| Summary: | This note studies the linearly damped generalized Hartree equation <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mo>±</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mrow><mo stretchy="false">(</mo></mrow><msub><mi>J</mi><mi>γ</mi></msub><msup><mrow><mo>∗</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mrow><mo>)</mo><mi>u</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>p</mi><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></mrow></semantics></math></inline-formula> Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula> norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>:</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mi>t</mi></mrow></msup><mi>u</mi></mrow></semantics></math></inline-formula> scatters in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula>, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and the last one is a damping term <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>a</mi><mi>u</mi></mrow></semantics></math></inline-formula>. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms.
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