An identification problem for an ultraparabolic integrodifferential equation
The problem studied by the author is the following: determine a pair of functions \(u:[0,T^*]^2 \times\Omega\to \mathbb{R}\) and \(h:[0,T^*]^2 \to\mathbb{R}\) such that \[ D_tu(t,s,x) +D_su(t,s,x)- {\mathcal A}u(t,s,x) =f(t,s,x)+ \int^t_0 d\tau \int^s_0 h(t-\tau, s-\sigma) {\mathcal B}u(\tau,\sigma, x)d\sigma \] for \((t,s,x) \in [0,T^*]^2 \times\Omega\), and \(u(t,0,x) =u_1(t,x)\), \((t,x)\in [0,T^*] \times \overline \Omega\); \(u(0,s,x) =u_2(s,x)\), \((s,x)\in [0,T^*] \times \overline \Omega\); \(u(t,s,x)=0\), \((t,s,x) \in[0,T^*]^2 \times\partial\Omega\); \[ \int_\Omega \varphi (x)u(t,s,x) dx=k(t,s), \quad(t,s) \in[0,T^*]^2, \] where \(\Omega\) is an open set in \(\mathbb{R}^n\), \({\mathcal A}\) and \({\mathcal B}\) are linear second-order differential operators with variable coefficients depending on \(x\in \overline\Omega\) only. A basic requirement is that \({\mathcal A}\) should be uniformly elliptic and, consequently, the differential operator \(D_t+D_s- {\mathcal A}\) turns out to be ultraparabolic. The aim of this paper consists of proving a local in time existence, uniqueness and continuous dependence result. The author proves the well posedness of an inverse problem related to an abstract integro-differential problem in two time variables with applications to ultraparabolic partial integrodifferential equations of Volterra type.
ultraparabolic partial integrodifferential equation of Volterra type
scientific article; zbMATH DE number 1334719
scientific article