Abstract: Consider the equation utt 􀀀
u + (x)(ut)m + q(x)u2 = 0; (x; t) 2 R4; ujt<0 = g(t 􀀀 x
 􀀀 R); (1) where x = (x1; x2; x3), (x) and q(x) are smooth finite functions with support in the ball B(R) = fx 2 R3j jxj < Rg, m > 1 is a real number and  is a unite vector. The function g(t) is supposed such that g(t) = 0 for t < 0 and g(0) = 0, g0(+0) = a > 0, g 2 C1[0;1). If (x) = 0 and q(x) = 0 then u(x; t) = g(t 􀀀 x
 􀀀 R) is the plane wave running in direction . A solution of problem (1) corresponds to falling down of the wave on the inhomogeneity located in B(R). The wave front t = x
 +R tangents to the boundary of ball B(R) at point x = 􀀀R in the moment of time t = 0. It is supposed that  = (') = (cos '; sin '; 0), ' 2 [0; ), and ' is the variable parameter of the problem. Related to this, by u(x; t; ') is denoted a solution of problem (1). Let S(R; ) = fx 2 R3j jxj = R; x
 > 0g be a part of the boundary of B(R) and g(t) be the given function. For the wave equation (1) the following inverse problem is studied. The inverse problem. Find (x) and q(x) for x 2 B(R) from the given trace of solutions to problem (1) on S(R; ) for variable  and for an interval of time, i.e. from the given function p(x; t; ') = u(x; t; '); 8' 2 [0; ); 8x 2 S(R; ); t 2 [x
 + R 􀀀 ; x
 + R + ]; (2) where  > 0 is an arbitrary small number. The main result is concluded in a reduction of the original inverse problem for coefficient (x) to the usual X-ray tomography problem and for q(x) to a new problem of the integral geometry. A stability estimate for solutions of the latter problem is stated.

Cite: Romanov V.G.
An inverse problem for a quasilinear wave equation
International conference “DAYS ON DIFFRACTION 2024” 10-14 Jun 2024