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A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation

Abstract

In this article, we first establish a global Carleman estimate for an ultrahyperbolic Schrödinger equation. Next, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the equation by some lateral boundary data.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her very valuable comments and suggestions.

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Correspondence to Fikret Gölgeleyen.

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Appendix

Appendix

In this section, we shall prove Lemma 1. We introduce

$$\begin{aligned} L_{0}w:=i\partial _{t}w+\varDelta _{y}w-\varDelta _{x}w=F, \end{aligned}$$
(5.1)

and

$$\begin{aligned} z\left( x,y,t\right) =e^{s\varphi }w\left( x,y,t\right) ,\ P_{s}z\left( x,y,t\right) =e^{s\varphi }L_{0}w. \end{aligned}$$
(5.2)

By (5.2), we have

$$\begin{aligned} P_{s}z= & {} e^{s\varphi }L_{0}w \\= & {} i\partial _{t}z-is\partial _{t}\varphi z+\varDelta _{y}z-\varDelta _{x}z-2s\left( \nabla _{y}\varphi \cdot \nabla _{y}z-\nabla _{x}\varphi \cdot \nabla _{x}z\right) \\&-\,s\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) z+s^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) z. \end{aligned}$$

We set

$$\begin{aligned} P_{s}z+is\partial _{t}\varphi z=P_{s}^{+}z+P_{s}^{-}z, \end{aligned}$$
(5.3)

where

$$\begin{aligned} P_{s}^{+}z= & {} i\partial _{t}z+\varDelta _{y}z-\varDelta _{x}z+s^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) z, \end{aligned}$$
(5.4)
$$\begin{aligned} P_{s}^{-}z= & {} -2s\left( \nabla _{y}\varphi \cdot \nabla _{y}z-\nabla _{x}\varphi \cdot \nabla _{x}z\right) -s\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) z, \end{aligned}$$
(5.5)

with the conventions \(z\cdot z^{\prime }=\sum \nolimits _{i=1}^{N}z_{i}z_{i}^{\prime }\) for all \(z=\left( z_{1},\dots ,z_{N}\right) \in \mathbb {C} ^{N}\), \(z^{\prime }=\left( z_{1}^{\prime },\dots ,z_{N}^{\prime }\right) \in \mathbb {C} ^{N}\).

Then we have

$$\begin{aligned} \left\| P_{s}z+is\partial _{t}\varphi z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}=\left\| P_{s}^{+}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}+\left\| P_{s}^{-}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}+2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }, \end{aligned}$$
(5.6)

where \(\mathrm{Re}\left( z\right) \) is the real part of z. Now, we calculate the last term in (5.6) by using (5.4) and (5.5), we obtain

$$\begin{aligned} 2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}+I_{7}+I_{8}, \end{aligned}$$
(5.7)

where

$$\begin{aligned} I_{1}= & {} -4s\mathrm{Re}\int _{Q_{T}^{\prime }}i\partial _{t}z\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}\right) dxdydt, \\ I_{2}= & {} -2s\mathrm{Re}\int _{Q_{T}^{\prime }}i\partial _{t}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dxdydt, \\ I_{3}= & {} -4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{y}z\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}\right) dxdydt, \\ I_{4}= & {} -2s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{y}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dxdydt, \\ I_{5}= & {} 4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{x}z\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}\right) dxdydt, \\ I_{6}= & {} 2s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{x}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dxdydt, \nonumber \\ I_{7}= & {} -4s^{3}\mathrm{Re}\int _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) z\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}\right) dxdydt, \\ I_{8}= & {} -2s^{3}\mathrm{Re}\int _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{ z}dxdydt, \end{aligned}$$

and \(\overline{z}\) is the conjugate of z.

Now, we shall estimate the terms \(I_{k}\), \(1\le k\le 8\) respectively. Here we set \(\varGamma _{x}=\partial D\times G^{\prime }\times (-T,T)\) and \(\varGamma _{y}=D\times \partial G^{\prime }\times (-T,T)\). By the integration by parts and the condition \(z\left( x,y,\pm T\right) =0\), we have

$$\begin{aligned} I_{1}= & {} -4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}i\partial _{t}z\nabla _{y}\varphi \cdot \nabla _{y}\overline{z}dxdydt+4s\mathrm{Re} \int \nolimits _{Q_{T}^{\prime }}i\partial _{t}z\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}dxdydt \nonumber \\= & {} -2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{y}\varphi )\cdot \nabla _{y}\overline{z}dxdydt-2s\mathrm{Im} \int \nolimits _{\varGamma _{y}}z\partial _{t}\overline{z}\left( \nabla _{y}\varphi \cdot \nu \right) dS_{y}dxdt \nonumber \\&+\,2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}z\varDelta _{y}\varphi \partial _{t}\overline{z}dxdydt+2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{x}\varphi )\cdot \nabla _{x}\overline{z}dxdydt \nonumber \\&+\,2s\mathrm{Im}\int \nolimits _{\varGamma _{x}}z\partial _{t}\overline{z}\left( \nabla _{x}\varphi \cdot \nu \right) dS_{x}dydt-2s\mathrm{Im} \int \nolimits _{Q_{T}^{\prime }}z\varDelta _{x}\varphi \partial _{t}\overline{z} dxdydt. \end{aligned}$$
(5.8)

In (5.8), we used the equality \(\mathrm{Re}\left( iz\right) =-\mathrm{Im} \left( z\right) \) and \(\mathrm{Im}\left( z\right) -\mathrm{Im}\left( \bar{z} \right) =2\mathrm{Im}\left( z\right) \), where \(\mathrm{Im}\left( z\right) \) denotes the imaginary part of \(z\in \mathbb {C} \).

Since \(\mathrm{Re}\left( iz\right) =\mathrm{Im}\left( \bar{z}\right) \), we get

$$\begin{aligned} I_{2}= & {} -2s\mathrm{Re}\int _{Q_{T}^{\prime }}i\partial _{t}z\overline{z}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt \nonumber \\= & {} -2s\mathrm{Im}\int _{Q_{T}^{\prime }}\partial _{t}\overline{z}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt. \end{aligned}$$
(5.9)

By the equality \(\mathrm{Re}z_{y_{j}}\bar{z}_{y_{i}y_{j}}=\frac{1}{2}\left( \left| z_{y_{j}}\right| ^{2}\right) _{y_{i}}\), we obtain

$$\begin{aligned} I_{3}= & {} -4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{y}z\nabla _{y}\varphi \cdot \nabla _{y}\overline{z}dxdydt+4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{y}z\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}dxdydt \nonumber \\= & {} 4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\sum \limits _{i,j=1}^{n}\varphi _{y_{i}y_{j}}\overline{z}_{y_{i}}z_{y_{j}}dxdydt-2s\int \nolimits _{Q_{T}^{ \prime }}\varDelta _{y}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt \nonumber \\&+\,2s\int _{\varGamma _{y}}\left( \partial _{\nu }\varphi \right) \left| \nabla _{y}z\right| ^{2}dS_{y}dxdt-4s\mathrm{Re}\int _{\varGamma _{y}}\left( \partial _{\nu }z\right) \nabla _{y}\varphi \cdot \nabla _{y}\overline{z} dS_{y}dxdt \nonumber \\&-\,4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\varphi _{x_{j}y_{i}}\overline{z} _{x_{j}}z_{y_{i}}dxdydt+2s\int \nolimits _{Q_{T}^{\prime }}\varDelta _{x}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt \nonumber \\&-\,2s\int _{\varGamma _{x}}\left( \partial _{\nu }\varphi \right) \left| \nabla _{y}z\right| ^{2}dS_{x}dydt \nonumber \\&+\,4s\mathrm{Re}\int _{\varGamma _{y}}\left( \partial _{\nu }z\right) \nabla _{x}\varphi \cdot \nabla _{x}\overline{z}dS_{y}dxdt. \end{aligned}$$
(5.10)

