Consider the following 4th order linear pde: $$ u_{xxyy}+a u_{xxy}+bu_{yy}+cu_y+du_x=0 $$ for $u(x,y)\in C^2\left([0,1]\times \mathbb{R},\mathbb{R}\right) $.
Is it possible to rewrite this equation in the form: $$ U_y(y)= \mathcal{L} U(y) $$ where $\mathcal{L}$ is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on $C^2\left([0,1],\mathbb{R}^n\right) $?