Abstract
In this paper, we investigate the problem for optimal control of a viscous generalized
\theta
-type dispersive equation (VG
\theta
-type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG
\theta
-type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.
Abstract
The aim of this paper is to study a quasistatic contact problem between an
electro-elastic viscoplastic body with damage and an electrically conductive
foundation. The contact is modelled with an electrical condition, normal
compliance and the associated version of Coulomb’s law of dry friction in
which slip dependent friction is included. We derive a variational
formulation for the model and, under a smallness assumption, we prove the
existence and uniqueness of a weak solution.
In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term
$h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$
Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space
$\mathcal {H}_{t}(\Omega )$
and the existence and regularity of the pullback attractors.
We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.
In this paper the following Cauchy problem, in a Hilbert spaceH, is considered:(I+λA)u″+A2u+[α+M(|A12u|2)]Au=fu(0)=u0u′(0)=u1Mandfare given functions,Aan operator inH, satisfying convenient hypothesis,λ≥0andαis a real number.Foru0in the domain ofAandu1in the domain ofA12, ifλ>0, andu1inH, whenλ=0, a theorem of existence and uniqueness of weak solution is proved.