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A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain. (English) Zbl 1065.35006

In the cone \(D = \{(x,t)\in \mathbb R^{n+1}\,\mid | x-x_0| <(t^{k+1} - t_0^{k+1})/(k+1)\}\) for the equation \[ Tu \equiv \frac{\partial^2 u}{\partial t^2} - t^{2k} \sum\limits_{j=1}^n \frac{\partial^2 u}{\partial x_j^2} = 0 \] the fundamental solution relative to the point \((x_0,t_0)\) is constructed. The integral distribution of the regular solution of the Cauchy problem for the equation \(Tu = f(x,t)\) with vanishing initial data \(u(x,0) = u_t(x,0) = 0\) is obtained. The estimate for the \(L_p\) norm of the solution of this problem by the \(L_q\) norm of the function \(f\) is established.

MSC:

35A08 Fundamental solutions to PDEs
35L80 Degenerate hyperbolic equations
35L10 Second-order hyperbolic equations
35M10 PDEs of mixed type
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References:

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