Solvability of a certain boundary value problem for pseudoparabolic equations of the forth order

S. G. Pyatkov

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 5 (2005) no. 3, pp. 43-56

Résumé

Under consideration is the equation $$ Mu=L_0(x,t,D_x)u_t+L_1(x,t,D_x)u=f(x,t),\quad(x,t)\in Q=G\times(0,T), $$ where $G\subset\mathbb{R}^n$ is a bounded domain with boundary $\Gamma$ and $L_0$, $L_1$ are elliptic operators of the second and forth order, respectively. The boundary conditions are of the form $$ u|_S=\varphi(x,t), \quad\frac{\partial u}{\partial n}\Bigl|_S=\psi(x,t), \quad u|_{t=0}=u_0(x), \quad S=\Gamma\times(0,T). $$ It is demonstrated that this problem is uniquely solvable in the weighted Sobolev space whose norm is defined by the equality $$ \|u\|^p=\sum_{|\alpha|\leqslant2}\|\rho D^\alpha u_t\|^p_{L_p(Q)}+\sum_{|\alpha|\leqslant4}\|\rho D^\alpha u\|^p_{L_p(Q)}, $$ where $\rho(x)$ is the distance from a point $x$ to $\Gamma$.