| Wang Chuanfang.[J].数学年刊A辑,1980,1(3-4):469~475 |
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| ON DISCRETE PHENOMENA IN THE MIXED PROBLEM |
| Received:October 31, 1979 Revised:January 17, 1980 |
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| On discrete phenomena in uniqueness of the initial value problem, F. Treves studied an interesting example and proved that the Oauohy problem
\[\left\{ \begin{array}{l}
{L_p}u = {u_{xx}} - {x^2}{u_{tt}} + p{u_t} = 0,t \ge 0;\u(x,0) = {u_t}(x,0) = 0,
\end{array} \right.\]
has non-triyial solutions if and only if p = 3, 5, …. Wang Guang-ymg and others
proved that the Oauohy problem
\[\left\{ \begin{array}{l}
{L_p}u = 0,t \ge 0;\u(x,0) = {\varphi _1}(x);{u_t}(x,0) = {\varphi _2}(x),
\end{array} \right.\]
and Goursat problem
\[\left\{ \begin{array}{l}
{L_p}u = 0,t \ge \frac{{{x^2}}}{2};\u(x,\frac{{{x^2}}}{2}) = {\varphi _3}(x),
\end{array} \right.\]
both have a unique solution if and only if p≠1, 3, 5, …. In this paper, we discuss in
detail the equation Lvu = 0 for discrete phenomena. We prove that solution of the mixed
problem
\[\left\{ \begin{array}{l}
{L_p}u = 0,x \ge 0,t \ge 0,\u(x,0) = \varphi (x),\{u_t}(x,0) = \psi (x),\u(0,t) = 0
\end{array} \right.\]
is not only existent but also unique, for р≠3, 7, 11，…，neither existence nor uniqueness could be proved in this problem, for p = 3, 7, 11，….，more precisely, only under
some compatibility condition can the solution exist for the equation \({L_p}u = 0\). |
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