are a bit much to ask for, expect maybe for special choices of the coefficient functions. What I said there is plenty of literature, I meant existence, uniqueness, numerical approximation etc. etc. $\endgroup$ Aug 24, 2016 at 19:02

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