×

Unicity of solution in an inverse problem for unknown source term in semilinear parabolic partial differential equation on bounded domain. (Chinese. English summary) Zbl 0986.35131

Summary: An inverse problem for the unknown source term in the semilinear parabolic partial differential equation on the bounded domain \(\Omega\subset\mathbb{R}^n\) is considered \[ u_t-Lu=\varphi (x,t)s(u)+ \gamma (x,t), \quad(x,t)\in \Omega\times (0,T),\;u(x,0)=u_0,\;x\in\Omega, \]
\[ {\partial u\over\partial n}|_{\partial \Omega\times (0,T)}=g(x,t),\quad u(x_0,t)=f(t), \;0<t<T. \] Here \(x_0\in \partial\Omega\) is fixed, \[ L=\sum^n_{i,j=1} a_{ij}(x) {\partial^2\over \partial x_i\partial x_j}+ \sum^n_{i=1} b_i(x){\partial \over \partial x_i}-c(x) \] is a uniform elliptic operator and \(0\leq c(x)\leq C_0\), \(u_0\geq 0\) is a constant. By establishing the monotoneous dependence of \(u(x_0,t)\) from \(s(u)\), the uniqueness of the solution is obtained.

MSC:

35R30 Inverse problems for PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations