Exact solutions of Cauchy problem for partial differential equations with double characteristics and singular coefficients

Abstract: Let $L_{a,b}\equiv(\partial_x-ax^k\partial_t)(\partial_x-bx^k\partial_t)+kbx^{k-1}\partial_t-\dfrac kx\partial_x$ 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 $\Omega$ be the first quadrant in the plane and $H_+$ the upper half-plane. Consider Cauchy problems
\cases L_{a,b}u=0 &\text{ in }\Omega \text{ or }H_+,
u(x,0)=\varphi_0(x), u_t(x,0)=\varphi_1(x)\quad &\text{ for }x\in\overline{\Bbb R_+}\text{ or }x\in{\Bbb R}\endcases \tag"$(P_1)$"
for $a>0$, $b>0$, and initial-boundary value problems
\cases L_{a,b}u=0 &\text{ in }\Omega\text{ or }H_+,
u(x, 0)=\varphi_0(x), u_t(x,0)=\varphi_1(x)\quad &\text{ for }x\in\overline{\Bbb R_+}\text{ or }x\in{\Bbb R},
u(0,t)=\psi_0(t) &\text{ for }t\in\overline{\Bbb R_+},\endcases \tag"$(P_2)$"

\cases L_{a,b}u=0 &\text{ in }\Omega\text{ or }H_+,
u(x,0)=\varphi_0(x), u_t(x,0)=\varphi_1(x)\quad &\text{ for }x\in\overline{\Bbb R_+}\text{ or }x\in\Bbb R,
\lim\limits\Sb (x,\tau)\to(0,t),x\ne0
(x,\tau)\in\Omega\text{ or }H_+\endSb \dfrac{u_x(x,\tau)}{x^k}=\psi_1(t)&\text{ for }t\in\overline{\Bbb R_+}\endcases \tag"$(P_3)$"
for $ab<0$ and
\cases L_{a,b}u=0 &\text{ in }\Omega\text{ or }H_+,
u(x,0)=\varphi_0(x), u_t(x,0)=\varphi_1(x)\quad &\text{ for }x\in\overline{\Bbb R_+}\text{ or }x\in\Bbb R,
u(0,t)=\psi_0(t), \lim\limits\Sb(x,\tau)\to(0,t),x\ne0
(x,\tau)\in\Omega\text{ or }H_+\endSb \dfrac{u_x(x,\tau)}{x^k} \!\!\! &=\psi_1(t)\quad  \text{ for }t\in\overline{\Bbb R_+}\endcases \tag"$(P_4)$"
for $a<0$, $b<0$. Under appropriate smoothness conditions on $\varphi_0$, $\varphi_1$, $\psi_0$ 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.

Keywords: exact solutions, Cauchy problem, double characteristics, singular coefficients

Classification (MSC91): 35C15, 35L99

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