Global existence of reaction-diffusion equations over multiple domains
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 = D?u + f (t, x, u)
?uk/?? =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 ??(k), D = diag(dk)is an m Ã? m positive de?nite diagonal matrix,
f : R Ã? RnÃ? Rm ? Rm, u0 is a componentwise nonnegative function, and each ?i is a bounded domain in Rn where ??i is a C2+?manifold such that ?i lies locally on one side of ??i and has unit outward normal ?. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ? ? Ã? [0,T ] and f (Â·, Â·, Â·) ? L?(? Ã? [0,T ) Ã? U ) for bounded U and the initial data u0 is continuous and nonnegative on ?.
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.
Advisor:Walton, Jay; Lowe, Bruce; Lee, D. Scott; Pasciak, Joe
School:Texas A&M University
School Location:USA - Texas
Source Type:Master's Thesis
Keywords:existence differential equations reaction diffusion
Date of Publication:12/01/2004