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I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...