Global existence of reaction-diffusion equations over multiple domains
Ryan, John Maurice-Car
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Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = DΔu + f (t, x, u) ∂uk/∂η =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside Ωσ(k), D = diag(dk)is an m ?? m positive deﬁnite diagonal matrix, f : R ?? Rn?? Rm → Rm, u0 is a componentwise nonnegative function, and each Ωi is a bounded domain in Rn where ∂Ωi is a C2+αmanifold such that Ωi lies locally on one side of ∂Ωi and has unit outward normal η. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ∈ Ω ?? [0,T ] and f (??, ??, ??) ∈ L∞(Ω ?? [0,T ) ?? U ) for bounded U and the initial data u0 is continuous and nonnegative on Ω. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f.