Let \(\emptyset \neq \Omega \subset \mathbb{R}^ n\) be a bounded domain with \(C^ 2\)-boundary and \(T > 0\). Of concern is the semilinear second order initial-boundary value problem \[ \begin{alignedat}{2} \partial_ tu(t,x) - A(t)u(t,x) & = f \bigl( t,x,u(t,x) \bigr), &\qquad (t,x) &\in (0,T) \times \Omega, \\ u(t,x) & = \int_ \Omega \Phi (x,y) u(t,y) dy, &\qquad (t,x) &\in (0,T) \times \partial \Omega, \\ u(0,x) & = u_ 0(x), &\qquad x &\in \Omega. \end{alignedat}\tag {\(*\)} \] Here, \(A(t) : = \sum_{i,j=1}^ n a_{i,j} (t, \cdot) \partial_ i \partial_ j + \sum_{i=1}^ n b_ i (t, \cdot) \partial_ i\) is uniformly elliptic, \(a_{i,j}\) and \(b_ i\) are uniformly Hölder-continuous, \(u_ 0 \in C(\text{cl} (\Omega))\) and \(f \in C ([0,T] \times \text{cl} (\Omega) \times \mathbb{R})\). \(\Phi\) is assumed to be continuous and \(\geq 0\) on \(\partial \Omega \times \text{cl} (\Omega)\), and \(\sup_{x \in \partial \Omega} \int_ \Omega \Phi (x,y)dy 0) (\forall t \in [0,T]\), \(x \in \text{cl} (\Omega)\), \(y,z \in [\vartheta (t,x), \eta (t,x)]) : f(t,x,y) - f(t,x,z) \geq - N(y - z)\). Also, uniqueness and exponential decay are obtained under additional hypotheses. The paper extends results of \textit{A. Friedman} [Q. Appl. Math. 44, 401-407 (1986; Zbl 0631.35041)] for the linear case and of \textit{K. Deng} [Q. Appl. Math. 50, 517-522 (1992; Zbl 0777.35006)].