Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:

$$u_t = grad[V(u)]$$

For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimensional faces of the $n+1$-th cube $[0,1]^{n} \times [0,T)$. Where by "$grad$" I mean the gradient w.r.t. to the spatial coordinates (excluding the last coordinate).

Are there necessary and sufficient conditions one can put on $V$ for which the above problem will always be well-posed for some choice of $T$? (for initial conditions in reasonable function spaces).

Are there similar conditions (perhaps more complicated) for the second order equation:

$$u_{tt} = grad[V(u)]$$

With the same type of boundary conditions?