On the problem of symmetrization of hyperbolic equations

V. Kostin

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Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

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We study a class of third order hyperbolic operators $P$ in $G = Ω \cap {0 \leq t \leq T}, Ω \subset ℝ^{n + 1}$ with triple characteristics on $t = 0$ . We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0 .$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$ .

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∗ Research partially supported by INTAS grant 97-1644 A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To...

Global time estimates for solutions to equations of dissipative type

Michael Ruzhansky, James Smith (2005)

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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.