From the equality \(\mathrm{Re}\overline{z}\nabla _{y}z=\frac{1}{2}\nabla _{y}\left( \left| z\right| ^{2}\right) \), we have

$$\begin{aligned} I_{4}= & {} -2s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{y}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dxdydt \nonumber \\= & {} -s\int _{Q_{T}^{\prime }}\varDelta _{y}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dxdydt \nonumber \\&+\,2s\int _{Q_{T}^{\prime }}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| \nabla _{y}z\right| ^{2}dxdydt \nonumber \\&+s\int _{\varGamma _{y}}\partial _{\nu }\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dS_{y}dxdt \nonumber \\&-\,2s\mathrm{Re}\int \nolimits _{\varGamma _{y}}\left( \partial _{\nu }z\right) \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dS_{y}dxdt. \end{aligned}$$
(5.11)

Moreover, since \(\mathrm{Re}z_{x_{j}}\overline{z}_{x_{j}y_{i}}=\frac{1}{2} (\left| z_{x_{j}}\right| ^{2})_{y_{i}}\), then

$$\begin{aligned} I_{5}= & {} 4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{x}z\nabla _{y}\varphi \cdot \nabla _{y}\overline{z}dxdydt-4s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{x}z\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}dxdydt \nonumber \\= & {} -4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\varphi _{y_{i}x_{j}}\overline{z} _{y_{i}}z_{x_{j}}dxdydt+2s\int \nolimits _{Q_{T}^{\prime }}\sum \limits _{i=1}^{m}\varDelta _{y}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&-\,2s\int _{\varGamma _{y}}\left( \partial _{\nu }\varphi \right) \left| \nabla _{x}z\right| ^{2}dS_{y}dxdt+4s\mathrm{Re}\int _{\varGamma _{x}}\left( \partial _{\nu }z\right) \nabla _{y}\varphi \cdot \nabla _{y}\overline{z} dS_{x}dydt \nonumber \\&+\,4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\sum \limits _{i,j=1}^{n}\varphi _{x_{i}y_{j}}\overline{z}_{x_{i}}z_{x_{j}}dxdydt-2s\int \nolimits _{Q_{T}^{ \prime }}\varDelta _{x}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&+\,2s\int _{\varGamma _{x}}\left( \partial _{\nu }\varphi \right) \left| \nabla _{x}z\right| ^{2}dS_{x}dydt \nonumber \\&-\,4s\mathrm{Re}\int _{\varGamma _{x}}\left( \partial _{\nu }z\right) \nabla _{x}\varphi \cdot \nabla _{x}\overline{z}dS_{x}dydt. \end{aligned}$$
(5.12)

By using the equality \(\mathrm{Re}\overline{z}\)\(\nabla _{x}z=\frac{1}{2} \nabla _{x}\left( \left| z\right| ^{2}\right) \), we write

$$\begin{aligned} I_{6}= & {} 2s\mathrm{Re}\int _{Q_{T}^{\prime }}\varDelta _{x}z\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dxdydt \nonumber \\= & {} -2s\int _{Q_{T}^{\prime }}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&+\,s\int _{Q_{T}^{\prime }}\varDelta _{x}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) )\left| z\right| ^{2}dxdydt \nonumber \\&-\,s\int \nolimits _{\varGamma _{x}}\partial _{\nu }\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dS_{x}dydt \nonumber \\&+\,2s\mathrm{Re}\int \nolimits _{\varGamma _{x}}\left( \partial _{\nu }z\right) \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dS_{x}dydt. \end{aligned}$$
(5.13)

From the relations \(\mathrm{Re}z\nabla _{y}\overline{z}=\frac{1}{2}\left( \nabla _{y}\left| z\right| ^{2}\right) \) and \(\mathrm{Re}z\nabla _{x} \overline{z}=\frac{1}{2}\left( \nabla _{x}\left| z\right| ^{2}\right) \), we see that

$$\begin{aligned} I_{7}= & {} -4s^{3}\mathrm{Re}\int _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) z\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z}\right) dxdydt \nonumber \\= & {} 2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt \nonumber \\&-\,2s^{3}\int _{\varGamma _{y}}\left( \partial _{\nu }\varphi \right) \left| z\right| ^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dS_{y}dxdt \nonumber \\&+\,2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{y}\varphi \cdot \nabla _{y}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt \nonumber \\&+\,2s^{3}\int _{\varGamma _{x}}\partial _{\nu }\varphi \left| z\right| ^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dS_{x}dydt \nonumber \\&-\,2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{x}\varphi \cdot \nabla _{x}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt. \end{aligned}$$
(5.14)

It is obvious that

$$\begin{aligned} I_{8}= & {} -2s^{3}\mathrm{Re}\int _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) z\overline{ z}dxdydt \nonumber \\= & {} -2s^{3}\int _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dxdydt. \end{aligned}$$
(5.15)

Hence, we can rewrite (5.7) as follows

$$\begin{aligned} 2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}+B_{0}, \end{aligned}$$

where

$$\begin{aligned} J_{1}= & {} -2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{y}\varphi )\cdot \nabla _{y}\overline{z}dxdydt+2s\mathrm{Im} \int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{x}\varphi )\cdot \nabla _{x}\overline{z}dxdydt, \\ J_{2}= & {} 4s\mathrm{Re}\sum \limits _{i,j=1}^{n}\int \nolimits _{Q_{T}^{\prime }}\varphi _{y_{i}y_{j}}z_{y_{j}}\overline{z}_{y_{i}}dxdydt-4s\mathrm{Re} \sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\int \nolimits _{Q_{T}^{\prime }}\varphi _{y_{i}x_{j}}z_{y_{i}}\overline{z}_{x_{j}}dxdydt, \\ J_{3}= & {} -s\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\varDelta _{y}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt, \\ J_{4}= & {} -4s\mathrm{Re}\sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\int \nolimits _{Q_{T}^{\prime }}\varphi _{y_{i}x_{j}}\overline{z} _{y_{i}}z_{x_{j}}dxdydt+4s\mathrm{Re}\sum \limits _{i,j=1}^{n}\int \nolimits _{Q_{T}^{\prime }}\varphi _{x_{i}x_{j}}\overline{z} _{x_{i}}z_{x_{j}}dxdydt, \\ J_{5}= & {} s\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\varDelta _{x}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt, \\ J_{6}= & {} 2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{y}\varphi \cdot \nabla _{y}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt\\&-\,2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{x}\varphi \cdot \nabla _{x}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt \end{aligned}$$

