Global time estimates for solutions to equations of dissipative type

Abstract

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.

Conservation laws and their hyperbolic regularizations

“…The assumption on the relaxation operator γ 1 in system (21) can be weakened if we assume that μ 2 = μ 2 (|ξ| 2 ), where μ 2 (0) = 0 and μ 2 (|ξ| 2 ) is a smooth stabilizing at the infinity function of the kink type, i.e., with the boundary layer, so that we can apply to the integral Fourier operator the L p − L q -estimates (see, e.g., [21,22]). More essential is the finding of the outer force by writing the conservation laws with dissipation for magnetic and electric currents.…”

Dispersive estimates for hyperbolic systems with time-dependent coefficients

“…The decay of solutions to the scalar higher order equations with constant coefficients of dissipative type was considered in [13]. Consequently, a comprehensive analysis of the dispersive and Strichartz estimates of scalar hyperbolic equations of higher orders with constant coefficients, of general form, as well as for hyperbolic systems with constant coefficients, with applications to nonlinear equations was carried out in [14].…”

Asymptotic integration and dispersion for hyperbolic equations

“…In particular, properties of characteristic roots are crucial in determining exact decay rates and the complete analysis is quite lengthy and involved. For example, in the case of equations of dissipative types analysed in [21] the decay rate is determined by properties of characteristics for small frequencies. A general analysis of this type is necessary for application to large systems, such as Grad systems in gas dynamics, or to Fokker-Planck equations, in which case the Galerkin approximation produces a sequence of scalar equations with orders going to infinity, see e.g.…”