Wellposedness and Scattering for Some Bi-inhomogeneous Schrödinger–Choquard Equation with Linear Damping

Abstract

Global existence and scattering are proved for some damped bi-inhomogeneous Schrödinger equation of Choquard type whenever the damping coefficient is large enough. For arbitrary damping, global existence of the solutions is claimed if the initial data belongs to some stable set.

Appendix

According to [25], see Appendix A, we have the so called Riesz potential inequality.

Lemma 7

Let $N\ge 1, q>1, 0<\alpha <\frac{N}{q}$ and $\frac{1}{r}=\frac{1}{q}-\frac{\alpha }{N}$. Then, $I_\alpha :L^q(\mathbb {R}^N)\rightarrow L^r(\mathbb {R}^N)$ is a bounded operator. Precisely, there exists $C_{N,\alpha ,q}>0$ such that

$$\begin{aligned} \Vert I_\alpha *f\Vert _r \le C_{N,\alpha ,q}\Vert f\Vert _q. \end{aligned}$$

Proof of Proposition 5

Elementary computations gives

$$\begin{aligned} 0<r<\frac{2N}{N-2}\Leftrightarrow 0<N\Bigg (\frac{1}{2}-\frac{1}{r}\Bigg )<1. \end{aligned}$$

Then

$$\begin{aligned} \Bigg \Vert \int _0^tU_\gamma (t-s)f(s)ds\Bigg \Vert _{L_T^\theta (L^r(|x|<1))}&\lesssim \Bigg \Vert \int _0^t e^{-\gamma (t-s)}U_0(t-s)f(s)ds\Bigg \Vert _{L_T^\theta (L^r(|x|<1))} \\&\lesssim \Bigg \Vert \int _0^T \frac{1}{|t-s|^{\frac{1}{\theta }+\frac{1}{\mu }}}\Vert f(s)\Vert _{r^\prime (|x|<1)} ds\Bigg \Vert _{L_T^\theta }\\&\lesssim \Bigg \Vert \int _0^T I_{\frac{1}{\mu ^\prime }-\frac{1}{\theta }}*\Vert f(s)\Vert _{r^\prime (|x|<1)} ds\Bigg \Vert _{L_T^\theta }. \end{aligned}$$

Applying Lemma 7 with $d=1$ and taking account of $\frac{1}{\theta }+\frac{1}{d}(\frac{1}{\mu ^\prime }-\frac{1}{\theta })=\frac{1}{\mu ^\prime }$, we obtain

$$\begin{aligned} \Bigg \Vert \int _0^tU_\gamma (t-s)f(s)ds\Bigg \Vert _{L_T^\theta (L^r(|x|<1))} \lesssim \Vert f\Vert _{L_T^{\mu ^\prime }(L^{r^\prime }(|x|<1))}. \end{aligned}$$

Similarly for integrals on $|x|>1$.

Proof of Corollary 3

If $i\dot{u}+\triangle u+i\gamma u=f$ with data $u_0$, then by using Proposition 3 and standard Strichartz estimates, one gets

$$\begin{aligned} \Vert u\Vert _{L_T^q(L^r)}&\lesssim \Vert U_\gamma (t)u_0\Vert _{L_T^q(L^r)}+ \Bigg \Vert \int _0^tU_\gamma (t-s)f(s)ds\Bigg \Vert _{L_T^q(L^r)}\\&\lesssim \Vert e^{-\gamma t}U_0(t)u_0\Vert _{L_T^q(L^r)}+\Bigg \Vert \int _0^te^{-\gamma (t-s)}U_0(t-s)f(s)ds\Bigg \Vert _{L_T^q(L^r)}\\&\lesssim \Vert e^{-\gamma t}U_0(t)u_0\Vert _{L_T^q(L^r)}+\Bigg \Vert \int _0^tU_0(t-s)(e^{-\gamma (t-s)}f(s))ds\Bigg \Vert _{L_T^q(L^r)} \\&\lesssim \Vert e^{-\gamma t}U_0(t)u_0\Vert _{L_T^q(L^r)}+ \Vert e^{-\gamma (t-s)}f(s)\Vert _{L_T^{\tilde{q}^\prime }(L^{\tilde{r}^\prime })} \\&\lesssim \Vert e^{-\gamma t}\Vert U_0(t)u_0\Vert _r\Vert _{L_T^q}+\Vert e^{-\gamma (t-s)}\Vert f(s)\Vert _{\tilde{r}^\prime }\Vert _{L_T^{\tilde{q}^\prime }} \\&\lesssim \Vert \Vert U_0(t)u_0\Vert _r\Vert _{L_T^q}+\Vert \Vert f\Vert _{\tilde{r}^\prime }\Vert _{L_T^{\tilde{q}^\prime }}\\&\lesssim \Vert U_0(t)u_0\Vert _{L_T^q(L^r)}+\Vert f\Vert _{L_T^{\tilde{q}^\prime }(L^{\tilde{r}^\prime })} \\&\lesssim \Vert u_0\Vert +\Vert f\Vert _{L_T^{\tilde{q}^\prime }(L^{\tilde{r}^\prime })}. \end{aligned}$$