and

$$\begin{aligned} B_{0}= & {} -2s\mathrm{Im}\int \nolimits _{\varGamma _{y}}z\partial _{t}\overline{z} \left( \nabla _{y}\varphi \cdot \nu \right) dS_{y}dxdt \\&+\,4s\mathrm{Re}\int \nolimits _{\varGamma _{y}}(\partial _{\nu }z)\left( \nabla _{x}\varphi \cdot \nabla _{x}\overline{z}-\nabla _{y}\varphi \cdot \nabla _{y}\overline{z}\right) dS_{y}dxdt \\&+\,2s\int \nolimits _{\varGamma _{y}}(\partial _{\nu }\varphi )\left( \left| \nabla _{y}z\right| ^{2}-\left| \nabla _{x}z\right| ^{2}\right) dS_{y}dxdt \\&-\,2s^{3}\int _{\varGamma _{y}}(\partial _{\nu }\varphi )\left| z\right| ^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dS_{y}dxdt \\&-\,2s\mathrm{Re}\int \nolimits _{\varGamma _{y}}(\partial _{\nu }z)\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dS_{y}dxdt\\&+\,s\int \nolimits _{\varGamma _{y}}\partial _{\nu }\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dS_{y}dxdt \\&+\,2s\mathrm{Im}\int \nolimits _{\varGamma _{x}}z\partial _{t}\overline{z}\left( \nabla _{x}\varphi \cdot \nu \right) dS_{x}dydt \\&+\,4s\mathrm{Re}\int _{\varGamma _{x}}(\partial _{\nu }z)\left( \nabla _{y}\varphi \cdot \nabla _{y}\overline{z}-\nabla _{x}\varphi \cdot \nabla _{x}\overline{z }\right) dS_{x}dydt \\&-\,2s\int \nolimits _{\varGamma _{x}}(\partial _{\nu }\varphi )\left( \left| \nabla _{y}z\right| ^{2}-\left| \nabla _{x}z\right| ^{2}\right) dS_{x}dydt \\&+\,2s^{3}\int _{\varGamma _{x}}(\partial _{\nu }\varphi )\left| z\right| ^{2}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dS_{x}dydt \\&+\,2s\mathrm{Re}\int _{\varGamma _{x}}(\partial _{\nu }z)\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \overline{z}dS_{x}dydt\\&-\,s\int _{\varGamma _{x}}\partial _{\nu }\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) \left| z\right| ^{2}dS_{x}dydt. \end{aligned}$$

Now, we estimate \(J_{k}\), \(1\le k\le 6\) and \(B_{0}\) using the following elementary properties of the weight function:

$$\begin{aligned} \begin{array}{cc} \partial _{t}\varphi =\left( -2\gamma \beta t\right) \varphi , &{} \varphi _{x_{i}x_{i}}=\gamma \varphi \left( 2+\gamma \psi _{x_{i}}^{2}\right) , \\ \varphi _{x_{i}y_{j}}=\gamma ^{2}\varphi \psi _{x_{i}}\psi _{y_{j}}, &{} \varphi _{x_{i}x_{j}}=\gamma \varphi \left( \psi _{x_{i}x_{j}}+\gamma \psi _{x_{i}}\psi _{x_{j}}\right) , \\ \varphi _{y_{i}y_{j}}=\gamma \varphi \left( \psi _{y_{i}y_{j}}+\gamma \psi _{y_{i}}\psi _{y_{j}}\right) , &{} \nabla _{x}\varphi =\gamma \varphi \nabla _{x}\psi , \\ \nabla _{y}\varphi =\gamma \varphi \nabla _{y}\psi , &{} \partial _{t}(\nabla _{x}\varphi )=\left( -2\gamma ^{2}\beta t\right) \varphi \nabla _{x}\psi ,\\ \partial _{t}(\nabla _{y}\varphi )=\left( -2\gamma ^{2}\beta t\right) \varphi \nabla _{y}\psi , &{} \varDelta _{x}\varphi =\gamma \varphi \left( \varDelta _{x}\psi +\gamma \left| \nabla _{x}\psi \right| ^{2}\right) , \\ \varDelta _{y}\varphi =\gamma \varphi \left( \varDelta _{y}\psi +\gamma \left| \nabla _{y}\psi \right| ^{2}\right) , &{} \varDelta _{y}\varphi -\varDelta _{x}\varphi =\gamma \varphi d_{1}\left( \psi \right) +\gamma ^{2}\varphi d_{2}\left( \psi \right) , \end{array} \end{aligned}$$

where

$$\begin{aligned} d_{1}\left( \psi \right)= & {} \varDelta _{y}\psi -\varDelta _{x}\psi , \\ d_{2}\left( \psi \right)= & {} \left| \nabla _{y}\psi \right| ^{2}-\left| \nabla _{x}\psi \right| ^{2}. \end{aligned}$$

Then, we have

$$\begin{aligned} J_{1}= & {} -2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{y}\varphi )\cdot \nabla _{y}\overline{z}dxdydt+2s\mathrm{Im} \int \nolimits _{Q_{T}^{\prime }}z\partial _{t}(\nabla _{x}\varphi )\cdot \nabla _{x}\overline{z}dxdydt \nonumber \\= & {} -2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\left( -2\gamma ^{2}\beta t\right) z\varphi \nabla _{y}\psi \cdot \nabla _{y}\overline{z}dxdydt \nonumber \\&+\,2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\left( -2\gamma ^{2}\beta t\right) z\varphi \nabla _{x}\psi \cdot \nabla _{x}\overline{z}dxdydt \end{aligned}$$
(5.16)

and

$$\begin{aligned} J_{2}= & {} 4s\sum \limits _{i,j=1}^{m}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\varphi _{y_{i}y_{j}}z_{y_{j}}\overline{z}_{y_{i}}dxdydt-4s\mathrm{Re} \sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\int \nolimits _{Q_{T}^{\prime }}z_{y_{i}}\overline{z}_{x_{j}}\varphi _{y_{i}x_{j}}dxdydt \nonumber \\= & {} 4s\sum \limits _{i,j=1}^{m}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \psi _{y_{i}y_{j}}z_{y_{j}}\overline{z}_{y_{i}}dxdydt+4s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| \nabla _{y}\psi \cdot \nabla _{y}z\right| ^{2}dxdydt \nonumber \\&-\,4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left( \nabla _{y}\psi \cdot \nabla _{y}z\right) \left( \nabla _{x}\psi \cdot \nabla _{x}\overline{z}\right) dxdydt. \end{aligned}$$
(5.17)

Before estimating \(J_{3}\), we can directly verify

$$\begin{aligned} \varDelta _{y}\left( \varphi d_{2}\left( \psi \right) \right)= & {} \left( \varDelta _{y}\varphi \right) d_{2}\left( \psi \right) +2\nabla _{y}\varphi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) +\varphi \varDelta _{y}\left( d_{2}\left( \psi \right) \right) \\= & {} \gamma \varphi \left( \varDelta _{y}\psi \right) d_{2}\left( \psi \right) +\gamma ^{2}\varphi \left| \nabla _{y}\psi \right| ^{2}d_{2}\psi +2\gamma \varphi \nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) \\&+\,\varphi \varDelta _{y}\left( d_{2}\left( \psi \right) \right) , \\ \varDelta _{y}\left( \varphi d_{1}\left( \psi \right) \right)= & {} \left( \varDelta _{y}\varphi \right) d_{1}\left( \psi \right) +2\nabla _{y}\varphi \cdot \nabla _{y}\left( d_{1}\left( \psi \right) \right) +\varphi \varDelta _{y}\left( d_{1}\left( \psi \right) \right) \\= & {} \gamma \varphi \left( \varDelta _{y}\psi \right) d_{1}\left( \psi \right) +\gamma ^{2}\varphi \left| \nabla _{y}\psi \right| ^{2}d_{1}\psi +2\gamma \varphi \nabla _{y}\psi \cdot \nabla _{y}\left( d_{1}\left( \psi \right) \right) \\&+\,\varphi \varDelta _{y}\left( d_{1}\left( \psi \right) \right) . \end{aligned}$$

