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Blowup criteria for the inhomogeneous nonlinear Schrödinger equation
Journal of Inequalities and Applications volume 2014, Article number: 55 (2014)
Abstract
In this paper, using the variational characteristic of the virial identity and a new estimate of the kinetic energy, we obtain a new sufficient condition for the existence of blowup solutions.
MSC:35Q55, 35B44.
1 Introduction
In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation (INSE)
where {i}^{2}=1; \mathrm{\u25b3}={\sum}_{j=1}^{N}\frac{{\partial}^{2}}{\partial {x}_{j}^{2}} is the Laplace operator in {\mathbb{R}}^{N}; u=u(t,x): [0,T)\times {\mathbb{R}}^{N}\to \mathbb{C} is the complex valued function and 0<T\le +\mathrm{\infty}; the parameter b\ge 0 and 2<p<\tilde{p} (we use the convention: \tilde{p}=+\mathrm{\infty} for N=2, \tilde{p}=\frac{2N}{N1}+\frac{2b}{N1} for N\ge 3); N\ge 2 is the space dimension. A few years ago, it was suggested that stable high power propagation can be achieved in a plasma by sending a preliminary laser beam that creates a channel with a reduced electron density, and thus reduces the nonlinearity inside the channel (see [1, 2]). In this case, the beam propagation can be modeled by the inhomogeneous nonlinear Schrödinger equation in the following form:
Recently, this type of inhomogeneous nonlinear Schrödinger equations has been widely investigated. When {k}_{1}\le K(x)\le {k}_{2} with {k}_{1},{k}_{2}>0 and p=2+\frac{4}{N}, Merle [3] proved the existence and nonexistence of blowup solutions to the Cauchy problem (1.3). When K(x)=K(\epsilon x)\in {C}^{4}({\mathbb{R}}^{N})\cap {L}^{\mathrm{\infty}}({\mathbb{R}}^{N}) with small ε and p=2+\frac{4}{N}, Fibich, Liu and Wang [2, 4] obtained the stability and instability of standing waves to the Cauchy problem (1.3).
We recall some known results on the blowup solutions for the classical nonlinear Schrödinger equation
Ginibre and Velo [5] showed the local wellposedness in {H}^{1}({\mathbb{R}}^{N}). Glassey [6] showed the existence of blowup solutions when the energy is negative and x{v}_{0}\in {L}^{2}({\mathbb{R}}^{N}). Ogawa and Tsutsumi [7] obtained the existence of blowup solutions in radial case without the restriction x{v}_{0}\in {L}^{2}({\mathbb{R}}^{N}). Weinstein [8] and Zhang [9] obtained the sharp conditions of global existence for critical and supercritical nonlinearity. Merle and Raphaël [10] showed the existence of blowup solutions without x{v}_{0}\in {L}^{2}({\mathbb{R}}^{N}) for p=2+\frac{4}{N}. Lushnikov [11] and Holmer et al. [12] obtained some sufficient conditions for existence of blowup for p=4 and N=3 basing on an estimate of the kinetic energy.
In this paper, we study blowup criteria for the Cauchy problem (1.1)(1.2) with p>\frac{2N+2b+4}{N}, where the nonlinearity x{}^{b}u{}^{p2}u includes an unbounded potential x{}^{b}. We note that (1.1) has scaling: {u}_{\lambda}(t,x)={\lambda}^{\frac{2+b}{p2}}u({\lambda}^{2}t,\lambda x) is a solution if u(t,x) is a solution. The scaleinvariant Lebesgue norm for this equation is {L}^{{p}_{c}}norm, where {p}_{c}=\frac{N(p2)}{2+b}. Since p>\frac{2N+2b+4}{N}, we have {p}_{c}=\frac{N(p2)}{2+b}>2 and we may call (1.1) a class of Schrödinger equations with {L}^{2}super critical nonlinearity. Chen and Guo [13] and Chen [14] showed the local wellposedness of the Cauchy problem (1.1)(1.2) in {H}_{r}^{1}={H}_{r}^{1}({\mathbb{R}}^{N}), where {H}_{r}^{1}({\mathbb{R}}^{N}) is the set of radially symmetric functions in {H}^{1}({\mathbb{R}}^{N}). Moreover, u(t,x) satisfies the following conservation laws:
and
Chen and Guo [13] also showed the sharp conditions of blowup and global existence of solutions to the Cauchy problem (1.1)(1.2) by the crossconstrained variational arguments. On the other hand, letting J(t):={\int}_{{\mathbb{R}}^{N}}x{}^{2}u(t,x){}^{2}\phantom{\rule{0.2em}{0ex}}dx, this can be interpreted as the average width of the initial distribution u. It follows from Chen and Guo’s results in [13] that we have the following proposition.
Proposition 1.1 Assume that {u}_{0}\in {H}_{r}^{1}, x{u}_{0}\in {L}^{2} and the corresponding solution u(t,x) of the Cauchy problem (1.1)(1.2) on the interval [0,T). Then, for all t\in [0,T), one has J(t):=\int x{}^{2}u(t,x){}^{2}\phantom{\rule{0.2em}{0ex}}dx<+\mathrm{\infty},
and
In the case Np2N2b4\ge 0, it follows from the last relation that
and for E({u}_{0})<0 the positivedefinite quantity J(t) becomes negative over a finite time by virtue of the above inequality. This means that a singularity appears in the solution of the given INSE. Indeed, applying Weinstein’s arguments [8] and the classical analysis identity
one has the following theorem (see also Chen and Guo [13]).
Theorem 1.2 Let N\ge 2, 0<b<N2 and \frac{2N+2b+4}{N}<p<\tilde{p} (where \tilde{p}=+\mathrm{\infty} for N=2, \tilde{p}=\frac{2N}{N2}+\frac{2b}{N1} for N\ge 3). Assume {u}_{0}\in {H}^{1}({\mathbb{R}}^{N}) and x{u}_{0}\in {L}^{2}({\mathbb{R}}^{N}) is radially symmetric. If the initial data satisfies either

