Calderón problem for nonlocal viscous wave equations: Unique determination of linear and nonlinear perturbations

The main goal of this article is to study a Calderón type inverse problem for certain viscous nonlocal wave equations. We show that the partial Dirichlet to Neumann map uniquely determines on the one hand linear perturbations and on the other hand homogeneous nonlinearities <i>f</i>(<i>u</i>) whenever the latter satisfy a certain growth assumption. As a preliminary step we discuss the well-posedness in each case, where for the nonlinear setting we invoke the implicit function theorem after establishing the differentiability of the associated Nemytskii operator <i>f</i>(<i>u</i>). In the linear case we establish a Runge approximation theorem in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup><mml:mover><mml:mi>H</mml:mi> <mml:mo>~</mml:mo></mml:mover> <mml:mi>s</mml:mi></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> , which allows us to uniquely determine potentials that belong only to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mi>∞</mml:mi></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> for some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi></mml:mrow> </mml:math> satisfying suitable restrictions. In the nonlinear case, we first derive an appropriate integral identity and combine this with the differentiability of the solution map around zero to show that the nonlinearity is uniquely determined by the Dirichlet to Neumann map. To make this linearization technique work, it is essential that we have a Runge approximation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup><mml:mover><mml:mi>H</mml:mi> <mml:mo>~</mml:mo></mml:mover> <mml:mi>s</mml:mi></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> instead of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>Ω</mml:mi> <mml:mi>T</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> at our disposal.