Initial-boundary value problems for the generalized \(n\)-dimensional IMB\({}_q\) equation (Q2755687)

The paper concerns the initial-boundary value problem for the \(\text{IMB}_q\) equation NEWLINE\[NEWLINE\begin{cases} u_{tt}-\nabla^2u_{tt}- \nabla^2u= \nabla^2g(u),\quad (x,t)\in \Omega\times (0,T),\\ u(x,t)=0,\quad (x,t)\in\partial \Omega\times [0,T),\\ u(x, 0)= \varphi(x),\quad u_t(x,0)= \psi(x),\quad x\in\overline \Omega. \end{cases}NEWLINE\]NEWLINE The local existence and uniqueness for both classical and generalized solution are studied. Besides, a sufficient condition for blowing up of solutions is also given.