Then, we obtain

$$\begin{aligned} J_{3}= & {} -s\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\varDelta _{y}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt \nonumber \\= & {} -s\int _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}\left( d_{1}\left( \psi \right) \left( \varDelta _{y}\psi \right) +\varDelta _{y}\left( d_{2}\left( \psi \right) \right) \right) dxdydt \nonumber \\&-\,s\int _{Q_{T}^{\prime }}\gamma ^{3}\varphi \left| z\right| ^{2}\left( d_{1}\left( \psi \right) \left| \nabla _{y}\psi \right| ^{2}+\left( \varDelta _{y}\psi \right) d_{2}\left( \psi \right) \right. \nonumber \\&\left. +\,2\nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) \right) dxdydt \nonumber \\&-\,s\int _{Q_{T}^{\prime }}\gamma ^{4}\varphi \left| z\right| ^{2}\left| \nabla _{y}\psi \right| ^{2}d_{2}\left( \psi \right) dxdydt. \end{aligned}$$
(5.18)

In (5.18), we used the equality \(\nabla _{y}\left( d_{1}\left( \psi \right) \right) =0\) and \(\varDelta _{y}\left( d_{1}\left( \psi \right) \right) =0\).

Next

$$\begin{aligned} J_{4}= & {} -4s\sum \limits _{j=1}^{m}\sum \limits _{i=1}^{n}\mathrm{Re} \int \nolimits _{Q_{T}^{\prime }}\varphi _{y_{i}x_{j}}\overline{z} _{y_{i}}z_{x_{j}}dxdydt+4s\sum \limits _{i,j=1}^{n}\mathrm{Re} \int \nolimits _{Q_{T}^{\prime }}\varphi _{x_{i}x_{j}}\overline{z} _{x_{i}}z_{x_{j}}dxdydt \nonumber \\= & {} -4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left( \nabla _{y}\psi \cdot \nabla _{y}\overline{z}\right) \left( \nabla _{x}\psi \cdot \nabla _{x}z\right) dxdydt \nonumber \\&+\,4s\sum \limits _{i,j=1}^{n}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \psi _{x_{i}x_{j}}\overline{z}_{x_{i}}z_{x_{j}}dxdydt \nonumber \\&+\,4s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| \nabla _{x}\psi \cdot \nabla _{x}z\right| ^{2}dxdydt. \end{aligned}$$
(5.19)

Since

$$\begin{aligned} \varDelta _{x}\left( \varphi d_{2}\left( \psi \right) \right)= & {} \left( \varDelta _{x}\varphi \right) d_{2}\left( \psi \right) +2\nabla _{x}\varphi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) +\varphi \varDelta _{x}\left( d_{2}\left( \psi \right) \right) \\= & {} \gamma \varphi \left( \varDelta _{x}\psi \right) d_{2}\left( \psi \right) +\gamma ^{2}\varphi \left| \nabla _{x}\psi \right| ^{2}d_{2}\psi +2\gamma \varphi \nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) \\&+\,\varphi \varDelta _{x}\left( d_{2}\left( \psi \right) \right) , \end{aligned}$$

we see that

$$\begin{aligned} J_{5}= & {} s\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\varDelta _{x}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt \nonumber \\= & {} s\int _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}\left( d_{1}\left( \psi \right) \left( \varDelta _{x}\psi \right) +\varDelta _{x}\left( d_{2}\left( \psi \right) \right) \right) dxdydt \nonumber \\&+\,s\int _{Q_{T}^{\prime }}\gamma ^{3}\varphi \left| z\right| ^{2}\left( d_{1}\left( \psi \right) \left| \nabla _{x}\psi \right| ^{2}\right. \nonumber \\&\left. +\left( \varDelta _{x}\psi \right) d_{2}\left( \psi \right) +2\nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) \right) dxdydt \nonumber \\&+\,s\int _{Q_{T}^{\prime }}\gamma ^{4}\varphi \left| z\right| ^{2}\left| \nabla _{x}\psi \right| ^{2}d_{2}\left( \psi \right) dxdydt. \end{aligned}$$
(5.20)

In (5.20), we used the equality \(\nabla _{x}\left( d_{1}\left( \psi \right) \right) =0\) and \(\varDelta _{x}\left( d_{1}\left( \psi \right) \right) =0\).

Since

$$\begin{aligned} \nabla _{y}\varphi \cdot \nabla _{y}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) =2\gamma ^{4}\varphi ^{3}d_{2}\left( \psi \right) \nabla _{y}\psi \cdot \nabla _{y}\psi +\gamma ^{3}\varphi ^{3}\nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) \end{aligned}$$

and

$$\begin{aligned} \nabla _{x}\varphi \cdot \nabla _{x}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) =2\gamma ^{4}\varphi ^{3}d_{2}\left( \psi \right) \nabla _{x}\psi \cdot \nabla _{x}\psi +\gamma ^{3}\varphi ^{3}\nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) , \end{aligned}$$

we have

$$\begin{aligned} J_{6}= & {} 2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{y}\varphi \cdot \nabla _{y}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt \nonumber \\&-\,2s^{3}\int _{Q_{T}^{\prime }}\left| z\right| ^{2}\nabla _{x}\varphi \cdot \nabla _{x}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) dxdydt \nonumber \\= & {} 4s^{3}\int _{Q_{T}^{\prime }}\gamma ^{4}\varphi ^{3}\left| z\right| ^{2}\left( d_{2}\left( \psi \right) \right) ^{2}dxdydt \nonumber \\&+\,2s^{3}\int _{Q_{T}^{\prime }}\gamma ^{3}\varphi ^{3}\left| z\right| ^{2}\nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) dxdydt \nonumber \\&-\,2s^{3}\int _{Q_{T}^{\prime }}\gamma ^{3}\varphi ^{3}\left| z\right| ^{2}\nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) dxdydt. \end{aligned}$$
(5.21)

