INITIAL BOUNDARY VALUE PROBLEM FOR SOBOLEV TYPE NONLINEAR EQUATIONS

Authors

  • H. A. Mamikonyan Chair of the Optimal Control Theory and Approximate Methods, YSU, Armenia

DOI:

https://doi.org/10.46991/PYSU:A/2006.40.2.033

Keywords:

initial boundary value problem, nonlinear operator

Abstract

In this paper following initial boundary value problem  is considered: $$\begin{cases}  A\left(\dfrac{\partial u}{\partial t}\right)+Bu=f, \\ u(0)=u_0, \\D^{\gamma}|_{\Gamma}=0, |\gamma|\leq m. \end{cases}$$ Operators $A$ and $B$ are nonlinear and have the following form: $$Au=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}A_{\alpha}(x, t, D^{\gamma}u), ~~Bu=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}B_{\alpha}(x, t, D^{\gamma}u), |\gamma|\leq m. $$

Conditions for functions $A_{\alpha}$ and $B_{\alpha}$ are obtained that lead to existence and uniqueness of solution of the problem in the spaces $L^p(0, T, ^0W^m_p), p\geq 2$.

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Published

2006-05-30

How to Cite

Mamikonyan, H. A. (2006). INITIAL BOUNDARY VALUE PROBLEM FOR SOBOLEV TYPE NONLINEAR EQUATIONS. Proceedings of the YSU A: Physical and Mathematical Sciences, 40(2 (210), 33–40. https://doi.org/10.46991/PYSU:A/2006.40.2.033

Issue

Section

Mathematics