On a problem without initial conditions for the equation of the third order with multiple characteristics (Q2742949)

Let \(Q=(0,l)\times(0,k)\times(T_0,T_1)\), where \(T_0,T_1\in \mathbb{R}\). The following third-order equation in the domain \(Q\) NEWLINE\[NEWLINE\frac\partial{\partial x}\Delta u+ \sum_{i=1}^2\left( a_i(x,y,t)\frac{\partial^iu}{\partial x^i}+b_i(x,y,t) \frac{\partial^iu}{\partial y^i}\right)+c(x,y,t)-u_t=f(x,y,t) NEWLINE\]NEWLINE with boundary value conditions \(u_{ x=0}=u_{x x=0}= u_{ x=l}=u_{ y=0}=u_{ y=k}=0\), but without initial conditions is considered. An analogue of the Saint-Venant's principle for the above problem is obtained. On the base of this principle the uniqueness theorem is proved, and some properties of the solution to the considered problem in an unbounded domain are studied.