Additionally, the boundary term is obtained as follows:

$$\begin{aligned} B_{0}= & {} -2s\mathrm{Im}\int \nolimits _{\varGamma _{y}}\gamma \varphi z\partial _{t} \overline{z}\left( \nabla _{y}\psi \cdot \nu \right) dS_{y}dxdt \nonumber \\&+\,4s\mathrm{Re}\int \nolimits _{\varGamma _{y}}\gamma \varphi \left( \partial _{\nu }z\right) \left( \nabla _{x}\psi \cdot \nabla _{x}\overline{z}-\nabla _{y}\psi \cdot \nabla _{y}\overline{z}\right) dS_{y}dxdt \nonumber \\&+\,2s\int \nolimits _{\varGamma _{y}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left( \left| \nabla _{y}z\right| ^{2}-\left| \nabla _{x}z\right| ^{2}\right) dS_{y}dxdt \nonumber \\&-\,2s^{3}\int _{\varGamma _{y}}\gamma ^{3}\varphi ^{3}\partial _{v}\psi \left| z\right| ^{2}d_{2}\left( \psi \right) dS_{y}dxdt \nonumber \\&-\,2s\mathrm{Re}\int \nolimits _{\varGamma _{y}}\left( \gamma \varphi d_{1}\left( \psi \right) +\gamma ^{2}\varphi d_{2}\left( \psi \right) \right) \overline{z }\left( \partial _{\nu }z\right) dS_{y}dxdt \nonumber \\&+\,s\int \nolimits _{\varGamma _{y}}\left( \left( \gamma ^{2}\varphi d_{1}\left( \psi \right) +\gamma ^{3}\varphi d_{2}\left( \psi \right) \right) \left( \partial _{\nu }\psi \right) \right. \nonumber \\&\left. +\gamma ^{2}\varphi \left( \partial _{\nu }\left( d_{2}\left( \psi \right) \right) \right) \right) \left| z\right| ^{2}dS_{y}dxdt \nonumber \\&+\,2s\mathrm{Im}\int \nolimits _{\varGamma _{x}}\gamma \varphi z\partial _{t} \overline{z}\left( \nabla _{x}\psi \cdot \nu \right) dS_{x}dydt \nonumber \\&+\,4s\mathrm{Re}\int _{\varGamma _{x}}\gamma \varphi \left( \partial _{v}z\right) \left( \nabla _{y}\psi \cdot \nabla _{y}\overline{z}-\nabla _{x}\psi \cdot \nabla _{x}\overline{z}\right) dS_{x}dydt \nonumber \\&-\,2s\int \nolimits _{\varGamma _{x}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left( \left| \nabla _{y}z\right| ^{2}-\left| \nabla _{x}z\right| ^{2}\right) dS_{x}dydt \nonumber \\&+\,2s^{3}\int _{\varGamma _{x}}\gamma ^{3}\varphi ^{3}\left( \partial _{\nu }\psi \right) \left| z\right| ^{2}d_{2}\left( \psi \right) dS_{x}dydt \nonumber \\&+\,2s\mathrm{Re}\int _{\varGamma _{x}}\left( \gamma \varphi d_{1}\left( \psi \right) +\gamma ^{2}\varphi d_{2}\left( \psi \right) \right) \overline{z} \left( \partial _{\nu }z\right) dS_{x}dydt \nonumber \\&-\,s\int _{\varGamma _{x}}\left( \left( \gamma ^{2}\varphi d_{1}\left( \psi \right) +\gamma ^{3}\varphi d_{2}\left( \psi \right) \right) \left( \partial _{\nu }\psi \right) \right. \nonumber \\&\left. +\gamma ^{2}\varphi \partial _{\nu }\left( d_{2}\left( \psi \right) \right) \right) \left| z\right| ^{2}dS_{x}dydt. \end{aligned}$$
(5.22)

By (5.16)–(5.22) we can write

$$\begin{aligned} 2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }= & {} 4s\sum \limits _{i,j=1}^{m}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \psi _{y_{i}y_{j}}z_{y_{j}}\overline{z}_{y_{i}}dxdydt \\&+\,4s\sum \limits _{i,j=1}^{n}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \psi _{x_{i}x_{j}}\overline{z}_{x_{i}}z_{x_{j}}dxdydt \\&+\,4s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| \nabla _{y}\psi \cdot \nabla _{y}z-\nabla _{x}\psi \cdot \nabla _{x}z\right| ^{2}dxdydt \\&+\,4s^{3}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{4}\varphi ^{3}\left| z\right| ^{2}\left( d_{2}\left( \psi \right) \right) ^{2}dxdydt+B_{0}+X_{1}+X_{2}, \end{aligned}$$

where

$$\begin{aligned} X_{1}= & {} 2s^{3}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{3}\varphi ^{3}\left| z\right| ^{2}d_{5}\left( \psi \right) dxdydt,\\ X_{2}= & {} -2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\left( -2\gamma ^{2}\beta t\right) z\varphi \nabla _{y}\psi \cdot \nabla _{y}\overline{z} dxdydt \\&+\,2s\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\left( -2\gamma ^{2}\beta t\right) z\varphi \nabla _{x}\psi \cdot \nabla _{x}\overline{z}dxdydt \\&-\,s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{4}\varphi \left| z\right| ^{2}\left( d_{2}\left( \psi \right) \right) ^{2}dxdydt-s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}d_{3}\left( \psi \right) dxdydt \\&-\,s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{3}\varphi \left| z\right| ^{2}d_{4}\left( \psi \right) dxdydt, \end{aligned}$$

and

$$\begin{aligned} d_{3}:= & {} d_{3}\left( \psi \right) =\left( d_{1}\left( \psi \right) \right) ^{2}+\varDelta _{y}\left( d_{2}\left( \psi \right) \right) -\varDelta _{x}\left( d_{2}\left( \psi \right) \right) , \\ d_{4}:= & {} d_{4}\left( \psi \right) =2\left( d_{1}\left( \psi \right) d_{2}\left( \psi \right) +\nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) -\nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) \right) , \\ d_{5}:= & {} d_{5}\left( \psi \right) =\nabla _{y}\psi \cdot \nabla _{y}\left( d_{2}\left( \psi \right) \right) -\nabla _{x}\psi \cdot \nabla _{x}\left( d_{2}\left( \psi \right) \right) . \end{aligned}$$

Since

$$\begin{aligned} \int \nolimits _{Q_{T}^{\prime }}4s\gamma ^{2}\varphi \left| \nabla _{y}\psi \cdot \nabla _{y}z-\nabla _{x}\psi \cdot \nabla _{x}z\right| ^{2}dxdydt\ge 0 \end{aligned}$$

and (3.4), we have

$$\begin{aligned} d_{2}^{2}=16(|x-x_{0}|^{2}-\alpha ^{2}|y-y_{0}|^{2})^{2}\ge 16\delta _{0}^{2}. \end{aligned}$$

Then, we see that

$$\begin{aligned} 2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }\ge & {} 8s\gamma \left( \int \nolimits _{Q_{T}^{\prime }}-\alpha \varphi |\nabla _{y}z|^{2}dxdydt+\int _{Q_{T}^{\prime }}\varphi |\nabla _{x}z|^{2}dxdydt\right) \nonumber \\&+\,64\delta _{0}^{2}\int _{Q_{T}^{\prime }}s^{3}\gamma ^{4}\varphi ^{3}|z|^{2}dxdydt \nonumber \\&+\,B_{0}+X_{1}+X_{2}. \end{aligned}$$
(5.23)

