We consider a heat equation with the non-linear right-hand side which depends on certain Volterra-type functionals acting on the unknown function and on its gradient. We give some natural sufficient conditions for the existence and uniqueness of solutions to this equation. The solution is obtained as a limit of a fast convergent sequence of successive approximations obtained by the quasi-linearisation method.
We prove a comparison theorem for an ODE and DAE system which arises from the method of lines. Under a Perron comparison condition on the functional dependence and a specific Lipschitz and (W+) condition on the classical argument, we obtain strong uniqueness criteria.
We consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.