MODIFIED QUASI-REVERSIBILITY METHOD FOR NONAUTONOMOUS SEMILINEAR PROBLEMS

We prove regularization for the ill-posed, semilinear evolution problem du/dt = A(t,D)u(t) + h(t,u(t)), 0 <= s <= t < T, with initial condition u(s) - chi in a Hilbert space where D is a positive, self-adjoint operator in the space. As in recent literature focusing on linear equations, regularization is established by approximating a solution u(t) of the problem by the solution of an approximate well-posed problem. The approximate problem will be defined by one specific approximation of the operator A(t,D) which extends a recently introduced, modified quasi-reversibility method by Boussetila and Rebbani. Finally, we demonstrate our theory with applications to a wide class of nonlinear partial differential equations in L-2 spaces including the nonlinear backward heat equation with a time-dependent diffusion coefficient.