Consider the following differential equation (DE) $$\dot{x}(t)=ax(t)+f(x(t))+u(t).$$ Let assume there is number $M$ such that the DE admits a local solution over some interval $[0,\tau]$ for all $|u(t)|\le M$.
Under this setting, can you find a counter examples in which the DE does not admit a solution over the same interval $[0,\tau]$ for $|u|>M$ or $u>M$ or $u<-M$?