Abstract
In this note, we consider a Schrödinger evolution equation with a power nonlinearity \(i |u|^\alpha u\) and a viscous damping term \(\nu \Delta u\). Then, we demonstrate that the Cauchy problem always admits the global existence of classical solutions with finite mass. Moreover, we can also observe that our proof is applicable for a nonlinear complex Ginzburg–Landau equation.
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1 Introduction
In this note, we consider the nonlinear Schrödinger equation with viscous damping
on \({\mathbb {R}}^d\), where \(\alpha , \nu >0\) and \(u_0\) is a prescribed \({\mathbb {C}}\)-valued function on \({\mathbb {R}}^d\). We shall seek for an unknown function \(u=u(t,x): [0,\infty ) \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\) governed by the Cauchy problem (1.1). Here, it should be noted that equation (1.1) is a particular case of the complex Ginzburg–Landau equation
where \(\kappa , \lambda , \mu , \gamma \in {\mathbb {R}}\) (see e.g. [1, 13]).
In the case of \(\nu =0\) in (1.1), some solutions of the purely nonlinear Schrödinger equation blow up in finite time under suitable assumptions on \(\alpha \) and \(u_0\). As is well known, if \(\alpha < 4/(d-2)_+ \), then the Cauchy problem is locally well-posed in the energy space \(H^1({\mathbb {R}}^d)\). In particular, if \(\alpha < 4/d\), then we obtain the maximal existence time \(T_* = + \infty \) for every \(u_0 \in H^1({\mathbb {R}}^d)\). On the other hand, if \(4/d \le \alpha < 4/(d-2)_+ \), then \(T_*=T_*(u_0) < \infty \) for every initial value \(u_0\) satisfying \(|x| u_0 \in L^2({\mathbb {R}}^d)\) and
That is, we have that \(\limsup _{t \uparrow T_*} ||u(t)||_{H^1} =+\infty \) (see for instance [6]). Later on, Cazenave et. al. [3, 4] extended the blowup result to the Ginzburg–Landau equation (1.2) in the case when \(\kappa =\mu \) and \(\nu = \lambda \) :
Our purpose of this note is to show that the Cauchy problem (1.1) always admits the global existence of classical solutions with finite mass due to the viscous damping term. More precisely, we shall establish the following result.
Theorem 1.1
Given any \(u_0 \in L^\infty ({\mathbb {R}}^d) \cap L^2({\mathbb {R}}^d)\), there exists a unique function \(u \in C_w([0,\infty ), L^\infty ({\mathbb {R}}^d)) \cap C([0,\infty ), L^2({\mathbb {R}}^d))\) with \(u|_{t=0}=u_0\),which is a classical solution of equation (1.1) on \((0,\infty ) \times {\mathbb {R}}^d\).
It seems that the standard strategy of compactness or monotonicity method is not available for deriving the global existence of such strong solutions at least when \(\alpha > 2d/(d-2)_+\) and \(\nu \) is sufficiently small (cf. [7,8,9, 11, 12]). Therefore, we need another approach to prove Theorem 1.1.
The outline of this note is as follows. In Sect. 2, we collect some preliminary results for demonstrating Theorem 1.1. In Sect. 3, we complete the proof of Theorem 1.1.
2 Preliminaries
Let \(A_\nu \varphi := (\nu +i) \Delta \varphi \) with domain \(H^2({\mathbb {R}}^d)\). Then, it is well known that \(A_\nu \) generates an analytic semigroup \((e^{t A_\nu })_{t \ge 0}\) on \(L^2({\mathbb {R}}^d)\), which is given by the explicit representation
In addition, \((e^{t A_\nu })_{t \ge 0}\) is a strongly continuous semigroup on \(L^p({\mathbb {R}}^d)\) for \(1 \le p < \infty \). Moreover, there is a constant \(C=C(d,\nu )>0\) such that
and
for all \(t >0\), where \(1 \le p \le r \le \infty \).
Our problem (1.1) can be converted into the integral equation
for an unknown function \(u=u(t)=u(t,\cdot ): [0,\infty ) \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\). By applying the standard fixed point argument to (2.4), we can immediately obtain the local-in-time solvability of (1.1) on \(L^\infty ({\mathbb {R}}^d)\). More precisely, we have the following.
