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# Dynamics of Partial Differential Equations

## Volume 9 (2012)

### Number 4

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

Pages: 369 – 381

DOI: https://dx.doi.org/10.4310/DPDE.2012.v9.n4.a4

#### Author

#### Abstract

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 $L_2$-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.

#### Keywords

parabolic equations, regularity, nonlinear equations, non-local equations

#### 2010 Mathematics Subject Classification

35K10, 35K20, 35K55, 35K58, 35K59

Published 7 January 2013