Given a system of ODE or PDE and a solution $u$ such that $$ \partial_t u = \mathcal{L}u , \quad u(t=0 ) = 0.$$ for some operator $\mathcal{L}$, one can ask the following question
Let $u$ be the solution of the PDE with the initial condition $u(t = 0) = u_0$. Determine a norm $\lVert \, \cdot \, \rVert$ and a threshold $\epsilon$ such that $\lVert u_0 \rVert < \epsilon$ implies $u \xrightarrow{t \to \infty} u_\infty$ in a precise sense.
For this question, one can find literature with techniques specific to this problem. I am interested in the following "forced" question:
Let $u$ be the solution of $$u_t = \mathcal{L}u + f, \quad u(t = 0) = 0. $$ Determine a norm $\lVert \, \cdot \, \rVert$ and a threshold $\epsilon$ such that $\lVert f \rVert < \epsilon$ implies $u \xrightarrow{t \to \infty} u_\infty$ in a precise sense.
Are there any references that deal with the above question, or something in the neighborhood or more generally asymptotic stability of forced PDEs?