Study of Some Non-coercive Quasilinear Parabolic Problem in Anistropic Sobolev Space

This paper is devoted to study the existence and regularity of renormalized solutions for the following quasilinear parabolic problem \( \left \{\begin {array}{ll} \displaystyle \frac {d u}{d t}-\sum _{i=1}^{N} D^{i}a_{i}(x,t,u, \nabla u)+|u|^{p_{0}-2}u=f(x,t,u,\nabla u) & \mbox{in} \quad \Omega \times (0,T), \\ u=0 & \mbox{on} \quad \partial \Omega \times (0,T),\\ u(0,x)=u_{0}(x) & \mbox{in}\quad \Omega , \end {array}\right .\) in the anisotropic parabolic Sobolev spaces \(L^{\vec {p}}(0,T;W_{0}^{1,\vec {p}}(\Omega ))\) , where the right-hand side is a Carathéodory function that satisfies only some growth condition, and the initial condition \(u_{0}\) belongs \(L^{1}(\Omega )\) .