Weighted in time energy estimates for parabolic equations with applications to non-linear and non-local problems

The paper suggests a modification of the contracting mapping method for non-linear and non-local parabolic equations. This modification is based on weighted in time energy estimates for the L2-norm of the solution of a parabolic equation via a weighted version of the H−1-norm of the free term such that the inverse matrix of the higher order coefficients of the parabolic equation is included into the weight. More precisely, this estimate represents the upper estimate that can be achieved via transformation of the equation by adding a constant to the zero order coefficient. The limit constant in this estimate is independent from the choice of the dimension, domain, and the coefficients of the parabolic equation.