(i)
E({u}_{0})<0,(1.8)

(ii)
E({u}_{0})=0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{J}^{\prime}(0)<0,(1.9)

(iii)
E({u}_{0})>0 and
{J}^{\prime}(0)<2\sqrt{2(Np2N2b)E({u}_{0})}J(0),(1.10)
then there exists 0<T<+\mathrm{\infty} such that the corresponding solution u(t,x) blows up in finite time T.
We remark that in the case E({u}_{0})>0, Np2N2b4\ge 0 both collapse and spreading of the initial disturbance are possible. Although the INSE is no longer applicable near the formation point of a singularity and dissipative or some other limiting mechanism come to play. It is very important to be able to predict the presence or absence of collapse for different classes of initial conditions. The sufficient conditions for existence of blowup solutions are given in [13] if either E({u}_{0})<0 or {J}^{\prime}(0)<0. A natural question arises whether there is a sufficient condition for existence of blowup solutions with E({u}_{0})>0 and {J}^{\prime}(0)>0.
In the present paper, motivated by the studies of the classical nonlinear Schrödinger equation (see [8, 11, 12]), we use variational characteristic of secondorder derivatives of the virial identity to catch up with the information of {\parallel \mathrm{\nabla}u\parallel}_{{L}^{2}}, and we obtain a new sufficient condition for the existence of blowup solutions to the inhomogeneous nonlinear Schrödinger equation (1.1). More precisely, let
Then we have the following theorem.
Theorem 1.3 Let N\ge 2, 0<b<N2 and \frac{2N+2b+4}{N}<p<min\{\frac{2b+2N+12}{N},\tilde{p}\}. Assume that {u}_{0}\in {H}^{1}({\mathbb{R}}^{N}) and x{u}_{0}\in {L}^{2}({\mathbb{R}}^{N}) is radially symmetric. If
where g(x) is defined by (1.11), then there exists 0<T<+\mathrm{\infty} such that the corresponding solution u(t,x) blows up in finite time T.
2 Notations and preliminaries
In this paper, we denote {L}^{q}({\mathbb{R}}^{N}), {\parallel \cdot \parallel}_{{L}^{q}({\mathbb{R}}^{N})}, {H}^{s}({\mathbb{R}}^{N}) and {\int}_{{\mathbb{R}}^{N}}\cdot \phantom{\rule{0.2em}{0ex}}dx by {L}^{q}, {\parallel \cdot \parallel}_{{L}^{q}}, {H}^{s} and \int \cdot \phantom{\rule{0.2em}{0ex}}dx, respectively. ℜz and ℑz are the real part and imaginary part of the complex number z, respectively. \overline{z} is denoted the complex conjugate of the complex number z. The various positive constants will be simply denoted by C.
For the Cauchy problem (1.1)(1.2), the space we work in is
which is a Hilbert space. Moreover, we define the energy functional E(u(t)) in {H}_{r}^{1} by
The functional E(u) is welldefined according to the Sobolev embedding theorem (see [15]). Chen and Guo [13] and Chen [14] showed the local wellposedness for the Cauchy problem (1.1)(1.2) in {H}_{r}^{1}, as follows.
Proposition 2.1 Let N\ge 2, b\ge 0 and 2+2b/(N1)<p<\tilde{p} (where \tilde{p}=+\mathrm{\infty} for N=2, \tilde{p}=\frac{2N}{N2}+\frac{2b}{N1} for N\ge 3). For any {u}_{0}\in {H}_{r}^{1}, there exists a unique solution u(t,x) of the Cauchy problem (1.1)(1.2) on the maximal time interval [0,T) such that u(t,x)\in C([0,T);{H}_{r}^{1}) and either T=+\mathrm{\infty} (global existence), or T<+\mathrm{\infty} and {lim}_{t\to T}{\parallel u(t,x)\parallel}_{{H}_{r}^{1}}=+\mathrm{\infty} (blowup). Furthermore, for all t\in [0,T), u(t,x) satisfies the following conservation laws:

