ABSTRACT

In order to extend application of our final approach, we next briefly discuss mKS-type equations with a third-order (also odd) nonlinear perturbation (dispersion) of the form

vt = −(−Δ)mv −ΔB1(|v|p) in IRN × IR+ (m ≥ 2). (113) Writing this PDE in a pseudo-parabolic form,

Pvt = (−Δ)m−1v +B1(|v|p), where P = (−Δ)−1 > 0, (114) and multiplying by v in L2(IRN ), we observe that, instead of a uniform L2bound, we are given an a priori H−1-bound: for uniformly bounded data

v0 ∈ L∞(IRN ) ∩H2m(IRN ), (115) the following holds:

‖v(t)‖−1 ≤ ‖v0‖−1 for t > 0. (116) Here, for simplicity, we assume that v0(x) also has exponential decay at infinity, so v(x, t) does for t > 0. In (114) and later on, we define (−Δ)−1w = g in a standard manner:

Δg = −w in IRN , g(x) → 0 as x → ∞. (117) For the solvability of this problem, we shall always assume that

w dx = 0. (118)

Clearly, this property holds for the divergence equation (113) with exponentially decaying solutions.