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

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 2, Tome 198 (1991), pp. 31-48

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 for operators $A$ and $K(u)$ and external force $F(t)$ the following assumptions are satisfies: 1) $A$ is linear bounded negative definite operator from $H_2$ onto $H_2$: $$ ||A||\leqslant c_4,\quad (Au,u)_{H_2}\leqslant-\gamma||u||^2_{H_2},\quad\gamma>0,\qquad{(8)} $$ 2) $K(u)$ is nonlinear operator acting from $H_2$ into $H_3$, and \begin{gather*} (K(u),u)_{H_2}\leqslant\varepsilon||u||^2_{H_2}\cdot||u||^\alpha_{H_1}+c_\varepsilon||u||^{1+\beta}_{H_1},\quad\forall\varepsilon>0,\ \alpha,\beta>0,\tag{9}\\ ||K(u)||_{H_3}\leqslant c_s||u||^{1+\alpha}_{H_2},\quad\forall u\in H_2,\tag{10}\\ ||K(u^1)-K(u^2)||_{H_2}\leqslant c_6(||u^i||_{H_2})||u^1-u^2||_{H_2};\tag{11} \end{gather*} 3) $F(t)\in H_2$, $\forall t\in\mathbb{R}^+$. In the paper four nonlocal problems for the equation (7)–(11) 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 on the axis $\mathbb{R}\equiv(-\infty,\infty)$ of solution of the equation (7)–(11) for the same conditions on the external force $F(t)$ which are supposed in the 1) (see Theorems 4–6); 3) 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 7–8). 4) Existence in the small of almost-periodic solution of the equation (7)–(11) with almost-periodic external force $F(t)$ (see Theorems 9–11). 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 $p\leqslant3$.