Existence of global smooth solution to a hyperbolic equation with singularity. (Chinese. English summary) Zbl 1054.35024
Summary: This paper considers the existence of a global smooth solution to the initial boundary value problem to a quasilinear hyperbolic equation
\[
u_t+\left(\frac 12 u^2\right)_x= -\frac{u^2}{x}\tag{1}
\]
with singularity at \(x=0\). Applying a function transformation, it transforms (1) into a regular hyperbolic equation. Then, by using the maximum principle of the first-order quasilinear hyperbolic system established by C. Zhu [Proc. R. Soc. Edinb., Sect. A 125, No. 6, 1277–1285 (1995; Zbl 0843.73033)], it obtains \(C^1\)-norm estimates to the corresponding problem and proves the existence of global smooth solution to the initial boundary value problem.
MSC:
35L60 | First-order nonlinear hyperbolic equations |
35L50 | Initial-boundary value problems for first-order hyperbolic systems |
35A20 | Analyticity in context of PDEs |