ABSTRACT
Thus, as in the linear case for n = 0, we study global asymptotic behavior (as t → +∞) and finite-time blow-up behavior (as t → T− < +∞) of solutions of QLSE (16). Overall, we are looking for similarity solutions of (16) of two “forward” and
Sturm’s “backward” types:
(i) global similarity patterns for t 1, and (ii) blow-up similarity ones with a finite-time behavior as t → T− < ∞. Both classes of such particular solutions of QSLE (16) are written in the
joint form as follows, by setting T = 0 in (ii):
u±(x, t) = (±t)−αf(y), y = x/(±t)β , where β = 1−αn2m , (178) for, respectively, ±t > 0, where similarity profiles f(y) satisfy the following nonlinear eigenvalue problems: in IRN ,
(NEP)± : B±n (α, f) ≡ −i (−Δ)m(|f |nf)± βy · ∇f ± αf = 0. (179)
Here, α ∈ IR is a parameter that stands, in both cases, for admitted real (!) nonlinear eigenvalues. Thus, exactly as in the linear case, the sign “+,” i.e., t > 0, corresponds to global asymptotics as t → +∞, while “−” (t < 0) yields blow-up limits t → T− = 0−, describing “micro-scale” structures of the PDE. In fact, the blow-up patterns are assumed to describe the structures of “multiple zeros” of solutions of the QLSE. As we have mentioned, this idea goes back to Strum’s analysis of solutions of the 1D heat equation performed in 1836 [377]; see [141, Ch. 1] for the whole history and applications of these fundamental Sturm’s ideas and two zero set theorems. Being equipped with proper “boundary conditions at infinity,” namely,
(global) B+n (α, f) : f(y) is “maximally” oscillatory as y → ∞, (180) (blow-up) B−n (α, f) : f(y) has a “minimal growth” as y → ∞, (181)
equations (179) produce two true nonlinear eigenvalue problems to study, which can be considered as a pair of mutually “adjoint” ones. Note that (181) actually also means that the admitted nonlinear eigenfunctions are not of a type of maximal oscillatory behavior at infinity that connects us with the issue (86) (which, however, is not sufficient, and a growth analysis at infinity should be involved in parallel, as shown below). Let us discuss, in greater detail, the meaning of those above conditions
at infinity. First, (180) means that, due to the type of nonlinearity |f |n, the oscillatory component such as (27) is admissible, with, of course, an extra generated algebraic factor of the WKBJ-type, which we do not specify hereby. This can be explained as follows: if f(y) has a standard WKBJ-type two-scale asymptotics
f(y) ∼ |y|δea|y|α as y → ∞, (182)
then, since |ea|y|α | = 1 for a ∈ i IR, substituting into (179+) yields the balance δn+ (2m− 1)(α− 1) = 1, (183)
i.e., different from the purely linear one as in (25), since, for n > 0, the exponent δ from the slower varying factor is involved (we do not calculate it here, which is a standard asymptotic procedure). Second, (181) also assumes, actually, a “minimal” growth at infinity. Namely,
quite similar to the linear problem for n = 0, the first two terms in (179)− generate a fast-growing bundle: as y → ∞ (as usual, we omit slower oscillatory components)
−i (−Δ)m(|f |nf)− β y · ∇f + ... = 0 =⇒ f(y) ∼ |y| 2mn . (184) On the other hand, two linear terms in (179)− lead to a different slower growth
... − β y · ∇f − αf = 0 =⇒ f(y) ∼ |y|−αβ ≡ |y| 2m|α|1+|α|n (185) (recall that α(0) = λ < 0). Since
2m n , (186)
this, in fact, means that (181) establishes a kind of “minimal” growth of admissible nonlinear eigenfunctions at infinity corresponding to (185). For n = 0, this implies a polynomial growth, and all the admissible (extended) eigenfunctions of B∗ turned out to be generalized Hermite polynomials (89). Note that, in self-similar approaches and ODE theory, such “minimal growth” conditions are known to define similarity solutions of the second kind; a term, which was introduced by Ya.B. Zel’dovich in 1956 [414], and many (but indeed easier) such ODE problems have been rigorously solved since. For quasilinear problems such as (179), condition (181) is incredibly more difficult. We, thus, cannot somehow rigorously justify that problem (179)−, (181) is well-posed and admits a countable family of solutions and nonlinear eigenvalues {α−γ (n)}. Actually, the homotopy deformation as n → 0+ is only our original intention to avoid such a difficult “direct” mathematical study of this nonlinear blow-up eigenvalue problem. All related aspects and notions used above and remaining unclear will be
properly discussed and specified. Of course, these conditions (180) and (181) remind us of the “linear” ones
associated with (29) for B and (86) for B∗, respectively, that were justified earlier for n = 0. Indeed, a better understanding of those conditions in the nonlinear case n > 0 demands much more difficult mathematics. However, one can observe that both (180) and (181) are just two asymptotic (not global) problems concerning admitted behaviors of solutions of (179) as y → ∞, so that, at this moment, we are in a position to neglect these and to face more fundamental issues to be addressed below. Note also that, at least in 1D or
and
for radially symmetric solutions in IRN , such asymptotic problems, for easier nonlinear ODEs, are easily solvable. Thus, for n = 0, equations (179), equipped with proper weighted L2 spaces,
take a very familiar form: the corresponding differential expressions are
B, B∗ : −i (−Δ)mf ± β y · ∇f ± αf = 0 in IRN . (187)
Then, we get the relation between α’s and λ’s from the spectrum σ(B) = σ(B∗):
for B : α = −λ+ N2m and, for B∗ : α = λ. (188)
Thus, our next goal is to show that, at least for small n > 0,
nonlinear eigenvalue problems (NEP)± admit countable sets of solutions Φ±(n) = {α±γ , f±γ }{|γ|≥0},
(189)
where, as usual and as it used to be in the linear case, γ is a multi-index in IRN to numerate the pairs. The last question to address is whether these sets
Φ±(n) of nonlinear eigenfunctions are evolutionary complete, (190)
i.e., describe all possible asymptotics as t → +∞ and t → 0−, respectively, on the corresponding compact subsets in the variable y in (178) in the CP for QLSE (16) with bounded integrable (and possibly compactly supported; any assumption is allowed) initial data. Thus, we perform a “homotopic deformation” of (16) as n → 0+ and reduce it to our linear equation (1), for which problems (189) and (190) are solved positively by non-self-adjoint spectral theory of the linear operator pair {B,B∗}.
