Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

In this paper, we will study the strongly nonlinear parabolic problems of the type \(\begin{aligned}\left\{ \begin{array}{ll} \displaystyle \frac{\partial b(u)}{\partial t} - \sum _{i=1}^{N} D^{i}(a_{i}(x,t,u,\nabla u))+ H(x,t,\nabla u) = f(x,t) \quad &{}\hbox { in }Q_{T}=\Omega \times (0, T),\\ u(x, t) = 0 &{}\hbox { on } S_{T}= \partial \Omega \times (0, T),\\ b(u(x, 0))= b(u_{0}(x)) &{}\hbox { in } \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial b(u)}{\partial t}-{\sum }_{i=1}^N{D}^i(a_i(x,t,u,\mathrm{\nabla }u))+H(x,t,\mathrm{\nabla }u)=f(x,t)}\hfill & \text{in}{Q}_T=\mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0\hfill & \text{on}{S}_T=\partial \mathrm{\Omega }\times (0,T),\hfill \\ b(u(x,0))=b(u_0(x))\hfill & \text{in}\mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(b(u_{0})\) $b(u_0)$ belongs to \(L^{1}(\Omega )\) $L^1(\mathrm{\Omega })$ , and the behavior of \(H(x,t,\xi )\) $H(x,t,\xi )$ satisfies the growth conditions. Our work focuses on establishing the existence and uniqueness of a renormalized solution for this quasilinear parabolic equation. Furthermore, we conclude some regularity results.