GLOBAL EXISTENCE AND OPTIMAL TEMPORAL DECAY-ESTIMATES FOR SYSTEMS OF PARABOLIC CONSERVATION-LAWS - II - THE MULTIDIMENSIONAL CASE

This paper is a continuation of our previous paper. It is concerned wi th the global existence and the optimal temporal decay estimates for t he Cauchy problem of the following multidimensional parabolic conserva tion laws [GRAPHICS] Here u(t, x) = (u(1)(t, x),...,u(n)(t, x))(t) is the unknown vector, f(j)(u) = (f(j1)(u),..., f(jn)(u))(t) (j = 1,2,... , N) are arbitrary n x I smooth vector-valued flux functions defined i n <(B)over bar (r)>((u) over bar), a closed ball of radius r centered at some fixed vector (u) over bar is an element of R-n, and D is a con stant, diagonalizable matrix with positive eigenvalues, Our results sh ow that if the flux function f(j)(u) satisfies f(j)(u)/\u - (u) over b ar\(s) is an element of L-infinity(<(B)over bar (r)>((u) over bar), R- n), j = 1, 2,..., N for some s > 2 + 1/N, (u) over bar is an element o f R-n, then for u(0)(x) - (u) over bar is an element of L-infinity boo lean AND L-1(R-N, R-n) with \\u(0)(x) - (u) over bar\\(L1(RN, Rn)) suf ficiently small, the above Cauchy problem () admits a unique globally smooth solution u(t, x) and u(t, x) satisfies the following temporal decay estimates. For each k = 0, 1, 2,... [GRAPHICS] Here D-k = Sigma( \alpha\=k)(partial derivative(\alpha\)/partial derivative x1(alpha 1). ..partial derivative x(N)(alpha N)). The above decay estimates are opt imal in the sense that they coincide with the corresponding decay esti mates for the solution to the linear part of the corresponding Cauchy problem. (C) 1998 Academic Press.