The existence of a unique global bounded classical solution of problem (1.1) is established under the assumption that the coefficients of the kinetic terms are large enough.
Based on the maximal Sobolev regularity, the existence of a unique global bounded classical solution of the problem is established under the assumption that both (mu_{1}) and (mu_{2}) are sufficiently large.
Assume that u is the bounded classical solution of (1.6) in (overline{Q}_{T_{varepsilon}} ), (Q_{T_{varepsilon}}=mathbb{R}^{2} times 0,T-varepsilon) ), for any given (varepsilonin(0,times 0,T-varepsilon
Thus, ((u,v,w)) is a global bounded classical solution to system (1.2). □.
In this section, we mainly investigate the existence of global bounded classical solutions to system (1.2) with nondegenerate diffusion.
Then system (4.1) admits a unique global bounded classical solution ((u_{varepsilon}, v_{varepsilon}, w_{varepsilon})).
When (n=2), (tau=1), and (xigamma-chialpha>0), the model (1.1) possesses a unique global bounded classical solution with any sufficient regular initial data (see [30, 31]).
We would like to point out that when (N=2), Theorem 1.2 holds for any (b_{1}>0), and then this fact, together with Theorem 1.1, implies that both (1.1) and (1.2) admit global and uniformly bounded classical solutions regardless of the sign and size of the advection rate χ.
We now turn to the existence of global bounded classical solutions.
It is proved in [1] that, when (N=1), (1.1) and its fully parabolic counter-part admit global classical solutions which are uniformly bounded, and when (Ngeq2), (1.1) admit global and uniformly bounded classical solutions provided that (chi<0) and (b_{1}) is sufficiently large.
(b) If there exists a positive, bounded, and classical solution of (7), then there exists a positive, bounded, and classical solution of left { textstylebegin{array}{l} u_{t}-triangle u=v^{p}+mu_{1} u^{r}, quad tinmathbb{R}, xinmathbb {R}^{n-1}, v_{t}-triangle v=u^{q}+mu_{2} v^{s}, quad tinmathbb{R}, xinmathbb{R}^{n-1}.