Since the signs of the terms of \(|\nabla _{x}z|^{2}\) and \(|\nabla _{y}z|^{2}\) are different, we need to perform another estimation:

$$\begin{aligned} 2\mathrm{Re}\left( P_{s}^{+}z+P_{s}^{-}z,\varphi z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }= & {} 2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}i\partial _{t}z\overline{z}\varphi dxdydt+2\mathrm{Re}\int \nolimits _{Q_{T}^{ \prime }}\varDelta _{y}z\overline{z}\varphi dxdydt \\&-\,2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\varDelta _{x}z\overline{z}\varphi dxdydt \\&+\,2s^{2}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}(\left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2})\varphi z\overline{z}dxdydt \\&-\,4s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\left( \nabla _{y}\varphi \cdot \nabla _{y}z-\nabla _{x}\varphi \cdot \nabla _{x}z\right) \varphi \overline{z }dxdydt \\&-\,2s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) z\varphi \overline{z}dxdydt \\= & {} K_{1}+K_{2}+K_{3}+K_{4}+K_{5}+K_{6}. \end{aligned}$$

We calculate the terms \(K_{j}\), \(1\le j\le 6\) as follows:

$$\begin{aligned} K_{1}= & {} 2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}i\partial _{t}z\overline{z} \varphi dxdydt \\= & {} -2\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\partial _{t}z\overline{z} \varphi dxdydt,\\ K_{2}= & {} 2\mathrm{Re}\int \nolimits _{\int \nolimits _{Q_{T}^{\prime }}}\varDelta _{y}z\varphi \overline{z}dxdydt \\= & {} \int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \left| z\right| ^{2}\varDelta _{y}\psi dxdydt+\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}\left| \nabla _{y}\psi \right| ^{2}dxdydt\\&-\,\int \nolimits _{\varGamma _{y}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{y}dxdt-2\int \nolimits _{Q_{T}^{\prime }}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt \\&+\,2\mathrm{Re}\int \nolimits _{\varGamma _{y}}\left( \partial _{\nu }z\right) \left( \varphi \overline{z}\right) dS_{y}dxdt,\\ K_{3}= & {} -2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\varDelta _{x}z\overline{z} \varphi dxdydt \\= & {} 2\int \nolimits _{Q_{T}^{\prime }}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt-\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \varDelta _{x}\psi \left| z\right| ^{2}dxdydt \\&-\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| \nabla _{x}\psi \right| ^{2}\left| z\right| ^{2}dxdydt+\int _{\varGamma _{x}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{x}dydt \\&-\,2\mathrm{Re}\int _{\varGamma _{x}}\left( \partial _{\nu }z\right) \left( \varphi \overline{z}\right) dS_{x}dydt,\\ K_{4}= & {} 2s^{2}\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\left( \left| \nabla _{y}\varphi \right| ^{2}-\left| \nabla _{x}\varphi \right| ^{2}\right) \varphi z\overline{z}dxdydt \\= & {} 2s^{2}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{3}\left( \left| \nabla _{y}\psi \right| ^{2}-\left| \nabla _{x}\psi \right| ^{2}\right) \left| z\right| ^{2}dxdydt \\= & {} 2s^{2}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{3}d_{2}\left( \psi \right) \left| z\right| ^{2}dxdydt,\\ K_{5}= & {} -4s\mathrm{Re}\,\int \nolimits _{Q_{T}^{\prime }}\varphi \overline{z}\left( \nabla _{y}\varphi \cdot \nabla _{y}z-\nabla _{x}\varphi \cdot \nabla _{x}z\right) \varphi dxdydt \\= & {} 2s\int \nolimits _{Q_{T}^{\prime }}\varphi ^{2}\gamma \left| z\right| ^{2}d_{1}\left( \psi \right) dxdydt+2s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{2}\left| z\right| ^{2}d_{2}\left( \psi \right) dxdydt \\&-\,2s\int _{\varGamma _{y}}\gamma \varphi ^{2}\left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{y}dxdt+2s\int _{\varGamma _{x}}\gamma \varphi ^{2}\left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{x}dydt,\\ K_{6}= & {} -2s\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\varphi z\overline{z} \left( \varDelta _{y}\varphi -\varDelta _{x}\varphi \right) dxdydt \\= & {} -2s\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi ^{2}\left( \varDelta _{y}\psi -\varDelta _{x}\psi \right) \left| z\right| ^{2}dxdydt \\&-\,2s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{2}\left| z\right| ^{2}\left( \left| \nabla _{y}\psi \right| ^{2}-\left| \nabla _{x}\psi \right| ^{2}\right) dxdydt \\= & {} -2s\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi ^{2}d_{1}\left( \psi \right) \left| z\right| ^{2}dxdydt-2s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{2}\left| z\right| ^{2}d_{2}\left( \psi \right) dxdydt. \end{aligned}$$

Then we obtain

$$\begin{aligned} 2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\left( P_{s}^{+}z+P_{s}^{-}z\right) \varphi \overline{z}dxdydt= & {} -2\int \nolimits _{Q_{T}^{\prime }}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt\nonumber \\&+\,2\int \nolimits _{Q_{T}^{\prime }}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&+\,B_{1}+X_{3}+X_{4}, \end{aligned}$$
(5.24)

where

$$\begin{aligned} X_{3}= & {} 2s^{2}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi ^{3}d_{2}\left( \psi \right) \left| z\right| ^{2}dxdydt, \\ X_{4}= & {} -2\mathrm{Im}\int \nolimits _{Q_{T}^{\prime }}\partial _{t}z\overline{z} \varphi dxdydt+\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \left| z\right| ^{2}d_{1}\left( \psi \right) dxdydt \\&+\,2s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}d_{2}\left( \psi \right) dxdydt. \end{aligned}$$

We note that

$$\begin{aligned} B_{1}= & {} -\int _{\varGamma _{y}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{y}dxdt+2\mathrm{Re}\int _{\varGamma _{y}}\left( \partial _{\nu }z\right) \left( \varphi \overline{z}\right) dS_{y}dxdt \\&-\,2s\int _{\varGamma _{y}}\gamma \varphi ^{2}\left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{y}dxdt+\int _{\varGamma _{x}}\gamma \varphi \left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{x}dydt \\&-\,2\mathrm{Re}\int _{\varGamma _{x}}\left( \partial _{\nu }z\right) \left( \varphi \overline{z}\right) dS_{x}dydt+2s\int _{\varGamma _{x}}\gamma \varphi ^{2}\left( \partial _{\nu }\psi \right) \left| z\right| ^{2}dS_{x}dxdt=0, \end{aligned}$$

since \(z=0\) on \(\varGamma _{x}\cup \varGamma _{y}\). We multiply (5.24) by \( -s\gamma \left( 4\alpha +\eta \right) \), then we have

$$\begin{aligned}&-\,2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}(4\alpha +\eta )\left( P_{s}^{+}z+P_{s}^{-}z\right) s\gamma \varphi \overline{z}dxdydt \nonumber \\&\quad =2\left( 4\alpha +\eta \right) s\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \left| \nabla _{y}z\right| ^{2}dxdydt -2\left( 4\alpha +\eta \right) s\int \nolimits _{Q_{T}^{\prime }}\gamma \varphi \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&\qquad +X_{5}+X_{6}, \end{aligned}$$
(5.25)

where we choose \(\eta >0\) later [see (5.28)] and

$$\begin{aligned} X_{5}= & {} -2(4\alpha +\eta )s^{3}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{3}\varphi ^{3}d_{2}\left( \psi \right) \left| z\right| ^{2}dxdydt,\\ X_{6}= & {} 2\left( 4\alpha +\eta \right) s\mathrm{Im}\int \nolimits _{Q_{T}^{ \prime }}\gamma \partial _{t}z\overline{z}\varphi dxdydt \\&-\left( 4\alpha +\eta \right) s\int \nolimits _{Q_{T}^{\prime }}\gamma ^{2}\varphi \left| z\right| ^{2}d_{1}\left( \psi \right) dxdydt \\&-\,2\left( 4\alpha +\eta \right) s^{2}\int \nolimits _{Q_{T}^{\prime }}\gamma ^{3}\varphi \left| z\right| ^{2}d_{2}\left( \psi \right) dxdydt. \end{aligned}$$