(i)
Conservation of mass: {\parallel u(t)\parallel}_{2}={\parallel {u}_{0}\parallel}_{2}.

(ii)
Conservation of energy: E(u(t,x))=E({u}_{0}).
In addition, by some basic calculations, we have the following lemma, which gives further insight in the dynamic criterion for collapse proposed by Lushnikov in [11].
Lemma 2.2 If V(t)>0 is the positive solution of the following differential equation:
and there exists 0<{T}_{1}<+\mathrm{\infty} such that {lim}_{t\to {T}_{1}}V(t)=0, then for the solution \tilde{V}>0 of the following differential equation:
there exists 0<{T}_{2}\le {T}_{1}<+\mathrm{\infty} such that {lim}_{t\to {T}_{2}}\tilde{V}(t)=0.
Proof Since the function {h}^{2}(t) is nonpositive, which pulls \tilde{V}(t) to zero more quickly than V(t) (see also [11]), one sees that the conclusion in Lemma 2.2 is true by the classical analysis identity (1.7). □
3 Proof of Theorem 1.3
Since x{u}_{0}\in {L}^{2}, we have xu(t,x)\in {L}^{2} by the local wellposedness. Taking J:=J(t)=\int x{}^{2}u{}^{2}\phantom{\rule{0.2em}{0ex}}dx, by Proposition 1.1, we get {J}^{\prime}(t)=4\mathrm{\Im}\int x\cdot \mathrm{\nabla}u\overline{u}\phantom{\rule{0.2em}{0ex}}dx and
It follows from some calculations that
Then we get \frac{N}{2}{\parallel u\parallel}_{{L}^{2}}^{2}\le {\parallel xu\parallel}_{{L}^{2}}{\parallel \mathrm{\nabla}u\parallel}_{{L}^{2}}, and by the fact that z{}^{2}=\mathrm{\Re}z{}^{2}+\mathrm{\Im}z{}^{2} we deduce that
Injecting (3.2) into (3.1), by the conservation laws, we deduce that
Letting J={A}^{\frac{8}{Np2N2b+4}} and rewriting (3.3) to remove the last term with {J}_{t}^{2}, we have
which has a simple mechanism analogy. Multiplying {A}_{t} in (3.4) and integrating with the time variable t, we get the corresponding mechanical energy
where
Restricting ourselves to the case E>0, and according to the assumptions on p, b and N we see that U(A) achieves its maximum {U}_{max} at {A}_{max} with
and
To facilitate the rest of the analysis, we introduce a rescaling. Define \tilde{J}(s) and \tilde{\epsilon}(s) by the relations
Thus, by (3.4), \tilde{A} satisfies the following differential inequality:
Applying Lemma 2.2, if we show that for the positive solution of the following differential equation:
there exists a time 0<{s}_{1}<+\mathrm{\infty} and {lim}_{s\to {s}_{1}}\tilde{A}(s)=0, then for the positive solution of the differential inequality (3.6), there exists a time 0<{s}_{2}\le {s}_{1}<+\mathrm{\infty} and {lim}_{s\to {s}_{2}}\tilde{A}(s)=0. Indeed, setting
we see that (3.5) converts to
It is obvious that the maximum of \tilde{U}(\tilde{A}(s)) is 1, which is attained by the maximum at \tilde{A}(s)=1. By the variational characteristic of \tilde{U}(\tilde{A}(s)), we claim that under one of the following conditions, \tilde{A}(s) vanishes in finite time, so does J(t) (which implies the solution u(t,x) blows up in finite time):

