Solutions in the large for certain nonlinear parabolic systems in arbitrary spatial dimensions

Abstract

We prove the global existence of smooth solutions for certain nonlinear systems of the form is a vector, are n x n smooth matrixes and D is a constant positive matrix. We assume the Cauchy data u0 satisfies , where a is a fixed vector, Aj(u)(j =1,2,. . .,N) defined in an r-ball about sufficiently small. The main techniques we used in this paper are motivated by the work of Hoff and Smoller11,19. Our results contain the former results of [11,19] as a special case and some new global existence results are established. As a by-product, our proofs indicate that the solutions we get are indeed analytic on t > 0. These regularity results, to the knowledge of the author, generalize the regularity results of Hoff and Smoller11,19.