ON SOME SUFFICIENT CONDITIONS FOR THE BLOW-UP SOLUTIONS OF THE NONLINEAR GINZBURG-LANDAU-SCHRÖDINGER EVOLUTION EQUATION

Investigation of the blow-up solutions of the problem in finite time of the first mixedvalue problem with a homogeneous boundary condition on a bounded domain of ndimensional Euclidean space for a class of nonlinear Ginzburg-Landau-Schrödinger evolution equation is continued. New simple sufficient conditions have been obtained for a wide class of initial data under which collapse happens for the given new values of parameters.


Introduction
In the present paper, the investigation of the blow-up of solutions of the problem for the first mixed-value problem of the Ginzburg-Landau-Schrödinger equation is continued.
In [8], the problem on blow-up of solutions of problem (1.1) is considered and in the case a 0 = λ 0 α − η = 0, where λ 0 is the first eigenvalue of the spectral problem (2.1), sufficient conditions on u 0 are suggested under which collapse happens for the given values of the parameters of (1.1a).The conditions on u 0 suggested in [8] are cumbersome.In the present paper, the simpler sufficient conditions on u 0 are offered under which in any value a 0 for given values of the parameters of (1.1a), the solutions of the problem (1.1) end with singularity.
The obtained results are stated in Theorems 3.1, 3.2, and 3.3.The proofs of these theorems are based on the Lemma 4.1, which is deduced from the equality for the solutions of the problem (1.1) and nontrivial solutions of the spectral problem (2.1) (Statement 4.2).

Notations
Let λ 0 be the first eigenvalue and v 0 (x) the corresponding first eigenfunction of the following problem: It is known that λ 0 > 0, v 0 (x) ∈ C 2 (Ω) ∩ C(Ω), and v 0 (x) > 0 for all x ∈ Ω (see, e.g., [13, page 434]).Without loss of the generality, we will consider that (2.2) We pass to the statement of the obtained results.

The results
We formulate the results in the form of the following three theorems.
Then the statement of Theorem 3.1 is valid, where in the case a 0 = 0; in the case a 0 = 0, t k = 1/ρχ 0 y ρ 0 .

Outline of the proof
) be the maximal solution of problem (1.1) in the sense that the interval [0,t max ) is a maximal interval of the existence of the solution for problem (1.1) from the indicated class.Clearly, t max is either finite or infinite.By proving the above stated theorems, we use the following lemma.
, let λ 0 be the first eigenvalue, and let v 0 (x) be a corresponding first eigenfunction of problem (2.1), satisfying the norm condition (2.2).On the interval [0,t max ), the following functions are defined: where z = a 0 + ib 0 .Then (1) in the case b 0 = 0 and k 1 = 0 for the function y(t) = sign(k 1 )(c 1 y 1 (t) + c 2 y 2 (t)) with the condition y 0 = sign(k 1 ) ỹ0 > 0 on the interval [0,t * ), where t * = min(t max ,t ρ ), inequality is valid: here, and χ 0 has been determined in Theorem 3.1 by formula (3.4); , the following differential inequality is valid: here χ * is determined by formula (4.3) with the condition y 0 = sign(k 1 ) ỹ0 > 0 on the interval [0,t max ), the following differential inequality is valid: where The above-mentioned lemma is proved on the ground of one suggestion.Now, we pass to the statement.
The proof of the first part of lemma is over.
Introducing the notations finally for Φ(t), we have the following expression: where Substituting this expression for Φ(t) into (4.15)for all t ∈ [0,t max ), we get the following equation: where Hence, under choosing λ = λ 0 and v(x) = v 0 (x) for all t ∈ [0,t * ), where t * = min(t max ,t ρ ) by virtue of (4.30), we conclude that y(t) strictly increases; therefore, y(t) ≥ y 0 .Further, by analogical considerations, which have been done in the proof of the first part of the lemma, we establish the following inequality: with the initial data y 0 = −sign(k 2 b 0 ) ỹ0 > 0 from which, obviously as in the proof of the first part of the lemma, one has where χ * is determined by formula (4.3) with its t ρ .The proof of the second part of the lemma is over.Lemma 4.1(3).

Proof of
, and b 0 = λ 0 β − ν = 0. Then from (4.15), we deduce that the following equation is true: with initial data y 0 = sign(k 1 ) ỹ0 > 0 from which, after separation of the variables by the well-known procedure, we conclude that for y(t), the following lower bound estimate is valid:  where R(t) = F 1/ρ (t).
We pay attention to these estimates.Function F(t) is defined and continues for all t ≥ 0. At the point t = 0, it has the value F(0) = 1.We calculate its value at the point t = t ρ .We have F(t ρ ) = 1 − y ρ 0 (1 − sinϕ)/χ, where χ has been determined in Theorem 3.1 by formula (3.3).By virtue of the condition put on y 0 , y 0 ≥ [χ/(1 − sinϕ)] 1/ρ , it follows that F(t ρ ) ≤ 0. Hence, the function F(t) in the segment [0,t ρ ] decreasingly intersects it at the unique point t k ∈ (0,t ρ ], which is the unique root in (0,t ρ ] of the trigonometric equation (Ω) → ∞ as t → t k .We obtained the contradiction as consequences of it.One has to state that t max ≤ t k , and therefore, the claim of Theorem 3.1 is true.The proof of Theorem 3.1 is over.