Universal estimates for parabolic equations and applications for non-linear and non-local problems

We obtain some "universal" estimates for $L_2$-norm of the solution of a parabolic equation via a weighted version of $H^{-1}$-norm of the free term. More precisely, we found the limit upper estimate that can be achieved by transformation of the equation by adding a constant to the zero order coefficient. The inverse matrix of the higher order coefficients of the parabolic equation is included into the weight for the $H^{-1}$-norm. The constant in the estimate obtained is independent from the choice of the dimension, domain, and the coefficients of the parabolic equation, it is why it can be called an universal estimate. As an example of applications, we found an asymptotic upper estimate for the norm of the solution at initial time. As an another example, we established existence and regularity for non-linear and non-local problems.