On the solvability conditions for the diffusion equation with convection terms

Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

“…, which was established in Lemma A3 of [20] under our assumptions on the potential function. Via (1.10) these perturbed plane waves satisfy homogeneous problem (1.7) when the wave vector k belongs to the sphere in three dimensions centered at the origin of radius √ a.…”

“…Note that the first one contains conditions on V (x) analogous to those stated in Assumption 1.1 of [18] (see also [19], [20]). …”

“…[9]), such that neither of these operators has a finite dimensional kernel and a closed image. Solvability conditions for operators of that kind have been studied extensively in recent articles for a single Schrödinger type operator (see [18]), the sums of second order differential operators (see [19]), the Laplacian operator with the drift term (see [20]). Non Fredholm operators arise as well while studying the existence and stability of stationary and travelling wave solutions of certain reaction-diffusion equations (see e.g.…”

“…As a consequence, the operator does not satisfy the Fredholm property, and solvability conditions of equation (1.4) are not known. We will derive solvability conditions for this equation using the technique developed in our preceding articles [18], [19], [20], [21], [22] and [23]. This method is based on the spectral decomposition of self-adjoint operators.…”

“…By means of the spectral theorem these functions form the complete system in L 2 (R 3 ). We introduce the following Sobolev type space (see also [20], [21])…”