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 à m positive definite diagonal matrix,
f : R à Rnà 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 ) à 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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/3312 |
| Date | 12 April 2006 |
| Creators | Ryan, John Maurice-Car |
| Contributors | Walton, Jay |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | 477152 bytes, electronic, application/pdf, born digital |
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