By adding (5.23) and (5.25) we have

$$\begin{aligned}&2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }-2\left( 4\alpha +\eta \right) s\mathrm{Re}\int \nolimits _{Q_{T}^{ \prime }}\gamma \left( P_{s}^{+}z+P_{s}^{-}z\right) \varphi \overline{z} dxdydt \nonumber \\&\quad \ge 2\eta s\int _{Q_{T}^{\prime }}\gamma \varphi \left| \nabla _{y}z\right| ^{2}dxdydt+\left( 8-8\alpha -2\eta \right) s\int _{Q_{T}^{\prime }}\gamma \varphi \left| \nabla _{x}z\right| ^{2}dxdydt \nonumber \\&\qquad +64\delta _{0}^{2}s^{3}\int _{Q_{T}^{\prime }}\gamma ^{4}\varphi ^{3}|z|^{2}dxdydt+B_{0}+X_{1}+X_{2}+X_{5}+X_{6}. \end{aligned}$$
(5.26)

Moreover, due to the fact that

$$\begin{aligned} \left\| P_{s}z+is\partial _{t}\varphi z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}=\left\| P_{s}^{+}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}+\left\| P_{s}^{-}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}+2\mathrm{Re}\left( P_{s}^{+}z,P_{s}^{-}z\right) _{L^{2}\left( Q_{T}^{\prime }\right) }, \end{aligned}$$

and

$$\begin{aligned}&-\,2\mathrm{Re}\int \nolimits _{Q_{T}^{\prime }}\left( 4\alpha +\eta \right) s\gamma \left( P_{s}z+is\partial _{t}\varphi z\right) \varphi \overline{z} dxdydt \\&\quad \le \left( 4\alpha +\eta \right) \left( \int _{Q_{T}^{\prime }}\left| P_{s}z+is\partial _{t}\varphi z\right| ^{2}dxdydt+\int \nolimits _{Q_{T}^{\prime }}s^{2}\gamma ^{2}\varphi ^{2}\left| z\right| ^{2}dxdydt\right) , \end{aligned}$$

we have

$$\begin{aligned}&\left( 4\alpha +\eta \right) \int \nolimits _{Q_{T}^{\prime }}\left| P_{s}z+is\partial _{t}\varphi z\right| ^{2}dxdydt \\&\quad \ge \int _{Q_{T}^{\prime }}\left| P_{s}^{+}z\right| ^{2}dxdydt+\int _{Q_{T}^{\prime }}\left| P_{s}^{-}z\right| ^{2}dxdydt+2\eta s\gamma \int _{Q_{T}^{\prime }}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt \\&\qquad +\left( 8-8\alpha -2\eta \right) s\gamma \int _{Q_{T}^{\prime }}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt+64\delta _{0}^{2}s^{3}\gamma ^{4}\int \nolimits _{Q_{T}^{\prime }}\varphi ^{3}\left| z\right| ^{2}dxdydt \\&\qquad +B_{0}+X_{1}+X_{2}+X_{5}+X_{6}. \end{aligned}$$

In the last inequality, we see that there exists a constant \(\gamma _{0}\) such that for arbitrary \(\gamma >\gamma _{0}\), the terms of \(X_{1}\) and \( X_{5}\) can be absorbed by \(64\delta _{0}^{2}s^{3}\gamma ^{4}\int \nolimits _{Q_{T}^{\prime }}\varphi ^{3}\left| z\right| ^{2}dxdydt\), and we have

$$\begin{aligned}&\left( 4\alpha +\eta \right) \int \nolimits _{Q_{T}^{\prime }}\left| P_{s}z+is\partial _{t}\varphi z\right| ^{2}dxdydt \\&\quad \ge \int _{Q_{T}^{\prime }}\left| P_{s}^{+}z\right| ^{2}dxdydt+\int _{Q_{T}^{\prime }}\left| P_{s}^{-}z\right| ^{2}dxdydt+2\eta s\gamma \int _{Q_{T}^{\prime }}\varphi \left| \nabla _{y}z\right| ^{2}dxdydt \\&\qquad +\left( 8-8\alpha -2\eta \right) s\gamma \int _{Q_{T}^{\prime }}\varphi \left| \nabla _{x}z\right| ^{2}dxdydt \\&\qquad +\,64\delta _{0}^{2}s^{3}\gamma ^{4}\int \nolimits _{Q_{T}^{\prime }}\varphi ^{3}\left| z\right| ^{2}dxdydt+B_{0}+X_{2}+X_{6}. \end{aligned}$$

Since \(\varphi >0\) on \(\overline{Q_{T}^{\prime }}\) for \(\gamma >\gamma _{0}\), there exist constants \(C_{1}=C_{1}\left( \gamma \right) \) and \( s_{1}=s_{1}\left( \gamma \right) \) such that for all \(s>s_{1}\),

$$\begin{aligned}&\left( 4\alpha +\eta \right) \int \nolimits _{Q_{T}^{\prime }}\left| P_{s}z+is\partial _{t}\varphi z\right| ^{2}dxdydt \\&\quad \ge \int _{Q_{T}^{\prime }}\left| P_{s}^{+}z\right| ^{2}dxdydt+\int _{Q_{T}^{\prime }}\left| P_{s}^{-}z\right| ^{2}dxdydt+C_{1}\left( \gamma \right) \eta s\int _{Q_{T}^{\prime }}\left| \nabla _{y}z\right| ^{2}dxdydt \\&\qquad +\,C_{1}\left( \gamma \right) \left( 8-8\alpha -2\eta \right) s\int _{Q_{T}^{\prime }}\left| \nabla _{x}z\right| ^{2}dxdydt+\,C_{1}\left( \gamma \right) s^{3}\int \nolimits _{Q_{T}^{\prime }}\left| z\right| ^{2}dxdydt \\&\qquad +\,B_{0}+X_{2}+X_{6}. \end{aligned}$$

