Global classical solutions for reaction-diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for problems of the form $\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)$, $\frac{\partial v}{\partial t} -b \Delta v=g(u,v)$ in \((0,+ \infty) \times \Omega\) where $ \Omega $ is an open bounded domain of class $ C^1$ in $\mathbb{R}^n$, $a >0$, $b >0$, $a \neq b$ and $f$, $ g $ are nonnegative continuously differentiable functions on $[0,+ \infty)\times [0,+ \infty)$ satisfying $f (0,\eta) = 0$, $g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}$ and $g(\xi,\eta) \leq \psi(\eta)f(\xi,\eta)$ for some $C >0$, $\alpha >0$ and $\beta \geq 1$ where $\varphi$ and $\psi$ are any nonnegative continuously differentiable functions on $[0,+ \infty)$ satisfying $\varphi(0)=0$ and $ \displaystyle \lim_{\eta \rightarrow +\infty} \eta^{\beta -1} \psi(\eta)=1$. The asymptotic behavior of the global solutions as $ t $ goes to $ + \infty $ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods