Geodesic
Solvability of a certain boundary value problem for pseudoparabolic equations of the forth order
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 5 (2005) no. 3, pp. 43-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\Gamma$. 
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     title = {Solvability of a certain boundary value problem for pseudoparabolic equations of the forth order},
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S. G. Pyatkov. Solvability of a certain boundary value problem for pseudoparabolic equations of the forth order. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 5 (2005) no. 3, pp. 43-56. http://geodesic.mathdoc.fr/item/VNGU_2005_5_3_a3/

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