Abstract
We study the uniform boundedness of solutions to reaction–diffusion systems possessing a Lyapunov-like function and satisfying an intermediate sum condition. This significantly generalizes the mass dissipation condition in the literature and thus allows the nonlinearities to have arbitrary polynomial growth. We show that two dimensional reaction–diffusion systems, with quadratic intermediate sum conditions, have global solutions which are bounded uniformly in time. In higher dimensions, bounded solutions are obtained under the condition that the diffusion coefficients are quasi-uniform, i.e. they are close to each other. Applications include boundedness of solutions to chemical reaction networks with diffusion.
Highlights
We study the uniform boundedness of solutions to reaction-diffusion systems possessing a Lyapunov-like function and satisfying an intermediate sum condition
The assumption (A4) is called an intermediate sum condition, in the sense that only one of the nonlinearties is assumed to be one-side bounded by a polynomial of order r, while for the others we just need a good ”cancellation”
In the main result, we show that, if the diffusion coefficients are quasi-uniform, meaning that if they are close to each other, one can obtain global strong solutions to (1)
Summary
For pN ≥ 4, such that ui ∈ Lq(Ω × (0, T )) for all q < pN. Step 5: Since p0 > 2, the sequence {pN } is strictly increasing and there exists pN0 ≥ 4 which makes pN0 = +∞. If K = 0 in (A3), the solution is bounded uniformly in time, i.e. It is again remarked that the condition on the closeness of the diffusion coefficients (8) depends only on r in (A4) and independent of the polynomial growth μ in (A5). In the recent work [CMT19], global existence and uniform-in-time bounds were obtained for (1) under mass dissipation condition (3) and quasi-uniform diffusion coefficients. We remark that the latter condition imposed in [CMT19] depends on the growth μ of the nonlinearities, and is much less general than (8). We write Cq,T to indicate that the constant depends on q, and might blow up to infinity when q → p
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