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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. LOMI, 1991 Volume 198, Pages 31–48 (Mi znsl5069)

This article is cited in 38 papers

Nonlocal problems for some class nonlinear operator equations arising in the theory Sobolev type equations

A. P. Oskolkov


Abstract: Let $H_i$, $i=0, 1, 2, 3$, are Hilbert spaces:
$$ H_3\subset H_2\subset H_1\subset H_0, \qquad{(1)} $$
and imbeddings are compact. Consider in $H_2$ nonlinear abstract equation
$$ \frac{du}{dt}=Au+K(u)+F(t),\quad t\in\mathbb{R}^+,\qquad{(7)} $$
and for operators $A$ and $K(u)$ and external force $F(t)$ the following assumptions are satisfies: In the paper four nonlocal problems for the equation (7)–(11) are studied: The examples of nonlinear dissipative Sobolev type equations (2)–(6) which are reduced to the abstract nonlinear equation (7)–(11) are given: equations of the motion of the Kelvin–Voight fluids (0.1), equations of the motion of the Kelvin–Voight fluids order $L=1,2,\dots$ (62) and (63), the system of the “Oskolkov equations” (64), semilinear pseudoparabolic equations (65) with $p\leqslant3$.

UDC: 517.94


 English version:
Journal of Soviet Mathematics, 1993, 64:1, 724–736

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