A hyperbolic inverse problem arising in the evolution of combustion aerosol
The authors consider the following hyperbolic problem arising in the theory of combustion aerosols: \[ (1)\quad D_ tu(x,t)+3k(t)D_ x(x^{2/3}u)(x,t)=C(u)(x,t),\quad (x,t)\in (1,+\infty)\times (0,T):=Q_ T; \] \[ (2)\quad u(x,0)=u_ 0(x),\quad x\in [1,+\infty);\quad (3)\quad u(1,t)=h(t),\quad t\in [0,T], \] where \[ C(u)=Y_+(x-2)\int^{x- 1}_{1}q(x-y,y)u(x-y,t)u(y,t)dy- u(x,t)\int^{+\infty}_{1}q(x,y)u(y,t)dy, \] \(Y_+\) denoting the Heaviside function. Under suitable assumptions on data h, \(u_ 0\) and kernel q they prove the existence in the large of a nonnegative solution to problem (1)-(3). As far as the uniqueness is concerned, a first result (in the large) requires the boundedness of q, while a second one (essentially local in time) allows the kernel q to be unbounded. The authors also consider the identification problem consisting in determining a triplet of functions u: \(Q_ T\to {\mathbb{R}}_+\), k,I: [0,T]\(\to {\mathbb{R}}_+\) satisfying, in addition to equation (1)-(3) (with \(u_ 0=0\) and \(h=(1/3)I/k)\), also the following information: \[ (4)\quad \int^{+\infty}_{1}x^ ju(x,t)dt=f_ j(t),\quad t\in [0,T],\quad j=1,2. \] Here \(f_ 1,f_ 2: [0,T]\to {\mathbb{R}}_+\) are prescribed functions. Concerning problem (1)-(4) a maximal in time existence result is proved together with a uniqueness-stability result.
scientific article; zbMATH DE number 4157084
scientific article