Global time estimates for solutions to equations of dissipative type

Abstract

Global time estimates of L p − L q norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass. The paper is devoted to the time decay of L p − L q norms of solutions to constant coefficients strictly hyperbolic equations of general form.It is known that such esti- mates lead to Strichartz estimates which are a powerful technique when dealing with nonlinear problems. We will assume that the principal part of the equation is strictly hyperbolic. The full equation may have variable multiplicities because of the lower order terms. One question of interest is to identify properties of such equations which determine the time decay rate of solutions. Another question of interest is what happens when there are multiple characteristic roots. Equations of higher orders appear in many applications. In particular, they arise as dispersion equations for hyperbolic systems, for example in the study of the Fokker- Planck equation and Grad systems in nonequilibrium thermodynamics. Moreover, in approximations of solutions to the Fokker-Planck equation the order of the cor- responding system tends to infinity. However, it turns out to still be possible to determine the decay rate of its solutions. The behaviour exhibited by these exam- ples is similar to the behaviour of the dissipative wave equation in the sense that characteristic roots lie in the complex upper half plane and come to the origin as single roots and at isolated points. That is why in this paper we will concentrate on equations of such type in Theorem 2.2, although we will also present a more general Theorem 2.1. Results described here are formulated for scalar equations. However, they can be easily extended to systems. They also yield the well-posedness results for semilinear equations. Details of such analysis will appear elsewhere.