The blow-up curve of solutions to one dimensional nonlinear wave equations with the Dirichlet boundary conditions

• Published in: Japan journal of industrial and applied mathematics Vol. 37; no. 1; pp. 339 - 363
• Main Authors: Ishiwata, Tetsuya, Sasaki, Takiko
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Computational Mathematics and Numerical Analysis; Dirichlet problem; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Original Paper; Wave equations; Wave equation; 35L05; Positive solutions; 35B09; 35B44; Blow-up
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-019-00399-7

In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for
∂
t
2
u
-
∂
x
2
u
=
|
u
|
p
-
1
u
and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem. In this paper, we consider the blow-up curve for
∂
t
2
u
-
∂
x
2
u
=
|
∂
t
u
|
p
-
1
∂
t
u
which does not have the variational structure. Nevertheless, we prove that the blow-up curve is not differentiable if the initial data changes its sign and satisfies some conditions.