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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:

u t-d 1 Δu=c-f(u,v)-αu,in + ×Ω,v t-d 2 Δv=g(u,v)-σv,in + ×Ω,u ν=v ν=0,on + ×Γ,u(0,x)=u 0 (x),v(0,x)=v 0 (x),(1)

where Ω n is an open and bounded domain with a C 1 boundary Γ, d 1 ,d 2 ,α,σ, are positive constants, c0 is a constant, and f,gC 1 ( + ×( + ) are nonnegative functions. ν 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:
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)
92D30Epidemiology