Strongly Quasilinear Parabolic Systems in Divergence Form with Weak Monotonicity

Document Type : Original Article


Department of Mathematics, Faculty of Sciences Dhar El Mehraz, B.P. 1796 Atlas, Fez-Morocco.


The existence of solutions to the strongly quasilinear parabolic system
\[\frac{\partial u}{\partial t}-\text{div}\,\sigma(x,t,u,Du)+g(x,t,u,Du)=f,\]
is proved, where the source term $f$ is assumed to belong to $L^{p'}(0,T; W^{-1,p'}(\Omega;R^m))$. Further, we prove the existence of a weak solution by means of the Young measures under mild monotonicity assumptions on $\sigma$.