Geodesic
Exact solutions of Cauchy problem for partial differential equations with double characteristics and singular coefficients
Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 9-24

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Let $ L_{a, b}\eq(\pa_x-ax^k\pa_t)(\pa_x-bx^k\pa_t)+kbx^{k-1}\pa_t -kx\pa_x $ be a family of operators with double characteristics and singular coefficients, where $a$, $b$ are reals with $ab\ne0$ and $a\ne b$, $k>0$ is an odd integer. Let $\ome$ be the first quadrant in the plane and $H_+$ the upper half-plane. Consider Cauchy problems \cases{L_{a,b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in\ov{{{\Bbb R}}_+} or x \in{{\Bbb R}}$ \cr} \eqno(P_1) for $a>0$, $b>0$, and initial-boundary value problems \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t) for $t\in\ov{{{\Bbb R}}_+}$,\cr} \eqno(P_2) \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x \in{{\Bbb R}}$, \cr\displaystyle\lim_{{(x, \tau) \to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}} \d{u_x(x, \tau)}{x^k}=\psi_1(t) for $t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_3) for $ab0$ and \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t), \displaystyle\lim_{{(x, \tau)\to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}}\d{u_x(x, \tau)}{x^k} \! \! \! = $\psi_1(t) \quad for t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_4) for $a0$, $ b0$. Under appropriate smoothness conditions on $\vp_0$, $ \vp_1$, $ \psi_0$ and $\psi_1$, we obtain different sufficient and necessary conditions for each problem to have classical solutions. Moreover, we obtain also explicit expressions of solutions in each case.
Let $ L_{a, b}\eq(\pa_x-ax^k\pa_t)(\pa_x-bx^k\pa_t)+kbx^{k-1}\pa_t -kx\pa_x $ be a family of operators with double characteristics and singular coefficients, where $a$, $b$ are reals with $ab\ne0$ and $a\ne b$, $k>0$ is an odd integer. Let $\ome$ be the first quadrant in the plane and $H_+$ the upper half-plane. Consider Cauchy problems \cases{L_{a,b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in\ov{{{\Bbb R}}_+} or x \in{{\Bbb R}}$ \cr} \eqno(P_1) for $a>0$, $b>0$, and initial-boundary value problems \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t) for $t\in\ov{{{\Bbb R}}_+}$,\cr} \eqno(P_2) \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x \in{{\Bbb R}}$, \cr\displaystyle\lim_{{(x, \tau) \to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}} \d{u_x(x, \tau)}{x^k}=\psi_1(t) for $t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_3) for $ab0$ and \cases{L_{a, b}u=0 in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t), \displaystyle\lim_{{(x, \tau)\to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}}\d{u_x(x, \tau)}{x^k} \! \! \! = $\psi_1(t) \quad for t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_4) for $a0$, $ b0$. Under appropriate smoothness conditions on $\vp_0$, $ \vp_1$, $ \psi_0$ and $\psi_1$, we obtain different sufficient and necessary conditions for each problem to have classical solutions. Moreover, we obtain also explicit expressions of solutions in each case.
DOI : 10.21136/MB.1996.125949
Classification : 35A05, 35C15, 35C99, 35L99
Keywords: exact solutions; Cauchy problem; singular coefficients; double characteristics 
Lu, Zhu-Jia. Exact solutions of Cauchy problem for partial differential equations with double characteristics and singular coefficients. Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 9-24. doi: 10.21136/MB.1996.125949
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