ASYMPTOTIC LIMIT AND DECAY ESTIMATES FOR A CLASS OF DISSIPATIVE LINEAR HYPERBOLIC SYSTEMS IN SEVERAL DIMENSIONS

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in, for instance, the velocity-jump processes in several dimensions. Given integers n,d >= 1, let A :=(A(1), ... , A(d)) is an element of (R (n x n))(d) be a matrix-vector and let B is an element of R- n x n be not necessarily symmetric but have one single eigenvalue zero, we consider the Cauchy problem for n x n linear systems having the form partial derivative(t)u + A .del(x) u + Bu = 0, (x,t) is an element of R-d x R+. Under appropriate assumptions, we show that the solution u is decomposed into u = u((1)) +u((2)) such that the asymptotic profile of u((1)) denoted by U is a solution to a parabolic equation, u((1)) - U decays at the rate t (-d/2(l/q-)(P/1)) (-1/2) as t -> +infinity in any L-P-norm and u((2)) decays exponentially in L-2 -norm, provided u(.,0) is an element of L-q (R-d) boolean AND L-2 (R-d) for 1 <= q <= p <= infinity. Moreover, u((1)) - U decays at the optimal rate t(-d/2) (()(1/q-1/p)-1) as t -> +infinity if the original system satisfies a symmetry property. The main proofs are based on the asymptotic expansions of the fundamental solution in the frequency space and the Fourier analysis.