We obtain solvability conditions in H6(ℝ3) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding article [18].
We prove local solvability in Gevrey spaces for a class of semilinear partial differential equations. The linear part admits characteristics of multiplicity k ≥ 2 and data are fixed in , 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class with respect to all variables.
We study the solvability of equations associated with a complex vector field in with or coefficients. We assume that is elliptic everywhere except on a simple and closed curve . We assume that, on , is of infinite type and that vanishes to a constant order. The equations considered are of the form , with satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of is , the equation is solvable in the category but not in the category....
We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove the existence of strong solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution.
A model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered and the existence and uniqueness theorem of the classic solution is proved.
Solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates, and full von Kármán systems are treated. The viscoelasticity can have the classical (``short memory'') form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.
A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain . On the boundary , the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.
A mathematical model of dissociative adsorption and associative desorption for diatomic molecules is generalized. The model is described by a coupled system of parabolic and ordinary differential equations. The existence and uniqueness theorem of the classical solution is proved.
Using a Hardy-type inequality, the authors weaken certain assumptions from the paper [1] and derive existence results for equations with a stronger degeneration.