Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces

Abstract

We consider nonlinear parabolic equations with nonlinear non–Lipschitz's term and singular initial data like Dirac measure, its derivatives and powers. We prove existence–uniqueness theorems in Colombeau vector space \( \fancyscript{G}_{{C^{1} ,W^{{2,2}} }} {\left( {\left[ {0,T} \right.} \right)},R^{n} ,n \leqslant 3 \). Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space \( \fancyscript{G}_{{C^{1} ,L^{{2}} }} {\left( {\left[ {0,T} \right.} \right)},R^{n} ,n \leqslant 3 \).