Initial boundary value problem for Sobolev type nonlinear equations

H. A. Mamikonyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 2 (2006), pp. 33-40

Résumé

In this paper following initial boundary value problem is considered. $$\left\{\begin{array}{l} A\left(\frac{\partial u}{\partial t}\right)+Bu=f,\\ u(0)=u_0,\\ D^{\gamma}u\Big|_{\Gamma}=0, |\gamma|\leq m, \end{array}\right.$$ Operators A and B are nonlinear and have the following forms $Au=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}A_{\alpha}(x,t,D^{\gamma}u),\quad Bu=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}B_{\alpha}(x,t,D^{\gamma}u),~~|\gamma|\leq m.$ Conditions for functions $A_{\alpha}(x,t,\xi_{\gamma})$ and $B_{\alpha}(x,t,\xi_{\gamma})$ are obtained that lead to existence and uniqueness of solution of the problem in the spaces $L^p(0,T,W^m_p),~р\geq 2$.