In this section, L is the elliptic Waldenfels operator as defined in (1).
Proof: Define υ t, x) := u t, x + ct) then (1) may be reformulated as Let M′ > max{ M, sup x u0 x)} and let z = z t, x) be the solution of Since M′ is a supersolution of the elliptic operator at the right-hand side of the differential equation, we know that z t < 0 (see, for instance Sattinger, 1973, p. 33).
The authors consider the bifurcation problem F ( x, λ ) = L x + ( λ − λ 0 ) x + R ( x ) = 0, where L is the elliptic self-adjoint operator on a suitable Banach space Y of functions, with another suitable Banach space of function X - the domain of L ⊂ Y, R : X → Y is a smooth map with R ( 0 ) = 0 and R ′ ( 0 ) = 0, λ 0 - is an eigenvalue of L of multiplicity n, x ∈ X and λ ∈ R.
A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C0-semigroup on Lp is established.
For the boundary value problem in Q = R n × ( 0, T ) : u t ( x, t ) + A u ( x, t ) = 0, u | t = 0 = 0, where A is the linear elliptic operator of orders 2p, the following classes of uniqueness are established: u ( x, t ) ≤ C ⋅ exp { k | x | 2 p 2 p − 1 } (Ladyzhenskaya [9] for one equation with coefficients depending only on t).
The elliptic operator is pseudomonotone because of hypotheses (A1), (A2), and (F1), and in view of Lemma 3.4 the operators and are bounded and pseudomonotone as well.
We also show that [IU] implies the condition [SP] (i.e., the constant function 1 is a small perturbation of the elliptic operator on D).
The leading coefficients of the elliptic operator are VMO functions.
