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

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 $p\leqslant3$.