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On the uniform boundedness of the solutions of systems of reaction-diffusion equations. (English) Zbl 1090.35095

The authors consider the following reaction-diffusion system, which may describes the dynamics of an epidemic disease:

$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}-{d}_{1}{\Delta }u=c-f\left(u,v\right)-\alpha u,\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{+}×{\Omega },\hfill \\ \frac{\partial v}{\partial t}-{d}_{2}{\Delta }v=g\left(u,v\right)-\sigma v,\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{+}×{\Omega },\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0,\hfill & \text{on}\phantom{\rule{4.pt}{0ex}}{ℝ}^{+}×{\Gamma },\hfill \\ u\left(0,x\right)={u}_{0}\left(x\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}v\left(0,x\right)={v}_{0}\left(x\right),\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\Omega }\subseteq {ℝ}^{n}$ is an open and bounded domain with a ${C}^{1}$ boundary ${\Gamma },$ ${d}_{1},{d}_{2},\alpha ,\sigma ,$ are positive constants, $c\ge 0$ is a constant, and $f,g\in {C}^{1}\left({ℝ}^{+}×\left({ℝ}^{+}\right)$ are nonnegative functions. $\frac{\partial }{\partial \nu }$ denotes differentiation in the direction of the outward normal.

The authors, under suitable conditions, and using ${L}^{p}$ arguments, prove the global existence and uniform boundedness of solutions for the system (1).

##### MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K57 Reaction-diffusion equations 35B45 A priori estimates for solutions of PDE 35B35 Stability of solutions of PDE 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 92D30 Epidemiology
##### Keywords:
global existence; epidemic disease