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On small energy stabilization in the NLKG with a trapping potential. (English) Zbl 1356.35142

The authors are interested in the following semilinear Cauchy problem for the Klein-Gordon equation in \(\mathbb{R}^4\): \[ \begin{aligned} u_{tt} - \Delta u + V(x) u + m^2 u + u|u|^2=0,\quad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),\\ \text{where } u_0 \in H^1(\mathbb{R}^3;\mathbb{C}),\quad u_1\in L^2(\mathbb{R}^3;\mathbb{C}),\quad V \in S(\mathbb{R}^3;\mathbb{R}). \end{aligned} \] Here, \(S(\mathbb{R}^3)\) denotes the space of Schwartz functions. The nonlinearity \(u|u|^2\) is analytic and absorbing. The potential \(V\) belongs to the class of short range potentials. The data are supposed to be complex-valued. It is already known that if the small data are supposed to be real-valued, then any small energy solution scatters to the origin. If the small data are supposed to be complex-valued, then \((u(t,\cdot),u_t(t,\cdot))\) can scatter to small standing waves. If the point spectrum of \( - \Delta + V(x)\) is empty (only the essential spectrum appears), then the scattering to zero of all small energy solutions is known. For this reason the authors are interested in the case that there exists a point spectrum and the data are complex-valued. So, the interaction of discrete and continuous modes become of interest.
For the discrete spectrum the authors assume that it is formed by \(n\) points \(e_1,e_2,\dots,e_{n-1},e_n\) with \(-m^2<e_1<\cdots< e_n <0\), where all these eigenvalues have the multiplicity one. Moreover, zero is no resonance. Then, the authors prove, that for small data the solution \((u(t,\cdot),u_t(t,\cdot))\) has the following representation: \[ (u(t,\cdot),u_t(t,\cdot)) =\Phi[z(t)] + (\eta(t,\cdot),\xi(t,\cdot)). \] Here
\(\Phi[z(t)]\) is determined by bounded states of the starting Cauchy problem;
\((\eta(t,\cdot),\xi(t,\cdot))\) scatters to \((w(u_+,v_+),d_t w(u_+,v_+))\) in the energy space \(H^1 \times L^2\), where \((u_+,v_+)\) are suitably chosen with \(\|(u_+,v_+)\|_{H^1 \times L^2} \leq C\varepsilon\) and \[ w(u_+,v_+) = K_0'(t)u_+ + K_0(t) v_+ \quad\text{with } K_0(t)= \frac{\sin(t \sqrt{-\Delta +m^2})}{\sqrt{-\Delta +m^2}}. \]
Finally, some norm estimates of terms appearing in the above representation for \((u(t,\cdot),u_t(t,\cdot))\) complete the paper.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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References:

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