LAUSR

Let {A, E} be the elliptic complex on a n-dimensional smooth closed Riemannian manifold X with the first order differential operators A and smooth vector bundles E over X. We consider nonlinear operator equations, associated with the parabolic differential operators ∂t + ∆, generated by the Laplacians ∆ of the complex {A, E}, in special Bochner-Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frechét derivative A i of the induced nonlinear mapping is continuously invertible and the map Ai is open and injective in chosen spaces. Introduction Let X be a Riemannian n-dimensional smooth compact closed manifold and E be smooth vector bundles over X . Denote by C Ei(X) the space of all smooth sections of the bundle E. Consider an elliptic complex of the first order differential operators A on X , 0 −→ C E0(X) A −−→ C E1(X) A −−→ · · · A −−−−→ C EN (X) −→ 0. (0.1) where A ◦ A ≡ 0. In this case it is equivalent to say that the Laplacians ∆ = (A)A + A(A), i = 0, 1, . . . , N , of the complex are the second order strongly elliptic differential operators on X where operator (A) is formal adjoint to A (see (1.1) below). For i < 0 and i ≥ N we assume that A = 0. Inspired by typical nonlinear problems of the Mathematical Physics, see, for instance [14], [27], we consider a family of nonlinear parabolic equations, associated with the complex {A, E}. With this purpose, we denote by Mi,j two bilinear bi-differential operators of zero order (see [5] or [25]), Mi,1(·, ·) : (E , E) → E, Mi,2(·, ·) : (E , E) → E. (0.2) We set for a differentiable section v of the bundle E N (v) =:Mi,1(A v, v) +AMi,2(v, v). (0.3) Let now XT be a cylinder, XT = X × [0, T ], where the time T > 0 is finite. Then, for any fixed positive number μ the operators ∂t + μ∆ i are parabolic on X × (0,+∞) (see [7]). Consider the following initial problem: given section f of the induced bundle E(t) (the variable t enters into this bundle as a parameter) and section u0 of the 2010 Mathematics Subject Classification. 58J10, 35K45.