Kirchhoff equation and p-Laplacian wave equation

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1.The fundamental solutions of the mixed problem with the single and general bo undary conditions have the L1-L\infty and Lp boundedness, if and only if the bo undary operator satisfies the strong Lopatinski conditions and the alpha condit ion. Moreover if we assume only the strong Lopatinski conditins without the alp ha condition, we can get the Lp boundednss of solutions for 1<p<infty. which ar e proved in: Kajitani and Baba: Time decay estimates of solutions to the mixed problem for heat equtions in a half space, Tsukuba J. Math.(2013).2.The time global solutions an the time asymptotic solutions to the Cauchy prob lem of the perturbed Kirhhoff equations in the hall space was obtained under th e conditions that the nitial data satisfy some integral conditions with respect to the time and thecoefficients satisfy nontrapping conditions.