Global existence of reaction-diffusion equations over multiple domains

by Ryan, John Maurice-Car

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 � m positive de?nite diagonal matrix, f : R � Rn� 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) ? ? � [0,T ] and f (·, ·, ·) ? L?(? � [0,T ) � 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.