Nonlocal problems for some class nonlinear dissipative Sobolev type equations

A. A. Kotsiolis ; A. P. Oskolkov ; R. D. Shadiev

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 11, Tome 199 (1992), pp. 91-113

Résumé

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 suppose that for operators $A$ and $K(u)$ and external force $F(t)$ the assumptions (8)–(12) are fulfilled. In the paper two nonlocal problems for the equation (7)–(12) are studied: 1. Existence in the large on the semiaxis $\mathbb{R}^+$ solution of the Cauchy problem (7)–(12) for distinct assumptions about external force $F(t): F(t)\in L_\infty(\mathbb{R}^+;H_2)$, $F(t)\in L_2(\mathbb{R}^+;H_2)$, $F(t)\in S_2(\mathbb{R}^+;H_2)$ (see Theorems 1–3). 2. Existence in the large time-periodic solutions of the equation (7)–(11), (15) with time-periodic external force $F(t)\in\tilde{L}_{2,\omega}(\mathbb{R}^+;H_2)$ and $F(t)\in\tilde{L}_{\infty,\omega}(\mathbb{R}^+;H_2)$ (see Theorems 6–7) The examples of nonlinear dissipative Sobolev type equations (2)–(6) which are reduced to the abstract nonlinear equation (7)–(11) are given: \item[] equations of the motion of the Kelvin–Voight fluids (50) (see Theorems 8–9), \item[] equations of the motion of the Kelvin–Voight fluids order $L=1,2,\dots$ (97) and (99), \item[] the system of the “Oskolkov equations” (90), (91), \item[] similinear pseudoparabolic equations (76) with $p\leqslant3$ and (85), (86) (see Theorems 10–11).