Nonlocal problems for a class of nonlinear dissipative Sobolev-type equations (Q1808234)

The authors consider the equation \[ {du\over dt}=Au+ K(u)+F(t),\;t\in\mathbb{R}_+, \tag{1} \] where the operator \(A\) is linear and bounded, and the operator \(K\) is nonlinear. In the paper the following two nonlocal problems for the equation (1) are studied: (i) Existence in the large of a solution for the Cauchy problem on the semiaxis \(\mathbb{R}_+\) under various assumptions on the external force \(F(t)\), (ii). Existence in the large of a solution \(\omega\)-periodic in \(t\), provided that the external force is \(\omega\)-periodic in \(t\). Examples are given for nonlinear dissipative equations of the Sobolev type that are reduced to the equation (1). These are the equations of the motion of a Kelvin-Voight fluid, the Oskolkov system, the Korteweg-de Vries-Bürgers alternative equation, and the semilinear pseudoparabolic equations.