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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: $$\cases \frac{\partial u}{\partial t} - d_1\Delta u = c - f (u,v)-\alpha u, &\text{in }\mathbb R^+\times\Omega, \\ \frac{\partial v}{\partial t} - d_2\Delta v= g (u,v) -\sigma v,&\text{in }\mathbb R^+\times\Omega, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu}=0,&\text{on }\mathbb R^+\times\Gamma, \\ u(0,x)=u_0 (x), \ \ v(0,x)=v_0 (x), \endcases\tag1$$ where $\Omega\subseteq{\mathbb R^n}$ is an open and bounded domain with a $C^1$ boundary $\Gamma,$ $d_{1}, d_{2}, \alpha, \sigma,$ are positive constants, $c\geq 0 $ is a constant, and $f, g\in C^1 (\mathbb R^+ \times (\mathbb R^+)$ 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).

35K50Systems of parabolic equations, boundary value problems (MSC2000)
35K57Reaction-diffusion equations
35B45A priori estimates for solutions of PDE
35B35Stability of solutions of PDE
35A05General existence and uniqueness theorems (PDE) (MSC2000)
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