Global existence of reaction-diffusion equations over multiple domains
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
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.