Global existence of reaction-diffusion equations over multiple domains

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Title: Global existence of reaction-diffusion equations over multiple domains
Author: Ryan, John Maurice-Car
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 = D & #916 ;u + f (t , x , u ) & #8706 ;uk / & #8706 ; & #951 ; =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 & #937 ; & #963 ; (k ) , D = diag (dk )is an m ? ? m positive de & #64257 ;nite diagonal matrix , f : R ? ? Rn ? ? Rm & #8594 ; Rm , u0 is a componentwise nonnegative function , and each & #937 ;i is a bounded domain in Rn where & #8706 ; & #937 ;i is a C2+ & #945 ;manifold such that & #937 ;i lies locally on one side of & #8706 ; & #937 ;i and has unit outward normal & #951 ; . Most physical processes give rise to systems for which f = (fk ) is locally Lipschitz in u uniformly for (x , t ) & #8712 ; & #937 ; ? ? [0 ,T ] and f ( ? ? , ? ? , ? ? ) & #8712 ; L & #8734 ; ( & #937 ; ? ? [0 ,T ) ? ? U ) for bounded U and the initial data u0 is continuous and nonnegative on & #937 ; . 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 .
URI: http : / /hdl .handle .net /1969 .1 /3312
Date: 2006-04-12

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Global existence of reaction-diffusion equations over multiple domains. Available electronically from http : / /hdl .handle .net /1969 .1 /3312 .

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