(a)
\tilde{\epsilon}(0)<1 and \tilde{A}(0)<1,

(b)
\tilde{\epsilon}(0)\ge 1 and {\tilde{A}}_{s}(0)<0.
Indeed, it follows from (3.7) and (3.9) that \tilde{\epsilon}(s)=\tilde{\epsilon}(0). (a) If \tilde{\epsilon}(0)<1, then \tilde{\epsilon}(t)<1 and \tilde{U}(\tilde{A}(s))<1 for t\in I (maximal existence interval). By the assumption \tilde{A}(0)<1, we deduce that \tilde{A}(s)<1 for t\in I. Moreover, we have
Using (3.7) and the classical analysis identity
we see that \tilde{A}(s) vanishes in finite time. On the other hand, for the second case (b), if \tilde{\epsilon}(0)\ge 1, then \tilde{\epsilon}(t)\ge 1. If follows from (3.9) that \tilde{U}(\tilde{A}(s))=1, which implies that \tilde{A}(s)=1 and
Using (3.7) and the classical analysis identity (3.10), we deduce that if {\tilde{A}}_{s}(0)<0, then \tilde{A}(s) vanishes in finite time. This completes the proof of claim (a) and (b). Now, we return to the proof of Theorem 1.3. If we define \tilde{J}={\tilde{A}}^{\frac{8}{Np2b2N+4}}, then (3.9) is equal to
Taking
we see that
Thus condition (a) is true if and only if
and condition (b) is true if and only if {\tilde{J}}_{s}(0)<h(\tilde{J}(0)). Collecting the above two conditions, we deduce that the solution u(t,x) blows up in finite time 0<T<+\mathrm{\infty} provided
Finally, substituting back J(t), we see that
which implies that (3.12) is equivalent to
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Acknowledgements
This work is supported partly by the National Natural Science Foundation of P.R. China grants 11226162 and 11371267.
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HY participated in the design of the study, SZ studied the virial identity, participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
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Yang, H., Zhu, S. Blowup criteria for the inhomogeneous nonlinear Schrödinger equation. J Inequal Appl 2014, 55 (2014). https://doi.org/10.1186/1029242X201455
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DOI: https://doi.org/10.1186/1029242X201455
Keywords
 inhomogeneous nonlinear Schrödinger equation
 blowup criteria
 virial identity