Study of nonlinear PDE with power nonlinearities

Let \( u' + Au = h(u,t) + f(x,t) \) with the initial condition \( u(x,0) = u_0(x) \), where \( u \in H \), \( u' := u_t := \frac{du}{dt} \), and \( H \) is a Hilbert space. The nonlinear term satisfies the estimate \( \|h(u,t)\| \le a\|u\|^p (1+t)^{-b} \), and the operator \( A \) satisfies the coercivity condition \( (Au,u) \ge \gamma(t)(u,u) \), where \( \gamma(t) = q_0(1+t)^{-q} \). Here, \( a, p, b, q_0, \) and \( q \) are positive constants. Sufficient conditions are established under which the solution exists and is either bounded or tends to zero as \( t \to \infty \).