Last time, I’ve written a paper which considered the following Reaction diffusion equations on

(1)

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

as .

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi