Last time, I’ve written a paper which considered the following Reaction diffusion equations on
Under condition and suitable assumptions on nonlinearity and external force , I’ve been able to prove the existence of a uniform attractor for the corresponding process.
Now I get a question: Why has to be positive? What happens if .
To answer this question, I should first state that “I need a boundedness uniformly in time on solutions ”
Now, I go back to where I need . Multiply (1) by then integrating over , after some computations, we get
If , then we easily get the boundness of in by Gronwall’s inequality, that means, for all .
If we don’t see any way to do that (at least by standard way).
I also state here a counter-example in one dimensional case: Consider
Here we have and . It’s easy to check that is a solution of the equation, and