Finding of Exact Solution to Nonlinear Operator Equation

The exact solution is found for the nonlinear operator equation of the form du(r,t)/dt = A[u(r,t)], u : Rⁿ x R₊ → R under condition u(r, 0) = φ(r) for operator A infinitely differentiable by Frechet. Examples of solving the Cauchy problems for equations of heat conduction, advection, ordinary differential equation with separable variables, and the Corteweg-de Fries equation.