Then we choose \(s_{2}=s_{2}\left( \gamma \right) >0\) such that \(\forall s>s_{2}\) all the terms of \(X_{2}\) and \(X_{6}\) can be absorbed into \( \left\| P_{s}^{+}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}\), \(\left\| P_{s}^{-}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}, \)\(C_{1}\left\| \nabla _{x}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}\), \(C_{1}\left\| \nabla _{y}z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}\) and \(C_{1}s^{3}\left\| z\right\| _{L^{2}\left( Q_{T}^{\prime }\right) }^{2}\). On the other hand, since

$$\begin{aligned} \left( 4\alpha +\eta \right) \int \nolimits _{Q_{T}^{\prime }}\left| P_{s}z+is\partial _{t}\varphi z\right| ^{2}dxdydt\le & {} \left( 4\alpha +\eta \right) \left( 2\int _{Q_{T}^{\prime }}\left| P_{s}z\right| ^{2}dxdydt\right. \\&\left. +C_{2}s^{2}\int \nolimits _{Q_{T}^{\prime }}\left| z\right| ^{2}dxdydt\right) , \end{aligned}$$

we have

$$\begin{aligned}&C_{3}\left( 4\alpha +\eta \right) \int _{Q_{T}^{\prime }}\left| P_{s}z\right| ^{2}dxdydt \nonumber \\&\quad \ge \int _{Q_{T}^{\prime }}\left| P_{s}^{+}z\right| ^{2}dxdydt+\int _{Q_{T}^{\prime }}\left| P_{s}^{-}z\right| ^{2}dxdydt+\eta s\int _{Q_{T}^{\prime }}\left| \nabla _{y}z\right| ^{2}dxdydt \nonumber \\&\qquad +\left( 8-8\alpha -2\eta \right) s\int _{Q_{T}^{\prime }}\left| \nabla _{x}z\right| ^{2}dxdydt+s^{3}\int \nolimits _{Q_{T}^{\prime }}\left| z\right| ^{2}dxdydt+B_{0} \nonumber \\ \end{aligned}$$
(5.27)

for sufficiently large \(s>0\).

Since (3.6), all the integrations on \(\varGamma _{y}\) vanish and \(\nabla _{y}z=0\), \(\nabla _{x}z=\left( \partial _{\nu }z\right) \cdot \nu \) on \(\varGamma _{x}\), we have

$$\begin{aligned} B_{0}= & {} -4\mathrm{Re}\int _{\varGamma _{x}}s\gamma \varphi \left( \partial _{\nu }z\right) \nabla _{x}\psi \cdot \nabla _{x}\overline{z}dS_{x}dydt+2\int \nolimits _{\varGamma _{x}}s\gamma \varphi \left( \partial _{\nu }\psi \right) \left| \nabla _{x}z\right| ^{2}dS_{x}dydt \\= & {} -8\int _{\varGamma _{x}}s\gamma \varphi \left| \partial _{\nu }z\right| ^{2}(x-x_{0})\cdot \nu dS_{x}dydt{+}4\int \nolimits _{\varGamma _{x}}s\gamma \varphi \left| \partial _{\nu }z\right| ^{2}(x-x_{0})\cdot \nu dS_{x}dydt \\= & {} -4\int \nolimits _{\varGamma _{x}}s\gamma \varphi \left| \partial _{\nu }z\right| ^{2}(x-x_{0})\cdot \nu dS_{x}dydt \\\ge & {} -4\int \nolimits _{\varGamma _{x}\cap \left\{ (x-x_{0})\cdot \nu \ge 0\right\} }s\gamma \varphi \left| \partial _{\nu }z\right| ^{2}(x-x_{0})\cdot \nu dS_{x}dydt. \end{aligned}$$

In (5.27), we choose \(\eta >0\) sufficiently small, such that

$$\begin{aligned} \left( 8-8\alpha -2\eta \right) >0, \end{aligned}$$
(5.28)

then we obtain

$$\begin{aligned}&\int _{Q_{T}^{\prime }}\left| P_{s}^{+}z\right| ^{2}dxdydt+\int _{Q_{T}^{\prime }}\left| P_{s}^{-}z\right| ^{2}dxdydt+s\int _{Q_{T}^{\prime }}\left| \nabla _{x}z\right| ^{2}dxdydt \\&\qquad +s\int _{Q_{T}^{\prime }}\left| \nabla _{y}z\right| ^{2}dxdydt+s^{3}\int \nolimits _{Q_{T}^{\prime }}\left| z\right| ^{2}dxdydt. \\&\quad \le C_{4}\int _{Q_{T}^{\prime }}\left| P_{s}z\right| ^{2}dxdydt+C_{4}s\int _{\partial D_{+}\times G^{\prime }\times (-T,T)}\left| \partial _{\nu }z\right| ^{2}dS_{x}dydt. \end{aligned}$$

Finally, we rewrite the last inequality with z instead of w. By the relations

$$\begin{aligned} \left| z\right| ^{2}= & {} e^{2s\varphi }\left| w\right| ^{2}, \,\left| \partial _{\nu }z\right| ^{2}=\left| \partial _{\nu }w\right| ^{2}e^{2s\varphi }\text { on }\partial D_{+}\times G^{\prime }\times (-T,T), \\ \left| \nabla _{x}we^{s\varphi }\right| ^{2}= & {} \left| \nabla _{x}z-s\lambda \varphi e^{s\varphi }w\nabla _{x}\psi \right| ^{2}\le 2\left| \nabla _{x}z\right| ^{2}+2s^{2}\lambda ^{2}\varphi ^{2}\left| \nabla _{x}\psi \right| ^{2}\left| z\right| ^{2},\\ \left| \nabla _{y}we^{s\varphi }\right| ^{2}= & {} \left| \nabla _{y}z-s\lambda \varphi e^{s\varphi }w\nabla _{y}\psi \right| ^{2}\le 2\left| \nabla _{y}z\right| ^{2}+2s^{2}\lambda ^{2}\varphi ^{2}\left| \nabla _{y}\psi \right| ^{2}\left| z\right| ^{2},\\ \left| L_{0}w\right| ^{2}\le & {} 2\left| Lw\right| ^{2} \\&+\,2\left| \sum \limits _{i=1}^{n}a_{i}\left( x,y,t\right) w_{x_{i}}{+}\sum \limits _{j=1}^{m}b_{j}\left( x,y,t\right) w_{y_{j}}{+}a_{0}\left( x,y,t\right) w\left( x,y,t\right) \right| ^{2}, \end{aligned}$$

we see that there exist positive constants \(C_{5}=C_{5}\left( \gamma \right) \) and \(s_{0}>s_{2}\left( \gamma \right) \) such that for all \(s>s_{0}\),

$$\begin{aligned}&\int _{Q_{T}^{\prime }}(\left| P_{s}^{+}w\right| ^{2}+\left| P_{s}^{-}w\right| ^{2})e^{2s\varphi }dxdydt \\&\qquad +\int _{Q_{T}^{\prime }}(s\left| \nabla _{y}w\right| ^{2}+s\left| \nabla _{x}w\right| ^{2}+s^{3}\left| w\right| ^{2})e^{2s\varphi }dxdydt \\&\quad \le C_{5}\int _{Q_{T}^{\prime }}\left| Lw\right| ^{2}e^{2s\varphi }dxdydt+C_{5}\int _{\partial D_{+}\times G^{\prime }\times (-T,T)}s\left| \partial _{\nu }w\right| ^{2}e^{2s\varphi }dS_{x}dydt. \end{aligned}$$

We complete the proof of Lemma 1.

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Gölgeleyen, F., Kaytmaz, Ö. A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation. Anal.Math.Phys. 9, 2171–2199 (2019). https://doi.org/10.1007/s13324-019-00326-6

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  • DOI: https://doi.org/10.1007/s13324-019-00326-6

Keywords

  • Ultrahyperbolic Schrödinger equation
  • Inverse problem
  • Stability
  • Carleman estimate

Mathematics Subject Classification

  • 35R30
  • 35B35
  • 35Q40