Proposition 2.1
There is a constant \(\varepsilon _0 = \varepsilon _0(d,\nu , \alpha ) >0\) such that for every \(u_0 \in L^\infty ({\mathbb {R}}^d)\), there exist a unique function \(u \in C_w([0,T], L^\infty ({\mathbb {R}}^d))\) for a time \(T \ge \varepsilon _0/||u_0||_{L^\infty }^\alpha \) with \(u(0)=u_0\), which is a classical solution of equation (1.1) on \((0,T) \times {\mathbb {R}}^d\).
In particular, for any \(T_0 \le T/2\), there is a constant \(C=C(d,\nu , \alpha ,T_0, M(T))>0\) with \(M(T):= \sup _{0 \le t \le T} ||u(t)||_{L^\infty }\) such that
Moreover, if in addition \(u_0 \in L^2({\mathbb {R}}^d)\), then \(u \in C([0,T], L^2 ({\mathbb {R}}^d))\) and
for all \(t \in [0, T]\).
Remark 2.1
By standard parabolic regularity theory, we know that the resulting solution u is a classical solution in \(C((0,T),C^3({\mathbb {R}}^d)) \cap C^1((0,T),C({\mathbb {R}}^d))\) with \(\nabla \partial _t u \in C ((0,T) \times {\mathbb {R}}^d)\). However, for small \(\alpha >0\), say, \(0< \alpha < 1\), one cannot expect these solutions to be in \(C((0,T),C^4({\mathbb {R}}^d))\) (cf. [5, Theorem 3.2 and A.1]).
Proof
For the uniqueness part, see [2, Lemma 9]. The estimate (2.5) is derived by applying the \(L^\infty \) smoothing estimate in (2.3) to the Cauchy problem (1.1) with the initial condition \(u|_{t=T-2T_0} = u(T-2T_0)\) via a singular Gronwall’s lemma [6, Lemma 8.1.1]. We omit the details of the proof of Proposition 2.1\(\square \)
Let us denote the Newtonian potential on \({\mathbb {R}}^d\) by
for a function \(f : {\mathbb {R}}^d \rightarrow {\mathbb {R}}\), where E(x) is the fundamental solution of the Laplacian \(\Delta \) on \({\mathbb {R}}^d\), i.e.,
Here, \(|{\mathbb {S}}^{d-1}|\) denotes the surface area of the unit ball in \({\mathbb {R}}^d\). It is well known that there are constants \(C=C(d, r, \rho )>0\) and \(C'=C'(d, r, \rho )>0\) such that
and
for any \(f \in L^r({\mathbb {R}}^d)\) supported on \(B(z ; \rho )\) (cf. for instance [10, Theorem 10.2]). Here, \(B(z; \rho )\) denotes the open ball centered at \(z \in {\mathbb {R}}^d\) of radius \(\rho > 0\).
Let us recall a dyadic partition of unity of Littlewood–Paley type: there exists a nonnegative, smooth, spherically symmetric function \(\varphi : {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) supported on the annulus \(\{2^{-1} \le |x| \le 2 \}\) such that
Let \(\phi = \{\phi ^n_z(x) \}_{z \in {\mathbb {R}}^d}^{n \in {\mathbb {Z}}}\) denote a family of localization functions on \({\mathbb {R}}^d\) given by
Note that \(\phi ^n_z=1\) on \(B(z; 2^{-n})\) and \(\phi ^n_z = 0\) on \({\mathbb {R}}^d \backslash B(z ; 2^{-n+1})\). Moreover, we have the generalization \(\phi = \{\phi ^n_z(x) \}_{z \in {\mathbb {R}}^d}^{n \in {\mathbb {R}}}\) via interpolation. For \(f \in L^1_{\mathrm {loc}}({\mathbb {R}}^d)\), let us introduce the operator
for \(x \in {\mathbb {R}}^d\). Then, we have the following identities
and
for \(f \in C^2({\mathbb {R}}^d)\). Moreover, we can observe from integration by substitution the following identities
and
with \(f_z^{-n} (x) := f(z + 2^{-n} x)\) for all \(f \in C^1({\mathbb {R}}^d)\).
