Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities. (English) Zbl 1397.35042
The authors study the following nonlinear equation with variable exponents:
\[
u_{tt}-\operatorname{div} (|\nabla u|^{r(\cdot)-2}\nabla u)+a|u_t|^{m(\cdot)-2}u_t=b|u|^{p(\cdot)-2}u
\]
coupled with initial conditions and homogeneous Dirichlet data. In the model, \(a, b>0\) and the exponents \(m\), \(p\), and \(r\) are given functions. The main result establishes blow-up of the solution in finite time for negative initial energy. Furthermore, blow-up is proven for certain solutions with positive energy as well.
Reviewer: Vanja Nikolić (Garching)
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L72 | Second-order quasilinear hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
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