| dc.description.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 Ã— [0,T ] and f (·, ·, ·) ∈ L∞(Ω Ã— [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. | en |