3 Proof of Theorem 1.1
In this section, we prove Theorem 1.1
Proof of Theorem 1.1
We argue by contradiction. Suppose that there exist some initial data \(u_0 \in L^\infty ({\mathbb {R}}^d) \cap L^2({\mathbb {R}}^d)\) such that the solution u obtained in Proposition 2.1 develops singularities at \(t=T_* < \infty \), that is,
Thus we can find a sequence of \((t_k)_{k \ge 1} \subset (0,T_*)\) with \(t_k \uparrow T_*\) such that
with an associated sequence \((m_k)_{k \ge 1} \in {\mathbb {N}}\).
Let us introduce the function
Since the identity (2.11) gives
with \(q^{(n)}_{\,z}(t) := \Delta ^{-1} (\phi ^n_z \Delta q(t))\), we deduce that
We thus estimate
for all \((n,z,t) \in {\mathbb {R}}^+ \times {\mathbb {R}}^d \times (0,T_*)\), where we used the bound \(|| u(t) ||_{L^2} \le || u_0 ||_{L^2}\) from (2.6) in the last inequality. Similarly, by using (2.14) and (2.9) with \(r=1/\theta \), we get
for every \((n,z,t) \in {\mathbb {R}}^+ \times {\mathbb {R}}^d \times (0,T_*)\). Hence, it follows from (3.4) that
for any \((n,z,t) \in {\mathbb {R}}^+ \times {\mathbb {R}}^d \times (0,T_*)\).
For the time sequence \((t_k)_{k \ge 1}\) in (3.2), let us choose a point \(x_k \in {\mathbb {R}}^d\) such that \(|u(t_k,x_k)| > 2^{m_k -1}\) for each \(k \ge 1\). Then for any sequence \((n_k)_{k \ge 1} \uparrow + \infty \), we have that
Let \(M_k:= 2^{m_k \alpha /2}\) and let us rescale the blowup solution u at \((t_k, x_k) \in (0,T_*) \times {\mathbb {R}}^d\) by
By our construction, \(U_{(k)}\) is a function defined on the rescaled interval \([-M_k^2 t_k, 0]\) such that
Furthermore, by the scaling invariance of equation (1.1), the function \(U_{(k)}\) is a solution of the Cauchy problem (1.1) on \([-M_k^2 t_k, 0] \times {\mathbb {R}}^d\) for any \(k \ge 1\). Then, we can deduce from the \(L^\infty \) estimate (2.5) that
for all \(k \gg 1\) with a constant \(C_\alpha =C(d, \nu , \alpha )>0\).
Define the function
By differentiation, we have
for all \(k \gg 1\). Furthermore, we see that the function
(cf. (3.3)) satisfies
since \(0< \theta < 1/d\) and \(2(1-\theta ) >1\).
We are now a position to complete the proof of Theorem 1.1. Since (3.6) implies
with \(n_k:= m_k(\alpha +2)/2\), we deduce from the inequality (3.5) that
On the other hand, we observe that
for all \(k \gg 1\), since (3.11). This contradicts (3.12). Hence, we have proved Theorem 1.1. \(\square \)
We finish this note to state that our argument is available to establish the global existence of classical solutions for the following complex Ginzburg–Landau equation
and furthermore, for the derivative-type nonlinear Schrödinger equation with viscous damping
Theorem 3.1
Consider the Cauchy problem for the equation (3.13) (resp. (3.14)) on \({\mathbb {R}}^d\). Given any \(u_0 \in L^\infty ({\mathbb {R}}^d) \cap L^2({\mathbb {R}}^d)\), there exists a unique function \(u \in C_w([0,\infty ), L^\infty ({\mathbb {R}}^d)) \cap C([0,\infty ), L^2({\mathbb {R}}^d))\) with \(u|_{t=0}=u_0\), which is a classical solution of equation (3.13) (resp. (3.14)) on \((0,\infty ) \times {\mathbb {R}}^d\).
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Hirata, D. Global existence of a nonlinear Schrödinger equation with viscous damping. J. Evol. Equ. 22, 39 (2022). https://doi.org/10.1007/s00028-022-00800-y
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DOI: https://doi.org/10.1007/s00028-022-00800-y