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:
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 $p\leqslant3$.