Qualitative behaviour of a class of pde

This dissertation addresses the initial and periodic boundary value problems for a fourth-order pseudo-parabolic equation given by \begin{equation*} u_t-a\Delta u_t-\Delta u+(-\Delta)^2u=-\nabla\cdot(|\nabla u|^{p-2}\nabla u),\quad (x,t)\in \Omega\times(0,T) \end{equation*} where the gradient non-linearity and the pseudo-term are as specified above. The problem is subject to the initial condition \begin{equation*} u(x,0)=u_0(x), \quad u_0 \in L^2(\Omega),\quad x\in \Omega, \end{equation*} and periodic boundary conditions \begin{eqnarray*} \forall x\in \Gamma_i , \;\;02$. A local existence-uniqueness result for mild solutions was established for any initial data in $L^2(\Omega)$. It was also demonstrated that mild solutions are weak solutions to the problem. Additionally, the existence of blow-up solutions was proven, and the blow-up time was shown to have a lower bound.